Keywordsandphrases.integerpoints,lattice,algorithms,polyhedra,toricvar - PDF document

Keywordsandphrases.integerpoints,lattice,algorithms,polyhedra,toricvarieties,generat-ingfunctions,valuations. ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA93Inmostcases,theformulaeonecangetareneithersoniceandsimpleasPick'sformula(Example1.2),norsotautologicalastheformulafromExample1.3.Weconsiderafewmoreexamples.Example1.4.Let
bethetetrahedronwiththevertices(00),(0),;b;0),and(0),wherea;bandarepairwisecoprimepositiveintegers.Thenthenumberoflatticepointsin
canbeexpressedas +a+b+c 4+1 + + +1 bc;aac;bab;c)+2wherep;q)istheDedekindsumdenedforcoprimepositiveintegersandp;q  and(())= 0ifhereasusualb
cisthe oorfunction.See[Mordell1951;Pommersheim1993;Dyer1991].Example1.5.Letbeanonemptyintegerpolytope.Forapositiveinteger,letdenotethedilatedpolytope.AsE.Ehrhartdiscovered[1977],thereisapolynomial),nowcalledtheEhrhartpolynomial,suchthatwhereFurthermore,=1and=vol),thevolumeof.Thefollowingreci-procitylawholds:dim(relint(forpositiveintegersThatis,thevalueofatanegativeintegerequals,uptoasign,thenumberofintegerpointsintherelativeinteriorof.See[Stanley1997,Section4.6],forexample.WewillarguethatbothExample1.4andExample1.5areusefulandbeautiful.Tonavigatetheseaof\latticepointsformulae"whichcanbefoundintheliteratureandwhicharetobediscoveredinthefuture,wehavetosetupsomecriteriaforbeautyandusefulness.Ofcourse,likeallsuchcriteria,oursispurelysubjective.Welookatthecomputationalcomplexityoftheformula.Fixthespace.Supposethatisarationalpolytope.Thereisanobviouswaytocountintegerpointsin:weconsiderasucientlylargebox;:::;;:::;dwhichcontains,andcheckintegerpointsfromonebyonetoseeiftheyarecontainedin.Inotherwords,thisisan\eective"versionoftheformulaofExample1.3.Wewillmeasurethe\usefulness"and\niceness"oftheformulaforthenumberoflattice ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA95Tosummarize,weapproacheveryformulainthispaperprimarilyfromthepointofviewofcomputationalcomplexity.Ofcourse,therearedierentphiloso-phiesthatareequallylegitimate.Thetopicofthispaperis\latticesandpolyhe-dra,"asopposedtoaclose,butsomewhatdierentinspirit,topic\latticesandconvexbodies."Thisiswhyweomitmanyinterestingresultsonintegerpro-gramming,latticereductionalgorithms,andcountinglatticepointsingeneralconvexbodies;see[Cooketal.1992;Lovasz1986;Grotscheletal.1993;Schrijver1986].Similarly,wedonotdiscussratherinterestingresultsconcerninglatticepointsinnonrationalpolyhedra[Skriganov1998].Ourapproachisalgebraicandwedon'tcoverrecentadvancesinprobabilisticmethodsofcounting,suchasthosein[Dyeretal.1993;1997](see[Bollobas1997]forasurvey).Inshort,thepaperpresents\analgorithmictheory,"oneofmanypossible.Thisareaoftheresearchhasbeenquiteactive.Alongwithsuchactivity,oneexpectsindependentdiscoveriesofcertainresults,andwithunequalpublicationdelays,thereisoftenconfusionaboutwhodidwhatrst.Wehavetriedtobeaccurateinthechronology,but,unfortunately,inaccuraciesarepossible.Thispaperismeanttobeasurvey.However,itdoescontainsomenewresults:Theorem4.4(especiallythesecondpart),Theorem5.3,Theorem9.6,resultsinSection10,andpossiblysomeresultsinSection7.Inaddition,someofthelinksinSection8arenew.Wheneverpossible,wehavetriedtoprovidethereaderwithsketchesofproofs.2.Preliminaries.AlgebraofPolyhedraThenumberofintegerpointsinapolytopeisavaluation;thatis,itsatis-estheinclusion-exclusionproperty.Thetheoryofvaluations,withthetheoryofthepolytopealgebraasitsbasis,wasdevelopedbymanyauthors;see[Mc-MullenandSchneider1983;McMullen1993]forasurvey.Severalinequivalentdenitionsandapproacheshavebeenused,eachhavingitsownadvantages.Forexample,onecaneitherchoosetoconsiderarbitrarypolytopes,ortoconsiderlatticepolytopesonly.Inaddition,onecandecideeithertoidentifyornottoidentifytwopolytopeswhichdierbya(lattice)translation.Also,valuationscanbedenedviatheinclusion-exclusionprinciplefortheunionoftwoorsev-eralpolytopes.See[KantorandKhovanskii1992;PukhlikovandKhovanskii1992a;Lawrence1988;McMullen1989;Morelli1993c;1993d].Hereweemployanapproachwhichisconvenientforus.Letbeaset.TheindicatorfunctionnA]:Rd!RofAisdenedbyyA](x)=1if0ifalgebraofpolyhedra)isthevectorspace(over)spannedbytheindicatorfunctions[]ofallpolyhedra.Thespace)isclosedunderpointwisemultiplicationoffunctions:foranytwofunctionsf;g),wehave ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA97Thesumiswell-denedsincethereareonlynitelymanynonzeroterms.Nowwearereadytoextend).Chooseapolytopecontainingtheoriginasaninteriorpoint,andletbethedilatationofbyafactorof.Forany0andany),wehaveeQ(t)]f2Pc(Rd)andwelet)=limItisveryeasytoseethatiswell-denedandthatitsatisestheconditionnP])=1foranynonemptypolyhedronTheEulercharacteristicallowsustointerpretvariousimportantvaluationsasintegraltransformswithrespecttoasameasure.See[KhovanskiiandPukhlikov1993].Theorem2.3.SupposethatisananetransformationThenthereisauniquevaluationsuchthathatP])=[foreachpolyhedronProof.Deneakernelx;y1if0ifThenforeachxedandeach),wehavex;y),sowecanapplytheEulercharacteristicon),whichwedenoteby(tostressthevariable).Nowweletwherex;yInadditiontopointwisemultiplication,thereisacommutativeandassociativebilinearoperation),whichwecallconvolution,becauseitcanbecon-sideredastheconvolutionwithrespecttotheEulercharacteristicasameasure.Manyauthors[Lawrence1988;McMullen1989;1993]considerasthetruemul-tiplicationinthealgebraofpolyhedra,andperhapsrightlyso,becauseithasmanyinterestingproperties.Definition2.4.Letandbepolyhedrain.TheMinkowskisumisdenedasP;yTheorem2.5.ThereisauniquebilinearoperationsuchthathatP]?[Q]=[foranytwopolyhedraProof.Fixadecomposition.Letbethelineartransformationx;yandlet)bethecorrespondingvaluationwhoseexistenceisassertedbyTheorem2.3.Forfunctionsf;gdenetheirouterproduct)by(x;y).Thenf?g ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA99forcertainrationalnumbersThennKi]=0:Proof.WeapplythevaluationofTheorem2.7tobothsidesoftheidentity.ThevaluationplaystheroleoftheFourierTransformwithrespecttotheEulercharacteristicasameasure.Thevaluationtransformspointwiseproductsintotheconvolutions:)andf?g)forf;gItsucestochecktheidentityforrK]anddC],whereK;Carecones.WehaveeK\C],D(f)=[)=[],and)=[[C]=[K+C]=[(Similarly,,K]
[C]=[]=[(f?gFinally,wedescribeanimportantvaluationonthepolytopealgebraassociatedwithavector.Letbeapolytopeandletavector.Letmax(u;P)=maxu;xbethemaximalvalueofthelinearfunctionu;xonthepolytopeandletu;x=max(u;Pbethefaceofwherethismaximumisattained.Theorem2.9.ForanythereisavaluationsuchthathatP])=[foranypolytopeSketchofproof.For0and0,denethekernelx;y1ifu;xand0otherwise.Thenfor),welet,wherex;y ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA101Example3.2.Denefourrationalpolyhedrain,and.Part3ofTheorem3.1impliesthat=1.Now sobyPart3,wemusthave)=1).Similarly, 1�x�1=�x sobyPart3wemusthave).ByPart4,Finally,since[]=[]+[[P0],Part1impliesthat)=0,whichisindeedthecase.Example3.3.Chooselinearlyindependentintegervectors;:::;uandlet=co;:::;ubetheconegeneratedby;:::;u.Inotherwords,0for;:::;kLet0for;:::;kbethe\fundamentalparallelepiped"generatedby;:::;u.Asiswell-known(see,forexample,[Stanley1997,Lemma4.6.7]),foreachintegerpoint,thereisauniquerepresentationwhereand;:::;aarenonnegativeintegers.Let1for;:::;kThenisanonemptyopenset,andforeachwehave wheretheseriesconvergesabsolutelyforeach.Part3ofTheorem3.1impliesthatwemusthave Animportantparticularcaseariseswhenthefundamentalparallelepipedcon-tainsonlyoneintegerpoint,theorigin.Thishappensifandonlyif;:::;uformabasisofthe-dimensionallatticeSpan;:::;u.Inthiscase,the ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA103rationalpolyhedrawithoutstraightlines.Dene)asalinearcombination)viatheinclusion-exclusionprinciple.Provingthat)iswell-denedboilsdowntoprovingthatwegetaconsistentdenitionof)ifitselfdoesnotcontainstraightlines.Thisistruesincethereisanonemptyopen,wherealltheseriesdening)and)convergeabso-lutely.Itthenfollowsthattheproperties(1){(3)aresatised,anditremainstocheck(4).Ifcontainsastraightline,thenforsomenonzerowehave.Therefore,)and)mustbeidenticallyzero.Themapcanbeextendedtoavaluation),sendingeveryfunctionintoarationalfunctionincomplexvariablessuchthatprovidedtheseriesconvergesabsolutely.Furthermore,if)forsome,thenFinally,thekernelofthisvaluationcontainsthesubspacespannedbytheindi-catorfunctionsofrationalpolyhedrawithstraightlines.WearegoingtopresentinTheorem3.5averyinterestingandimportantcorollarytoTheorem3.1.ItwasprovedbyM.BrionbeforeTheorem3.1anditsrstproofusedalgebraicgeometry[Brion1988].Elementaryproofswerepublishedin[Lawrence1991;PukhlikovandKhovanskii1992b;Barvinok1993]andelsewhere.First,weneedadenition.Definition3.4.Letbeapolyhedronandletbeavertexof.Thesupportingtangentconecone(P;v)ofisdenedasfollows:supposethat;:::;misarepresentationofasthesetofsolutionsofasystemoflinearinequalities,whereand.Letbethesetofconstraintsthatareactiveon.Thencone(P;vOfcourse,theconecone(P;v)doesnotdependonaparticularsystemofin-equalitieschosentorepresent.Ifisarationalpolyhedronthencone(P;visarationalpointedconewithvertex.Moregenerally,ifisaface,wedenecone(P;Fwhereisthesetofinequalitiesthatareactiveon.Ifdimanddim,theapexofcone(P;F)isa-dimensionalanesubspacein ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA105Therefore,e,
(�1)dXv2Vert(
))K�v](modItremainstoshowthat[cone(

K�v](modThelastassertionfollowsbycomparingcone(
)andviaachainofinter-mediatecones,where2f�and;:::;").Notethatcone(
and.Oncewegett
Xv2Vert(
)[cone(
)](modforanysimplex
,usingtriangulationswecanshowthattP]Xv2Vert([cone(P;v)](modforanypolytopeSupposethatisanarbitrarypolyhedron.Asiswellknown,canberepresentedastheMinkowskisum,whereisapolytope,isaconewithoutstraightlines,andisasubspacein.If,thenhasnovertices,and)=0,sothestatementofthetheoremistrue.If,wemaywriteeQ]Xv2Vert([cone(Q;v)](modandbyCorollary2.6,6,P][Q+K]Xv2Vert([cone(Q;v](modEachvertexisavertexof,andcone(P;v)=cone(Q;v.Further-more,avertexisavertexofifandonlyiftheconecone(Q;vdoesnotcontainstraightlines.Theproofnowfollows.Following[BrionandVergne1997c],insteadof),onecanconsideravaluation))withthevaluesinthespaceofformalLaurentpowerseriesin;:::;x)withrationalcoecients:Thisleadstoessentiallythesametheory:thespace))hasthenaturalstructureofamoduleovertheringofpolynomialssx1;:::;x].Aformalpowerseriesisidentiedwitharationalfunction),where)provided.Forexample,theseries))isidentiedwithzero,since)=0.Inthisapproach, ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA107Theorem4.2.FixThereexistsapolynomialtimealgorithmwhichgivenarationalpolyhedralconecomputesunimodularconesnumbers2f�suchthathatK]=Xi2I"i[Ki]:InparticularthenumberofconesinthedecompositionisboundedbyapolynomialintheinputsizeofSketchofproof.Usingtriangulation,wereducetheproblemtothecaseofasimplecone=co;:::;u,where;:::;uarelinearlyindependentintegerpoints(weleaveasidelower-dimensionalcones,whichcanbetreatedinasimilarway).Weintroducetheindexind()whichmeasureshowfarisfrombeingunimodular:ind(^


^isthevolumeoftheparallelepipedspannedbythegenerators;:::;u.Onecanshowthatind()isthenumberofintegerpointsinthefundamentalparallelepipedof(cf.Example3.3).Thustheindexofaconeisapositiveintegerwhichequals1ifandonlyifisunimodular.Onecanshowthatlogind()isboundedbyapolynomialintheinputsizeof.Wearegoingtoiterateaprocedurewhichreplaces[]byalinearcombinationof[],wherearerationalconeswithsmallerindices.Consideraparallelepipedind(;:::;dWeobservethatiscentrallysymmetricandhasvolume2.Therefore,bytheMinkowskiconvexbodytheorem(see,forexample,[Lagarias1995]),thereisanonzerointegerpoint(suchavectorcanbeconstructedecientlyusing,forexample,integerprogrammingindimension:see[Schrijver1986]).For;:::;d,let=co;:::;u;w;u;:::;u,thenind(^


^^


^^


^^


^ind(ind(Furthermore,thereisadecompositionnK]=Xj2J"j[Kj]+XF"F[F];whererangesoverlower-dimensionalfacesof,and2f�.Ifweiteratethisprocedure,weobservethattheindicesoftheconesinvolveddecreasedoublyexponentially,whereasthenumberofconesincreasesonlyexponentially.Therefore,iteratingtheprocedure(loglogind())times,weendupwitha ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA109Notethattheareunimodular,providedthattheareunimodular.Iftheconeisdenedbylinearinequalitiesin,thenthecomplexityofthealgorithmandtheresultingformulais,whereistheinputsizeof.ThecomplexityofthealgorithmofTheorem4.2,asstated,canbeasbadasisgivenasaconichullofintegervectorsin,andisgivenbyasetoflinearinequalities.ThesavingsincomputationalcomplexitycomesfromthefactthatifweiteratetheprocedureofTheorem4.2asstated,thenumberofconesineverystepgrowsbyafactorof2.Ifwediscardlower-dimensionalcones,thenumberofconesineverystepgrowsbyafactorofonly.Wearereadytostatethemainresultofthissection.Wenotonlycomputetheexpressionfor),butwealsodescribehowitchangeswhenthefacetsofaremovedparalleltothemselvessothatthecombinatorialstructureofdoesnotchange.Theorem4.4.FixThereexistsapolynomialtimealgorithmwhichforagivenrationalpolyhedronfor;:::;mwherecomputesthegeneratingfunctionintheform where2f�;:::;bisabasisofforeachSupposethatthevectors;:::;marexedandthevaryinsuchawaythatthecombinatorialstructureofthepolyhedron;:::;staysthesameThentheexponentsinthedenominatorofeachfractionremainthesamewhereastheexponentsinthenumeratorchangewithwherethearelinearfunctionsissuchthatisanintegerpolytopethenforeachpairi;jThecomputationalcomplexityofthealgorithmforndingwhereistheinputsizeofInparticularthenumberoftermsProof.LetVert()bethesetofverticesof.ForVert(),letbethesetofthoseinequalitiesthatareactiveat,andletP;v)=cobetheconichullofthenormalsofthefacetscontaining.Thenforthetangentconecone(P;v),wehavecone(P;vP;v ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA111asprovidedbyTheorem4.4,thenthepoint=(1;:::;1)isapoleofeachfractionintherepresentation(4.4.1).Thiscanbecanbehandledbytakinganappropriateresidueorbycomputingthevalueof)atapointclose=(1;:::;1)androundingtheanswertothenearestintegerasin[DyerandKannan1997].Weusethe\residue"approach,suggestedin[Brion1988]andalsousedin[Barvinok1994b],wheretherstpolynomialtimealgorithmforcountingintegerpointsinarationalpolytopewasconstructed.Definition5.1.Considerthefunction;:::; exp(+1complexvariablesand;:::;.Itiseasytoseethatisanalyticinaneighborhoodoftheorigin=0andthereforethereexistsanexpansion;:::;;:::;wheretd;:::;)isahomogeneouspolynomialofdegree,calledthe-thToddpolynomial;:::;.Itiseasytocheckthattdisasymmetricpoly-nomialwithrationalcoecients;see[Hirzebruch1966]or[Fulton1993,Section5.3]).Algorithm5.2(Apolynomialtimealgorithmforcountingintegerpointsinrationalpolytopeswhenthedimensionisfixed).Supposethedimensionisxedandisarationalpolytope.WeuseTheorem4.4tocompute Weconstructavectorsuchthatl;b=0foreachand.Todothiseciently,weconsiderthe\momentcurve")=(1;;;:::;.Foreach,thefunctionisanonzeropolynomialofdegreeatmost1in,andthusthisfunctionhasatmost1zeros.Therefore,wecanselectfromthesetofintegervectors(0)(1);:::;g,where+1.Let;:::;).For0,letexp(;:::;exp(andletl;bandl;a.Then=lim)=limexp(l;a exp(l;bexp(l;b=lim exp( exp(exp( ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA113Remark(Relationtointegerprogrammingalgorithms).Integerpro-grammingisconcernedwithoptimizingagivenlinearfunctiononthesetofintegerpointsinagivenrationalpolyhedronin.Byusingastandardtrickofdichotomy,onecanreduceanintegerprogrammingproblemtoasequenceoffeasibilityproblems:givenarationalpolyhedron,decidewhethercontainsanintegerpoint,andifso,ndsuchapoint.IntegerProgrammingisdicult(NP-hard)ingeneral,butitadmitsapolynomialtimealgorithmifthedimensionisxedandnotapartoftheinput.TherstintegerprogrammingalgorithmhavingpolynomialtimecomplexityinxeddimensionwasconstructedbyH.W.Lenstra[1983];see[Grotscheletal.1993;Lovasz1986;Schrijver1986]forasurveyandsubsequentimprovements.Ofcourse,ifwecancountintegerpointsin,wecandecidewhether.ThecatchisthatinoneofthecrucialingredientsofAlgorithm5.2,namelyintheproofofTheorem4.2,werefertointegerprogramminginxeddimension.Therefore,employingAl-gorithm5.2tosolveanintegerprogrammingproblemmayseemtoresultinaviciouscircle.However,asM.DyerandR.Kannan[1997]show,onecanavoidthedependenceonLenstra'salgorithminTheorem4.2byusinganappropriatelatticereductionalgorithm.HenceAlgorithm5.2givesrisetoanewlinearpro-grammingalgorithm,whosecomplexityispolynomialtimewhenthedimensionisxedandwhichuseslatticereduction(cf.[Lenstraetal.1982]),butdoesnotuse\rounding"ofagivenconvexbodyin,whichisthesecondmainin-gredientinLenstra'salgorithmanditssubsequentimprovements(see[Grotscheletal.1993;Lovasz1986]).Thisroundingseemstobequitetime-consuminganditwouldbeinterestingtondoutifAlgorithm5.2cancompetewiththeknownintegerprogrammingalgorithms.Ontheotherhand,Lenstra'salgorithmcanbenaturallyextendedto\convexinteger"problems,whereasAlgorithm5.2heavilyusesthepolyhedralstructure.Weconcludethissectionwithadescriptionofaverygeneral\trick"whichsometimesallowsonetocountlatticepointsecientlyevenifthedimensionislarge.Remark(Changingthelattice).Supposethatisasublatticeofaniteindex.Letbetheduallattice.Therefore,andFor;:::;xandanyvector;:::;,deneilexp(2i;:::;exp(2iThenthevalueofil=expl;mdependsonlyonthecosetrepresentedby ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA1156.Existenceof\LocalFormulae"TheresultsofSections4and5provideasatisfactorysolutionofthecountingproblemwhenthedimensionoftheambientspaceisxed.Ifthedimensionisallowedtogrow,thealgorithmscanbecomelessecientthanstraightforwardenumeration.Ifthedimensionisallowedtogrow,theproblemof\ecientcounting"seemstobeill-posed,sincemuchdependsontheparticularsofthepolytope.Forexample,itbecomesrelevantwhetherthepolytopeisgivenbythelistofitsverticesorbythelistofitsfacets.Inthissection,weexplainourapproachtowhatthe\right"countingproblemiswhenthedimensionisallowedtogrow.P.McMullen[1983]provedthatthenumberofintegerpointsinarationalpolytopecanbeexpressedasalinearcombinationofthevolumesofthefacesofthepolytope,wherethecoecientofvol(),whereisafaceof,dependsonlyonthetranslationclassmodofthesupportingconecone(P;F)of(seeDenition3.4).Theorem6.1(\LocalFormula").Foreveryrationalconeonecandenearationalnumbersuchthat(i)Thefunctionisinvariantunderlatticetranslationsforany(ii)Foranyrationalpolytopevol(cone(P;Fwherethesumistakenoverallfacesvol(isthevolumeofmeasuredintrinsicallyinitsanespanTheorem6.1immediatelyimpliesthatthenumberofintegerpointsinthedilatedpolytope,whereispositiveinteger,isaquasipolynomial,wherethe)areperiodicfunctions.Ifisintegral,thenwegetagenuinepolynomial,namelytheEhrhartpolynomialof,seeExample1.5.Thefunctionfailsbadlytobeunique.Essentially,onecangettheexistenceusingaHahn{Banachtypereasoning:see[McMullen1993].Inthenextsection,wesketchamoreconstructiveapproach,alsoduetoMcMullen,viawhatwecallthe\CombinatorialStokesFormula"(theformulabelongstoMcMullen,whereasitsnameisourinvention).Inmanyimportantcases,canbechosentobeavaluationonrationalcones.Notethat,ifdimanddim,theapexofcone(P;F)isadimensionalanesubspace.Thustheconeisjusttheproductofa(dimensionalpointedconewiththeapexattheoriginandarationalsubspace. ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA117Thisexplainswhyitisconvenienttoconsiderrightfromthebeginningagenerallattice(discreteadditivesubgroupof),ratherthanonlyDefinitions7.1.Fixalattice.Apolytopeiscalledalattice-polytopeprovidedthatitsverticesbelongto.Apolytopeiscalledrational-polytopeprovidedthatforsomepositiveintegerisalatticepolytope.Let)bethesubspace(subalgebra)of)spannedbytheindicatorfunctions[]ofrationalpolytopes.Avaluationiscalledsimpleprovided([])=0whendimPdvaluationiscalled-invariantprovided([])=([])forany.Avaluationiscalledcentrallysymmetricprovided([])=([])foreverypolytope.Oftenwewrite()insteadof([]).7.2(Conesandangles).Letbeacone.Letbetheunitspherecenteredattheapexof.Wedene)tobethesphericalmeasureoftheintersectionnormalizedinsuchawaythatthesphericalmeasureofthewholesphereis1.Wealsoagreethatif=0,then(0)=1.Clearly,)=0ifisnotafull-dimensionalcone.Theintrinsicmeasure)isdenedasthesphericalmeasure)intheanehullof,where=dimFinally,letbea-dimensionalpolyhedron,andletbea-dimensionalfaceof.TheexteriorangleP;F)ofistheintrinsicmeasureofthenormalconeof(seeDenition6.3);thatis,P;FP;Fisa-dimensionalpolyhedralcone,thenK;F)=1wherethesumistakenoverallnonemptyfaces;see[McMullen1975].Theproof,alsoduetoMcMullen,isimmediate:foreverypoint,letbethe(unique)pointclosesttointheEuclideanmetric.Thenthesummandsof(7.2.1)correspondtoadissectionofintopieces,eachofwhichconsistsofthosepointssuchthat)isintherelativeinteriorofagivenfaceAnimportantexampleofasimplelattice-invariantvaluationrelatedtolatticepointcountingariseswhenwecountlatticepointswiththeir\solidangles."Example7.3(ThesolidanglevaluationForapolyhedronandapoint,denethesolidangleP;xinthefollowingway:let)denotetheballcenteredatofradius.WeletP;x)=limvol( vol(where\vol"istheusualvolumein.Forexample,ifthenP;x)=0.Similarly,ifdimPdthenP;x)=0forany.Furthermore,P;x)=1ifandonlyifisintheinteriorof.If-dimensionalandliesinthe ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA119Sketchofproof.Withoutlossofgenerality,weassumethat=.Weproceedbyinductionon.For=0,theresultisclear.Supposethat1,andconsiderasahyperplanein(thelastcoor-dinateis0).Wedeneavaluation onby ()=((;1]).Clearly, isasimple-invariantvaluationin,sowecanapplytheinductionhypothesisto .Letbethecorrespondingnumber,bethecorrespondingfunction,andletbethecorrespondingfamilyofvaluationsforrationalhyperplanes.Ifisarationalhyper-plane,thenisarationalhyperplaneinwhosenormalvectoris.Usingtheinductionhypothesis,onecanshowthatthevaluationscanbeextendedtovaluationsinsuchawaythatt;1])=)foranyrationalpolytope.Hencewehaveconstructedvaluationsonthehyperplaneswhosenormalvectorsarein.Forarationalpolytopelet )=(volwherethesumistakenoverallfacetsofwhosenormalvectorsareinUsingTheorem2.9,onecanshowthat isinfactasimple-invariantvaluation.Itisclearthat ;1])=0,whereisalatticepolytopein.Since isasimplevaluation,weconcludethat )=0ifisa\latticeprism"m"m;n],wherem;nNowwearereadytodeneforanyrationalhyperplaneandforall.Letbearationalanehyperplaneandletbearationalpolytope.Translatingbyalatticevector,ifnecessary,wecanalwaysassumethatisintheupperhalfspaceof(thehalfspacewithpositivelastcoordinate).Letbetheprojectionofdownonto,andlet()=conv((Q0)bethe\skewedprism"withbottomfacetandtopfacet.Let Onecanshowthat)iswell-denedandthatthefamilyis-invariant,sinceifwechoosealatticetranslation,thedierencebetweenthevalues ontheskewedprisms()and()willbethevalueof onarightprismsmm;n],whichiszero.Letdenotethelastcoordinateofavector=0wecanthinkofasavectorin.Dene1if1if)if=0.NowthetheoremfollowssinceeP]=XF(nF)[()]modulolower-dimensionalpolytopeswherethesumistakenoverallfacets ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA121Corollary7.7canbeconsideredasa\101st"generalizationofPick'sformula(Example1.2).Indeed,allfacetsofapolygonarecentrallysymmetric.Pick'sformulaisequivalenttosayingthatifwecounteveryintegerpointinapolygonwithweightequaltotheangleatthispoint,wegettheareaofthepolygon.Aninterestingexampleofapolytopewithcentrallysymmetricfacesispro-videdbyazonotope,thatis,theMinkowskisumofnitelymanylatticeinter-vals.Toremovevariousnormalizingfactors,itisconvenienttomeasurevolumesofpolytopesintrinsically,withrespecttoagivenlattice.Namely,xalattice.Supposethatisalatticepolytopeandsupposethat=dimWithoutlossofgenerality,wemayassumethattheanehullcontainstheorigin.Thenisalatticeofrank,andwenormalizethevolumeformininsuchawaythatdet=1.Corollary7.8.LetbeanintegerzonotopethatistheMinkowskisumofnitelymanyintegerintervalsThenthenumberofintegerpointsinisexpressedbytheformulavol(P;FwherethesumistakenoverallfacesP;FistheexteriorangleofandthevolumeofafaceismeasuredintrinsicallywithrespecttothelatticeProof.Sinceeveryfaceofazonotopeiscentrallysymmetric,theresultfollowsfromCorollary7.7andformula(7.3.1).Ofcourse,itisquiteeasytondasimplealternativeproofwhichdoesnotuseTheorem7.4.Forafull-dimensionallatticeparallelepiped(thatis,theMinkowskisumoflinearlyindependentintervals)wehave()=vol(),sincelatticetranslatesoftilethespaceandboththevolumeandthevaluationaresimpleandlattice-invariant.Sincealatticezonotopecanbedissectedintolatticeparallelepipeds(see,forexample,[Ziegler1995,Lecture7]),weget)=vol()foranylatticezonotope.Sinceeveryfaceofalatticezonotopeisalatticezonotopeitself,weuse(7.3.1)tocompletetheproof.Theboundaryofaconvexpolytopecanberepresentedasaunionoflower-dimensionalpolytopes.Therefore,wecanapplytheStokesformulaofTheorem7.4recursively,rsttothepolytope,thentoitsfacets,thentoitsridges,andsoforth.Wewillendupwithadecompositioninvolvingvolumesoffacesandsome\local"functions,dependingonlyonthesupportingconesatthefaces(see[McMullen1978/79]).ThususingTheorem7.4,onecanproveTheorem6.1rstforthesimplevaluation,andthen,applying(7.3.1),forthenumberofintegerpoints.Example7.9(CoefficientsoftheEhrhartpolynomial).Foranyra-tionalpolytope,thereisapositiveintegersuchthatthedilatation ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA123Unlikethecoecients),thenumbers)aremonotone,thatis,)ifandareintegerpolytopeswith[Stanley1993].Inparticular,)arealwaysnonnegative.However,theyarenothomogeneous.FromtheproofofTheorem7.4,itisnotatallclearwhatthefunctionsinThe-orem6.1mightlooklike.Theproblemappearssomewhateasierforexponentialvaluations(Example7.5).Sincethemaintermis0unless=0weconcludethatifthehighesttermoftheEhrhart(quasi)polynomialofthevaluation,thenisorthogonaltosome-dimensionalfaceofApossibleapproachwouldbetorelatetheexponentialvaluationsofEx-ample7.5andthesolidanglevaluation(seeSection7.3).Letexp detexpexpl;xwhere0isaparameter.Itistheneasytoshow(see[Barvinok1992])thatThuswegetadecompositionofintotheFourierseriesofthevaluations)=limexpThisapproachwasdiscoveredindependentlyandisbeingdevelopedbyR.DiazandS.Robins[1999].8.UsingAlgebraicGeometrytoCountLatticePointsInrecentyears,manyauthors[BrionandVergne1997a;CappellandShaneson1994;Ginzburgetal.1999;KantorandKhovanskii1993;Inrri1992;Morelli1993a;1993b;Pommersheim1993;1996;1997;PukhlikovandKhovanskii1992b]havestudiedtheproblemoflatticepointenumerationusingthesubjectoftoricvarieties.Inthissection,wedescribesomeoftheseresults.Inparticular,weshowhowthevaluationintroducedinSection3playsakeyroleinvariousformulasforcountinglatticepointsandfortheclosely-relatedproblemofndingtheToddclassofatoricvariety.Toricvarieties,thoughveryspecialandsomewhatsimplefromthepointofviewofalgebraicgeometry,provideapowerfullinkbetweenthetheoryoflatticepolytopesandalgebraicgeometry.Earlyresearchersintheeldoftoricvarietiesrealizedthatndingaformulaforatoricvariety'sToddclass,acharacteristicclasslivinginhomology,wouldyieldaformulaforthenumberoflatticepointsinanintegralpolytope.Detailsofthisconnection,adirectresultoftheRiemann{RochTheorem,aresketchedfollowingthestatementofAlgorithm8.7.Foramorecompletediscussion,see[Fulton1993,Section5.3].Thisconnectionbetweenpolytopesandtoricvarietieshasbeenveryfruitful,especiallyoverthe ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA125Proposition8.2.Theassignmentoftoconessatisesthefollowingproperties(i)(Invarianceunderlatticeautomorphismsareequivalentunderanautomorphismofwhichpreservestheprescribedorderingoftheraysofthen(ii)(AdditivityundersubdivisionK;K;:::;Kare-dimensionalra-tionalsimpleconeswhoseindicatorfunctionssatisfyyK]=1[K1]+


+l[Kl]moduloconesofsmallerdimensionthen(iii)(SummationformulaLetbethefundamentalparallelepipedforandlet;:::;ubetheprimitivegeneratorsfortheconeThen Thusthefunctionisregularattheoriginandhencedenesapowerseriesinthevariables;:::;ySketchofproof.Invarianceunderlatticeautomorphismscanbeeasilycheckedfromthedenitionof;itisalsoaconsequenceofthesummationformula(Part3above),whichisclearlyinvariantunderlatticeautomorphisms.Tocheckadditivityundersubdivisions,letK;K;:::;Kbeasabove,andapplyCorollary2.8,whichimpliesthattheindicatorfunctions[]ofthedualconessumtotheindicatorfunctionofmodulostraightlines.ButbyPart2ofTheorem3.1,vanishesonconescontainingstraightlines.Thedesirediden-tityisthenaconsequenceoftheadditivityformulaforgiveninPart1ofTheorem3.1.ThesummationformulafollowseasilyfromtheformulaofExample3.3andthedenitions,bearinginmindthat.Finally,theassertionaboutthepowerseries)followsfromthesummationformulafor)andthefactthefunction isregularat=0.Onefurtherusefulpropertyofisitscompatibilityalongcommonfaces:Lemma8.3.Iftheconesmeetatacommonfacethepowerseriesagreewhenrestrictedtothatisifinwesettozeroallvariablescorrespondingtoraysoutsideofweobtainidenticalpowerseriesintheremainingvariables ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA127Theorem8.6.ForaxeddimensionthereexistsapolynomialtimealgorithmwhichgivenacompletefancomputesuptodegreeThustheToddclasscanbeexpressedasapolynomialinthetorus-invariantdivisorsinpolynomialtimeSketchofproof.TheideaisverysimilartotheproofofTheorem4.4.Again,thekeyistheresultof[Barvinok1994b],statedasTheorem4.2above,thatapolynomialtimealgorithmexiststowriteanarbitraryrationalconeasthedierenceofunimodularcones.TocomputetheToddclass,oneappliesthistheoremtoall-dimensionalconesof,andusestheadditivitypropertyofProposition8.2toexpressthepowerseriesuptodegree.Atthispointwearedone,sincetheseexpressionsfordetermineThewell-knowndictionarybetweenpolytopesandtoricvarietiesallowsustousethisresulttoobtainapolynomial-timealgorithmforcomputingthenumberoflatticepointsinanintegralconvexpolytope.Thefollowingalgorithmappearedin[Pommersheim1997]:Algorithm8.7.Givenasimpleintegralconvexpolytope,thefollowingalgorithmcomputesthenumberoflatticepointsin(1)Supposethatisrepresentedasthesolutiontothenitelymanyinequalitiesasfollows:;:::lwherethearetheraysoftheinnernormalfanof(2)Using(8.6),computetheToddpowerseries;:::;y)indegreeuptoDenotethisdegreepolynomialby(3)Let=expandletbethedegreepartoftheproduct(4)IntheStanley{Reisnerring,letbetheidealgeneratedbythesetChooseanyvertexof,andlet;:::;vbethecorrespondingconeof.ComputenormalformsofandofwithrespecttoanyGrobnerbasisfortheideal.Thedegreepartofthequotientknowntobeaone-dimensionalvectorspace.Thusthesetwonormalformsmaybedividedtoproducearationalnumber.Thenumberoflatticepointsisthennf( ind()nf( ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA129WenotethatthisvariantofAlgorithm8.7maybeusedtocomputeagivencoecientintheEhrhartpolynomialofindependentlyoftheothercoecients.IfweareinterestedonlyinthetopEhrhartcoecients,wemustconsideronlythosemonomialsinofdegreeatmost,andonlythoseconesofdimensionatmost.Itfollowsthatforxed,theprocedureabovereducesinpolynomialtimethecomputationofthetopEhrhartcoecientsofanintegralpolytopetothecomputationofthevolumesofitsfaces.ThisprovidesapartialanswertoProblem6.2.Question8.8.Therstpolynomial-timealgorithmforcountinglatticepointsinpolytopeswasthatofTheorem4.4(originallyin[Barvinok1994b]),whichinvolvednotoricvarieties.Algorithm8.7,thoughbasedonalgebraicgeometry,appearsquitesimilarin avor.Inparticular,bothalgorithmsareultimatelybasedonsubdividingconesintounimodularcones,andbotharelinkedquitecloselytothevaluationintroducedinTheorem3.1.Inshouldbenoted,how-ever,thattheoriginalalgorithmisbasedonsubdivisionsofthetangentconesofthepolytope,whereasAlgorithm8.7naturallyinvolvessubdivisionsofthedualcones.Theseobservationsmotivateseveralquestions:Howexactlyarethesealgorithmsrelated?Whatistheprecisenatureofthedualityinvolvedhere?Ifoneweretoimplementthesealgorithms,whatfeaturesofeachalgorithmwouldbeadvantageous?ThepowerseriesalsoplaysakeyroleinthelatticepointformulaofBrionandVergne[1997b,Theorem2.15],whichisageneralizationoftheformulaofPukhlikovandKhovanskii[1992b].Intheseratherremarkableformulas,thepowerseriesisconsideredasaninnite-orderdierentialoperator,calledtheTodddierentialoperator.Adeformedpolytopeiscreated,withallfacetsmovedindependently,butparalleltotheoriginalfacets.Thevolumeofthisdeformedpolytopeiscalculatedasafunctionofthedisplacements.ApplyingtheTodddierentialoperatortothisfunctionyieldsthenumberoflatticepointsinthepolytope.Moregenerally,oneobtainsanEuler{Maclaurinformula,expressingthesumofanypolynomialfunctionoverthelatticepointsinapolytopeastheToddoperatorappliedtotheintegralofthepolynomialfunctionoverthedeformedpolytope.Wenowstatetheseformulasprecisely.Asabove,assumethatisasimpleintegralconvexpolytope.Supposethat;:::lwherethearetheraysoftheinnernormalfanof.For;:::;h,wedenethedeformedpolytope)by;:::lTheorem8.9.LetbeasaboveletbeitsinnernormalfanandletbeapolynomialfunctiononDenotebytheintegralofoverthedeformed ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA131Then;:::;uasadegreerationalfunctiononlivesnaturallyinRat(GrAssuchitcoincideswithMorelli's;:::;uSketchofproof.Wecancheckequalityoftherationalfunctionsaboveonunimodularconeseasily.ByProposition8.2,Part3,;:::;u thedegree0partofwhichequalsthe-thToddpolynomialtd;:::;u).Thisequals),asdiscussedin[Morelli1993b,followingstatementofTheorem4].Equalityforallsimpleconesthenfollowsfromadditivityundersubdivisions,satisedbybothMorelli's,and(cf.Proposition8.2,Part2.)ItisalsopossibletorelateMorelli's)forasimple-dimensionalconewithkd,tothecoecientsoftheLaurentseries.Adiscussionofthisconnectionmaybefoundin[Pommersheim1997].9.GeneralizedDedekindSumsandCountingLatticePointsInExample1.4,wesawtheclassicalDedekindsumappearinaformulaforthenumberoflatticepointsinacertaintetrahedron.InthissectionweexploretheimportantrolethatDedekindsumsandtheirhigher-dimensionalgeneralizationsplayinlatticepointformulas.Inparticular,followingBrion{Vergne,weshowthatthehigher-dimensionalDedekindsumsintroducedbyZagier[1973]appearascoecientsinthepowerseriesofSection8.ItthenfollowsfromtheresultsofSection8thattheseimportantsumsofZagierarecomputableinpolynomialtimewhenthedimensionisxed.TheclassicalDedekindsumrstappearedlongagoinDedekind'sworkonthe-function,andsincethenhasariseninavarietyothercontexts.Forexample,Hirzebruchconnectedthesesumswithgeometrybyshowingthattheyappearnaturallyinformulasforthesignatureofcertainsingularquotientspaces.The1951formulaofMordell(Example1.4)markedtherstappearanceofDedekindsumsinaformulaforlatticepoints.WenowrelatetheclassicalDedekindsumtothepowerseriesintroducedinSection8.Thisrelation,lessthanadecadeold,putstheMordellformulainamuchmoregeneralandgeometriccontext.Letbeatwo-dimensionalconeinalattice.Allsuchconesaresimplicial,andinfactisequivalentbyalatticeisomorphismtotheconep;q,with0pq.Inthiscase,theconeissaidtohavetypep;q).Itiseasilycheckedthat(whichequalsind())isuniquelydeterminedby,andisdetermineduptomultiplicativeinversesmodulo.(Theconeisequivalenttobyalatticeisomorphismthatswapstheraysof.)This ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA133polygongivesarelationamongtheDedekindsumscorrespondingtoitsangles(see[Pommersheim1993,Theorem8].)OneeasilyndslatticetrianglesthatdemonstrateDedekind'sreciprocityrelation,Rademacher'sthree-termlaw,aswellasnewrelationsfortheclassicalDedekindsum.Wenowturnourattentiontothehigher-dimensionalDedekindsumsintro-ducedbyZagier.Motivatedbytopologicalconsiderations,Zagiermadethefol-lowingdenition:Definition9.3.Letbeapositiveinteger,andlet;:::;pbeintegersprimeto,witheven.ThehigherdimensionalDedekindsum;:::;pisdenedby;:::;p)=(cot cot Notethatifisodd,thesumaboveclearlyvanishes.Itshouldbenotedthatonecanalsodenethehigher-dimensionalDedekindsumintermsofthehyperboliccotangent:;:::;pcoth coth Thename\higher-dimensionalDedekindsum"isjustiedinpartbytheequal-ity Thatis,when=2,thehigher-dimensionalDedekindsumreducestoaclassicalDedekindsum.ZagiershowedthathissumssatisfyareciprocityrelationwhichgeneralizesthereciprocityformulafortheclassicalDedekindsum:Theorem9.4.Let;:::abecoprimeintegersThen ;:::;;:::;a;:::;a Here;:::;aisthe-polynomialdenedasthecoecientofthepowerseriesexpansionof tanhLiketheclassicalreciprocitylaw,thistheoremmaybeprovedusingtheconnec-tionwithToddclasses:seetheremarkfollowingTheorem9.5.WhileTheorem9.4iseasilyseentobeageneralizationofDedekind'sclassicalreciprocitylaw,thereisanimportantdierence.Aswehaveseen,theclassicalreciprocitylawallowsforecientcomputationoftheclassicalDedekindsum.However,inthecaseofthehigherdimensionalsums,itisnotatallclearhow ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA135BymultiplyingouttheproductappearinginTheorem9.5,oneobtainsanex-pressionforthiscoecientasasumofhigher-dimensionalDedekindsums.The\leadingterm"isthe-dimensionalDedekindsum;:::;p1).TheformulaabovecanbeusedtogiveasimplegeometricproofofZagier'sreciprocityformula(Theorem9.4).In,taketheunimodularconespannedbythestandardunitvectors.Subdivideintoconesbyintroducingtheraythrough(;:::b),wherethe'sarepairwisecoprimepositiveintegers.Applyingtheadditivityformulafor(Proposition8.2)tothissubdivisionyieldsarelationonthecoecientsinthepowerseriesassociatedtothesecones.SinceTheorem9.5identiescertainofthesecoecientsasDedekindsums,wegetarelationamongDedekindsums.ThisrelationiseasilyseentoyieldZagier'shigherreciprocitylaw.WenowshowthattheresultsofSection8provideforecientcomputationofthehigher-dimensionalDedekindsum,aresultnotobviousfromthereciprocitylaw(Theorem9.4).Theorem9.6.Forxeddimensionthereisapolynomialtimealgorithmwhichgivenintegers;:::prelativelyprimetocomputesthehigher-dimensionalDedekindsum;:::;pSketchofproof.Firstnoticethatwecanreducetothecaseinwhich=1.Thisisbecauseoftheeasilyveriedidentity:;:::;p;:::;pwhereisamultiplicativeinverseofmoduloApplyingTheorem8.6,weseethattheproductofTheorem9.5iscomputableinpolynomialtimeforxeddimension.Thisproduct,whenexpanded,canbeexpressedasthesumof2higher-dimensionalDedekindsums(manyofwhicharezero).Ofthesesums,allbuttheleadingterm,;:::;p1)areDedekindsumsofdimensionlessthan.Byinduction,thesemaybecomputedinpoly-nomialtime.Thus,;:::;p1)itselfmaybecomputedinpolynomialtime.Theorem9.5linksacertaincoecientinthepowerserieswiththehigher-dimensionalDedekindsum.ThisDedekindsumappearsintheveryspecialcasewhenallfacetsoftheconeareunimodular.Forgeneralcones,thegeneralcoecientinrepresentsasignicantfurthergeneralizationofZagier'ssums.Thesecoecientsalsoadmitcotangentformulas,usingtheideasoftheproofofTheorem9.5.Infact,thesemoregeneralcotangentsumscanbeseenintheremarkablelatticepointformulaofDiazandRobins[1997].Thisveryexplicitformulaex-pressesthenumberoflatticepointsinanarbitrarysimplexintermsofcotangentsums.Thesumsthatappearareeasilyseentomatchthecotangentsumsthatappearascoecientsinthepowerseries.ThecotangentformulasDiazand ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA137Thesetcontainsallsucientlylargepositiveintegers,sotheinitialintervalisthemaininterest.Wecanwrite forsomepolynomial)andsomepositiveinteger.ThepolynomialencodesinacompactwayinformationregardingtheinitialintervalofSupposethata;b,andarecoprimepositiveintegers.Let;; bethesetofallnonnegativeintegercombinationsofa;b,andThenfor1,wehave forcertainpositiveintegers,and.Thisinterestingfactwasdiscovered,apparently,byG.Denham[1996].Thisresultfollowsalsofromageneralresultfrom[PeevaandSturmfels1996].Thepaper[SzekelyandWormald1986]containsanelementaryproofthatthenumberoftermsinthenumeratorisatmost12,whichistwiceasmanyasthesharpbound.Forexample,if=23,=29,and=44,then Asinthepreviousexample,thesetcontainsallsucientlylargepositiveintegers,sotheinitialintervalofistheinterestingpart.Itcanbeencodedbyashortpolynomial.Question.Whatinformationregardingcanbeextractedfrom)andwhatoperationsonsetsgivenbytheirgeneratingfunctions)canbecarriedouteciently?Forexample,bysubstituting=(1;:::;1),onecangetthenumberofpointsin(since=(1;:::;1)isthepoleofeveryfraction,onemayneedtocomputeanappropriateresidue,asinSection5).OurnextresultisthatonecanecientlyperformBooleanoperationsonsetsgivenbytheirfunctions).Infact,weproveamoregeneralstatement:theHadamardproductofshortrationalfunctionscanbecomputedinpolynomialtime.Theorem10.2.Supposethatarerationalfunctionsincom-plexvariables;:::;x xai1)xa(xi2I2ixqi where;:::;a;:::;b ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA139ByTheorem4.4,theseriesm;nwhere;:::;;:::;)andcanbecomputedinpolynomialtimeasarationalfunction).Thefunction)isobtainedbyspecializing)for=(1;:::;1)and.OnemayhavetoresolvesingularitiesasinSection5.Remark.Ifthefunctionsandaresucientlygeneric,thatis,ifthevectors;:::;a;:::;bspan,wewillhavedimand Corollary10.3.FixThereexistsapolynomialtimealgorithmwhichforanynitesetsgivenbytheirgeneratingfunctions xai1)xa(S2;xi2I2ixqi where;:::;a;:::;bcomputesthegeneratingfunctionssS2;x);f(S1\S2;x),andf(S1nS2;x).Proof.Chooseagenericvector,suchthatc;a=0andc;bforallvectorsand.Bymultiplying,ifnecessary,thedenominatorandthenumeratorofeachfractionbyanappropriatemonomial,wecanalwaysassumethatc;a0andc;b0foralland.Thentheset1for;:::;kand1for;:::;kisanonemptyopensetin,andforevery,thereareLaurentexpansions:andThecorollarynowfollowsfromTheorem10.2,sinceeS2;x)=f(S1;x)+f(S2;x)�f(S1\S2;x);f(S1nS2;x)=f(S1;x)�f(S1\S2;x):Themostintriguingquestioniswhethertheprojectionofasetwithashortgeneratingfunctionisasetwithashortgeneratingfunction. ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA141,apolynomialtimealgorithmforthisproblemwasconstructedbyR.Kannan[1992].Sometimesitiseasytoseethattheprojectionindeedhasashortgeneratingfunction.Example10.4.Letbearationalpolytope,andletthesetofintegerpointsin.Thenprojectionhasageneratingfunctionwhosecomplexityispolynomialintheinputsizeof.(Notethatthesetofintegerpointsinarationalpolytope.)Let=(0;:::;beavectorandlet).For),thepreimageisthesetofintegerpointsintheinterval.Itthenfollowsthat)ifandonlyifthereisexactlyoneintegerpointsuchthat.Hence,isthespecializationofthegeneratingfunctionwhen=1.UsingTheorem4.4,weconcludethat)canbecomputedinpolynomialtime.Specialization=1mayrequireresolutionofsingularitiesasinSection5(iftheexpression)containsfractionswithamonomial(1)inthedenominator).HencewegetapolynomialtimealgorithmforcomputingDefinition10.5.Fixthedecomposition,andletbetheprojectiononthesecondsummand.Forasetandapoint,wedenethebern;mWehave,andwecanconsiderthegeneratingfunctionOnecanprovethatifisdescribedbyashortrationalfunction,thebers)aredescribedbya\consistent"systemofshortrationalfunctionsaschanges.Weusethetermopenpolyhedrontomeantherelativeinteriorofapolyhedron.Theorem10.6.Letbeanitesetwiththegeneratingfunction Thenthespacecanberepresentedasadisjointunionofopenpolyhedraandforeverythereexistvectors;:::;bsuchthatforevery wherearerationalnumbersandInotherwordsaslongasstayswithinthedenominatorsofthefractionsdonotchange ANALGORITHMICTHEORYOFLATTICEPOINTSINPOLYHEDRA143AcknowledgmentBarvinokwaspartiallysupportedbytheAlfredP.SloanResearchFellowship,byNSFgrantsDMS9501129andDMS9734138,andbytheMathematicalSci-encesResearchInstitutethroughNSFgrantDMS9022140.PommersheimwaspartiallysupportedbyNSFgrantDMS9508972.WewishtothankMichelBrion,StavrosGaroufalidis,SinaiRobins,andBerndSturmfelsforusefulconversations.References[Barvinok1992]A.Barvinok,\ComputingtheEhrhartpolynomialofaconvexlatticepolytope",preprintTRITA-MAT-1992-0036,RoyalInstituteofTechnology,Stockholm,October1992.[Barvinok1993]A.I.Barvinok,\Computingthevolume,countingintegralpoints,andexponentialsums",DiscreteComput.Geom.:2(1993),123{141.[Barvinok1994a]A.I.Barvinok,\ComputingtheEhrhartpolynomialofaconvexlatticepolytope",DiscreteComput.Geom.:1(1994),35{48.[Barvinok1994b]A.I.Barvinok,\Apolynomialtimealgorithmforcountingintegralpointsinpolyhedrawhenthedimensionisxed",Math.Oper.Res.:4(1994),[BetkeandKneser1985]U.BetkeandM.Kneser,\ZerlegungenundBewertungenvonGitterpolytopen",J.ReineAngew.Math.(1985),202{208.[Bollobas1997]B.Bollobas,\Volumeestimatesandrapidmixing",pp.151{182inFlavorsofgeometry,editedbyS.Levy,Math.Sci.Res.Inst.PublicationsCambridgeUniversityPress,NewYork,1997.[Brion1988]M.Brion,\Pointsentiersdanslespolyedresconvexes",Ann.Sci.EcoleNorm.Sup.(4):4(1988),653{663.[BrionandVergne1997a]M.BrionandM.Vergne,\AnequivariantRiemann{Rochtheoremforcomplete,simplicialtoricvarieties",J.ReineAngew.Math.(1997),[BrionandVergne1997b]M.BrionandM.Vergne,\Latticepointsinsimplepolytopes",J.Amer.Math.Soc.:2(1997),371{392.[BrionandVergne1997c]M.BrionandM.Vergne,\Residueformulae,vectorpartitionfunctionsandlatticepointsinrationalpolytopes",J.Amer.Math.Soc.:4(1997),[CappellandShaneson1994]S.E.CappellandJ.L.Shaneson,\Generaofalgebraicvarietiesandcountingoflatticepoints",Bull.Amer.Math.Soc.:1(1994),[Cooketal.1992]W.Cook,M.Hartmann,R.Kannan,andC.McDiarmid,\Onintegerpointsinpolyhedra",Combinatorica:1(1992),27{37.[Denham1996]G.Denham,\TheHilbertseriesofacertainmodule",1996.Unpub-lishedmanuscript.[DiazandRobins1997]R.DiazandS.Robins,\TheEhrhartpolynomialofalatticepolytope",Ann.ofMath.(2):3(1997),503{518.Erratumin:1(1997),237. 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150GRAHAMDENHAMANDPHILHANLONasfollows: Letbetheweightassignedtothehyperplane.ThenthematrixisgivenExample1.2.Animportantexamplethatwewillreturntoseveraltimesisthearrangementconsistingofthehyperplanesgivenby;:::;xNotethatconsistsofthere ectinghyperplanesfortherootsystemandsowedenotethisarrangementby.Twopoints(;:::;x)and;:::;y)areinthesameregionofthecomplementifandonlyiftherela-tiveordersoftheircoordinatesarethesame.So,thepermutationsinindextheregionsofthecomplementviathecorrespondence;:::;xLetdenotetheweightassignedtothehyperplane.For;;entryinistheproductofallsuchthatappearstotheleftoftheone-lineformofbuttotherightofintheone-lineformof.Anotherwayofsayingthisisthat;istheproductofallsuchthataninversionof.Inparticular,ifallparametersaresetequalto,thenentryof,where)denotesthenumberofinversionsofExample1.3.Foreach,letdenotethehyperplaneinwhichconsistsofallvectorswitha0inthe-thcoordinate,andconsiderthearrangement;:::;O.Inthiscase,)hassize2andtheindividualregionscanbeindexedbysequences;:::;s),whereeachiseither+1or1.Thesequencecorrespondstotheregionwhichcontainsallvectors(;:::;xwhere0if1and0if=1.GivensequencesS;T,thesetofhyperplanesseparatingandisequaltothesetofsuchthatthe-th 152GRAHAMDENHAMANDPHILHANLONwhereistheVarchenkomatrixforthearrangement(describedinourExample1.2above)withparametersVarchenkoandSchechtmangoontoshowthatthekernelofthiscontragredientformdescribestheSerrerelationsforthequantumKac{MoodyLiealgebraThemaintopicofthispaperwillbelinearalgebraicpropertiesofthematrices.WearegoingtoreportonanumberofresultsandlistanumberofopenquestionsthathavetodowiththenullspaceofandtheSmithnormalformof.Inaddition,wewilldescribeanewapplicationoftheVarchenkomatricestotheproblemofcomputingtheBettinumbersoftheMilnorbreofthearrangement2.TheNullspaceoftheMatricesInthissection,wewillconsiderthenullspaceoftheVarchenkomatrices.ThereadermayrecallthatthenullspacesareimportantintheworkofVarchenkoandSchechtmanastheyencodetheanaloguesoftheSerrerelationsforquantumKac{MoodyLiealgebrasinthecasethatthearrangementis.InthissectionwewillsurveysomeknownresultsandstatesomeopenproblemsrelatingtothenullspaceoftheVarchenkomatrix.Astartingpointforthestudyofthenullspaceofisthedeterminantof.Notethatdet()isapolynomialinthehencewillvanishforcertainchoicesofthe.Thefollowingresult,duetoVarchenko,givesanelegantfactorizationofdet().Asanimmediateconsequenceofthistheorem,oneobtainsacharacterizationofthosevaluesoftheparametersforwhichhasanontrivialnullspace.Theorem2.1[Varchenko1993,Theorem1].Thedeterminantofthebilinearformofthecongurationdet(whereisthesetofnonemptyintersectionsinwhereistheprod-uctoftheoverhyperplanescontainingandwhereisanonnegativeintegerdescribedexplicitlyin[Varchenko1993].Infact,Varchenkogivestwomethodstocomputetheexponents).Therst,whichprecedesthestatementofhistheoremismoregeometric,thesecondwhichcomeslaterinthepaperismorecombinatorial.Forcompleteness,wewillbrie ydescribethesecondmethod.Tocompute),rstchooseahyperplanewhichcontains.Then)ishalfthenumberofregionswhichhavethepropertythatistheminimalintersectioncontaining \H. 154GRAHAMDENHAMANDPHILHANLONWhenyoustudythenullspaceofforvaluesofthehyperplaneweightswheredet()=0,therstquestionthatarisesishowtodeterminethedimensionofthenullspace.Deformationtheoryargumentsshowthatdimensionofthenullspaceisnolargerthanthesumoftheexponents)takenoverintersectionswith=1.Togetmoreinformationingeneralisdicult.However,acaseofspecialinterestinwhichmoregeneralstatementscanbemadeisthecasewhereallhyperplaneweightsareequaltothesamenumber.Inthiscase,onecandenetheSmithnormalformofoverrq]whichcontainstheinformationnecessarytodeterminethedimensionofthenullspace.TheSmithnormalformoverrq]isthetopicofthenextsectioninthispaper.Aswewillseeinthenextsection,itispossibleforthedimensionofthenullspaceof),foracomplexnumber,tobestrictlysmallerthanthedeformationtheoryboundstatedabovewhichinthiscaseisequaltothemultiplicityofasadivisorofdet(Thequestionofdimensioncanbeslightlybroadenedtoaskforthe\structure"ofthenullspace(ormoregenerallythecokernel)meaninganaturalchoiceofbasisforthenullspace(cokernel)aswell.PROBLEM1.Letbeanarrangementandletbeachoiceofparametersforwhichthedeterminantofvanishes.Findanaturalbasisforthenullspace(orforthecokernel)ofintermsofsomekindofcombinatorialandgeometricinformationaboutthearrangementandtheparametersOnecouldaskwhatsortofinformationmightgointoasolutiontoProblem1.Theorem2.1suggeststhatthenecessaryinformationmayincludefactsaboutthecombinatorialstructureof)andinparticularatintersectionsforwhich=1.ThepreviousexamplesuggestsanotherinterestingquestionthatonecanaskaboutthenullspaceoftheVarchenkomatrices.Supposethatanitegroupactsasagroupofanelineartransformationsandpreservesthearrangement.Supposealsothattheweightassignedtohyperplanesis-invariant.Thenthegroupactsontheregions)andthisactioncommuteswith.Soonecanaskforthe-modulestructureofthenullspaceofPROBLEM2.Letbeanarrangementandletbeanitegroupofane-lineartransformationswhichpreservetheset.Inadditionassumethattheweightingonhyperplanesisinvariantundertheactionof.Whatcanyousayaboutthe-modulestructureofthenullspaceofWhatsortofsolutionmightyouhopetondforProblem2?AsstatedaboveinProblem1,youwouldliketodeterminethenullspaceofintermsofsomekindofcombinatorialandgeometricinformationrelatedtothelattice)andthechoiceofparameters.YouwouldhopeforananswertoProblem2whichextendsthisideatoincludenotjustcombinatorialinformationabout)butalsoinformationabouthowactson 156GRAHAMDENHAMANDPHILHANLONAsnotedearlier,Problems1and2areparticularlyinterestinginthiscasebecausethenullspaceofhasaninterpretationintermsofquantumKac{MoodyLiealgebrasviaresultsofSchechtmanandVarchenko.Alsointhiscase,thereisanaturalgroupofsymmetriesofthearrangement,namelythegroup.SowewillassumethattheweightingonhyperplanesisinvariantunderwhichinthiscaseisequivalenttotheirallhavingthesamevalueHanlonandStanley[1998]havestudiedthe-modulestructureofthenull-spaceofforthisarrangementwithweightonallhyperplanes.Weendthissectionbycataloguinganumberofresultsandopenproblemsthatappearintheirwork.TherststepistounderstandwhatTheorem2.1saysinthiscaseaboutthedeterminantof.TheintersectionlatticeofisthepartitionlatticeItturnsoutthattheexponent)is0unlessthepartitionhasexactlyonenontrivialblock.Ifhasonenontrivialblockofsize,then2)!(+1)!.Sointhiscase,Theorem2.1specializesto:Corollary2.3.LetbethearrangementandletallhyperplaneshaveweightThendet(2)!(+1)!(2{1)ThisfactorizationofthedeterminantinthiscasewasrstprovedbyZagier[1992],whocameacrossthematrixinthiscaseinanentirelydierentcontext.InviewofCorollary2.3,weneedonlyconsidervaluesofthatareoftheformis=j;:::;n.Therstresultthatappearsin[HanlonandStanley1998]concernsthespecializationofthatformthatisinsomesensethemostextreme.ThisresultisstatedintermsofthemoduleLiewhichistherepresentationofonthemultilinearpartofthefreeLiealgebra.Therehasbeenagreatdealofstudyofthisrepresentationinpartbecauseitplaysaroleinmanydiversemathematicalsituations.Theorem2.4[HanlonandStanley1998,Theorem3.3].Leti=nThenker=ind(LieLieAninterestingfeatureofTheorem2.4istheappearanceofthemodule=ind(LieLieItisanoldresultthatLieiscontainedinind(Lie).Sothisisagenuine(ratherthanvirtual)modulewhichisoftencalledtheWhitehousemodule.ThisnamereferstoSarahWhitehousewhocameacrosstherepresentationinquiteadierentcontext.TherepresentationappearsearlierintheworkofKontsevich[1993].Whitehouse,togetherwithAlanRobinson,investigatedthehomologyofthespaceofhomeomorphicallyirreducibletreeswhichhavelabelled 158GRAHAMDENHAMANDPHILHANLONWenextturntoProblem3andseewhatisknownhereinthespecialcaseof,withallhyperplaneweightsequalto.HanlonandStanley[1998]haveprovedaresultthatgivesasolutiontoProblem3,butinordertostateitweneedsomenotation.Foreachpartition,letbethevirtualcharacterwhichhasvalue0onconjugacyclassesnotindexedbyandwhichhasvalue ontheconjugacyclassindexedbywhereistheorderofthecentralizerinofapermutationwithcycletypeItiswell-knownthatwecanwriteexplicitlyaswhereistheirreduciblecharacterofindexedby(see[JamesandKerber1981]or[Sagan1991]formoreontheirreduciblerepresentationsof).ItiseasytoseethatthevirtualcharactersspantherepresentationspaceofThefollowingresult,whichwasoriginallyconjecturedbyStanley,giveanelegantsolutiontoProblem3inthiscase:Theorem2.6[HanlonandStanley1998],Theorem3.7.ForeachletdenotedetThen(i)unlessisoftheformforsomel;d(ii)foralll;dandallNotethat(a)and(b)reducethedeterminationof)tothecaseswhereoftheform(iii)1)!,where,intheexponentontheright-handside,denotesthenumber-theoreticMobiusfunction.(iv)3.TheSmithNormalFormofLetnotationbeasintheprevioussection,sothatisanarrangementofhyperplanesandistheVarchenkomatrixofthearrangement.Inthissectionwewillassumethatallparametersareequaltoaparameter.SoisthematrixwithrowsandcolumnsindexedbyregionswhoseR;SentryisR;SwhereR;S)isthenumberofhyperplaneswhichseparateandWecanspecializeVarchenko'sTheorem(Theorem2.1)bysettingalltogetaformulaforthedeterminantofdet(where)isthenumberofhyperplanesinwhichcontain.Aconsequenceofthisformulaisthatdet()vanishesonlyatqbeingcertainrootsofunity. 160GRAHAMDENHAMANDPHILHANLONThecomputationoftheSmithnormalformoftheVarchenkomatricesarisesasournextproblemofinterestinthispaper.PROBLEM4.(i)DeterminetheSmithnormalformofintermsofsomeinformationaboutthearrangement(ii)IsSNF()determinedjustbytheintersectionlatticeofAsidefromtheintrinsicinterestofcomputingtheSmithnormalformofthematrices,thisproblemisdirectlyrelevanttothedeterminationofthenullspaceinthecasethatallparametersaresetequalto.Asnotedinthelastchapter,factsabouttheSmithnormalformofwereneededbyDenham[1999]toproveaconjectureofHanlonandStanley.Inthenextsection,wewilldescribeanotherapplicationoftheSmithnormalformof(duetoDenham),thistimetocomputingthehomologyoftheMilnorbreofthearrangementNotethattheSmithnormalformofisdeterminedbythenumbersforeveryandevery.SoanalternativeformulationofProblem4isthefollowing.PROBLEM4Determinethenumbers0andforeachEquivalently,determinethenumbers0andforeachOnegeneralresultisknownthatgivesanelegantpartialsolutiontoProblem4Theorem3.3[DenhamandHanlon1997].Foranyarrangementandanynonnegativeintegeristhe-thBettinumberofthearrangementEquivalently whereisthecharacteristicpolynomialofthelatticeistherankExample3.4.ToillustrateaninstanceofProblem4,weconsiderthearrange-ments.Foreachintherange25thereisachartbelowwhichgivesthenumbers.Tounderstandthesecharts,assumethatisxed.ByCorollary3.1weknowthatifandonlyifisaprimitive-throotofunityforsomethatdivides1)andsome.Thesenumbersindextherowsofthechartsbelow.InviewofProposition3.2(b)wedonotincludetheform2whereisodd.Sothenumberthatappearsinthed;jentryofthechartbelowisthenumberofentriesinSNF()thatareexactlydivisibleby(i=dWiththesenotationalconventions,theSmithnormalformsofthearrange-mentsforsmallvaluesofareasfollows:=2:=01111=3:=01211323510 162GRAHAMDENHAMANDPHILHANLONthisactioncommuteswiththeactionsofbothanddq].Soactsonthespacesforeachandeach.Problem1asksonetodeterminethedimensionsofthe,orequivalentlytodeterminethecharacteroftheidentityactingoneach.Onecanaskmoregenerallyabouttheir-modulestructuresofthePROBLEM5.Determinethe-modulestructureofthespacesintermsofsomeinformationabouttheactionofonthearrangementAsimpler,butstillinteresting,variantofProblem5istodeterminethe-modulestructureoftheentirecokernelofExample3.5.Toillustratetheseconceptsandproblemswedoanexample.Letbethetwo-dimensionalarrangementwhereistheline=0,istheline=1andistheline1.Letbethegroupoforderfourgeneratedbyand,whereisthere ectionofacrosstheline=0andisre ectionofacrosstheline=0.Indextheregionsinthecomplementofthearrangement;:::;Rsothatarethoseinthehalf-plane0arrangedfromtoptobottomandarethoseinthehalf-plane0arrangedfromtoptobottom.The-matrixofthisarrangementwithrespecttothisorderingisqqqTheSmithnormalformofisgivenby)=diagsothatthecokernelhasdimension14(over.)Abasisfortheimageofgivenbythesixvectors=(1;q;q;q=(11+21+2=(1=(1;q;=(1=(1 164GRAHAMDENHAMANDPHILHANLONForlocalsystemsofrankone,variousauthorshavestudiedthehomology),forthemostpartinconnectionwithgeneralizedhypergeometricfunc-tions:see[Kohno1986;Gel'fand1986],orthereferencesprovidedin[Varchenko1995].Whenthelocalsystemistrivial,onerecoverstheordinaryhomology),whichisisomorphictotheWhitneyhomologyofthelatticeWhenthedeningpolynomialisreal,thereisastrongdeformationretractontoSalvetti'sCW-complex[Salvetti1987].Inthiscase,thereareexplicitchaincomplexesthatcalculate)and;N;).Thematrixappearsaspartofanimportantchainmapbetweenthetwocomplexes.Ourinterestinlocalsystemsoverthearrangement'scomplementcomesfromaspecialcase.Itiswellknownthat,ifisacentralarrangement,therestrictionofthedeningpolynomialtothecomplement,,givesthestructureofabrebundleover[Milnor1968].Thetypicalbre(1)isacomplexmanifoldofdimension1,knownastheMilnorbreofthepolynomial(orofishomotopicallyequivalenttoaninnitecycliccoveringofsoitssingularhomologyisforalocalsystemmt;tIfthepolynomialisreal,then,wecanexploitthecomplexesdenedby(4{1)tostudythehomologyoftheMilnorbre(Section4C).Milnor[1968]consideredarbitrarypolynomialmapsthatvanishat0.Inparticular,heshowedthatifthepolynomialhasanisolatedsingularityat0,thentheMilnorbreofhasthehomotopytypeofabouquetofspheres.TheMilnorbreofdeningpolynomialsofarrangements,then,continuetobeofinterestbyformingarestrictedfamilyofexamplesinwhichthesingularityisnotisolated.AlthoughthehomotopytypeoftheMilnorbreofarealarrangementisdeterminedbythefacelatticeofanarrangement,anexplicitdescriptionofthehomologyisonlyknowninspecialcases;see[CohenandSuciu1998;OrlikandRandell1993].Moreover,itisnotknownif,ingeneral,theintersectionlatticeofacomplexhyperplanearrangementdeterminestheMilnorbre'shomology.InSection4A,weintroducetheVarchenko{Salvettichaincomplexes.Sec-tion4Bindicatestheroleofthematrices,and4CspecializestheconstructiontoapplytotheMilnorbre.4A.Salvetti'scomplex.Inordertodescribethegeneralsetup,recallthatthefundamentalgroup)isgeneratedbyloopsaroundeachhyperplane[Randell1982].Letbea(complex)localcoecientsystemdenedbyarepresentationEnd().Foreach,let).Werequireinwhatfollowsthattheimageofbeabelian.Subjecttothisconstraint,onecanchooseanysetofcommutingendomorphismsEnd(),sincethen 166GRAHAMDENHAMANDPHILHANLONTheboundarymaps)and)aregivenbyP;RP;QR;QQ;QR)(4{2)andP;RP;QQ;R(4{3)Proposition4.2[Varchenko1995].Thechaincomplexwithboundarymap(4{2)computesthehomologyofthecomplementwithlocalcoecientsystemThechaincomplexwithboundarymap(4{3)computestherelativehomologyofthepairwithlocalcoecientsystem;N;4B.Relationtothematrix.Foranyface,let)denotethesmallestsubspaceofthatcontains.For0,deneamap)byP;R))=R;SP;Sonthebasisof)givenin(4{1).Here,R;S)=0whereandareregionssatisfyingR;S.Notethatthesubarrangementequals)=0,anditsregionscanbeidentiedwiththoseregions.Withthisinmind,onendsthatwheretheweightsofthehyperplanesofaretakentobeOnecanverifythat;thatis,)isachainmap[Varchenko1995].4C.ApplicationtotheMilnorbre.Inthissection,isareal,centralarrangement.The(complex)homologyoftheMilnorbreofisisomorphictoot;t]),where)actsonnt;t]byforall.Sincethedeterminantsofthematricesarenonzeroover 21 ],thechainmapisaninjection,andtheshortexactsequencecoker0(4{4) 168GRAHAMDENHAMANDPHILHANLONwithaconstructionfromD.B.Fuks.Inthiscase,),whereisthederivedsubgroupofthepurebraidgroup.AsinSection2,onecanattempttomakeusethe-modulestructure.UsingFuks'complex,E.V.Frenkel[1988]hasfoundanexpressionforthe-invariantpartofInaparallelvein,CohenandSuciu[1998]constructacomplexthatisusefulforgrouphomologycalculationsforsupersolvable,ratherthanreal,arrangements.ThereaderisinvitedtondapatterninourcomputationsofthehomologyoftheMilnorbreforthearrangement,inparallelwithourExample3.4.Thetablebelowshowsthecharacteristicpolynomialofthemonodromyoperator);inparticular,theBettinumbersofarejustthedegreesofthepolynomials.Thesepolynomialswerealsocalculatedfor5in[CohenandSuciu1998]. 2345 1�t1�t1�t1�t1 )(1)(1 t)3t3t6)2t)t2)23 t)t2)2t6 1�t1�t1 t)t) )(1 t)t3)t5)6t)t3) t)t3)t5)6tt)t3) 5.FactorizationsofInthissection,wewillgivetwocombinatorialrecipesforwritingthematrixofan-dimensionalarrangementasaproduct.Therst,giveninSection5B,showsthatdiag(;:::;Awhereisinvertibleasamatrixofpolynomialsintheweights;:::;aandthedimensionofisthethBettinumberofthearrangement'scomplexcomplement.Thesecondfactorizationisonlydenedforcentralarrangements.Itdependsonanorderofthehyperplanes,;:::;HThefactorizationhastheform PROPERTIESOFTHESCHECHTMAN{VARCHENKOBILINEARFORMS171For)and),let R;Swhere denotesthetopologicalclosureof)shouldbethoughtofasthetopologicalspaceconsistingofthewallsaroundchamberthatdonotseparatefrom,inthedeconedarrangement.Lemma5.3.LetbeacentralarrangementandtakeasdenedaboveLetbetheMobiusfunctionoftheposet;Z;RThenthereducedEulercharacteristicofDefinition5.4.Foranyhyperplane,deneamapofsets)byletting ,theintersectionofallhyperplanescontaining Itisshownin[Denham1999]thattheMobiusfunctionof;Z;R)hasasimpledescriptionintheseterms.Proposition5.5.ForunlessMoreoverletbethecodimensionofisboundedotherwiseisunboundedotherwise5B.Inductionondimension.Ourrstdecompositionisnotunique:itde-pendsonanarbitrarychoiceofa agof atsingeneralpositionwithrespecttothearrangement.Bythis,wemeananesubspacesforwhich


codim,andforifcodim�Xn;otherwise,codim()=codim,for1Letfor0dn,andletconsistofthesingleregioncontainingthepointLeta;bbethesubmatrixofwithrowsandcolumnsindexedbyandrespectively.For0dncanbeidentiedwiththematrixofthearrangement,andLemma5.2applies.Itstatesthatwhereisthematrixwithentriesi;ji;j PROPERTIESOFTHESCHECHTMAN{VARCHENKOBILINEARFORMS173Proof.Forconvenience,wereorderthebasisofthespaceofregionssothat)appearsrstinorder.Let2,andlettm]beabijectionforwhichand).Letbethecorrespondingpermutationmatrix,R;i,for)anddm].InthenotationofLemma5.2,SinceR;S),wehaveand.UsingLemma5.2,itfollowsthatataH 0]MHQ:Thepropositionrelatestwomatrices,onewiththehyperplanesweightedar-bitrarily,andtheotherwithonehyperplanegivenweightzero.Onecanapplythepropositiontoeachhyperplaneinsuccessiontoobtainthefollowing.Theorem5.8.Let;:::;HbearealcentralarrangementofhyperplanesForletbeaparalleltranslateofThenwhereisamatrixoververa0;:::;aExplicitly(i)S;S)=1,(ii)R;SR;S=maxR;S(iii)R;SotherwiseHereagainistheMobiusfunctionofwiththesignchosensothatExample5.9.ConsiderthearrangementfromExample1.2.Orderthehy-perplanes,andordertheregions123321.Then,where010010and100001 PROPERTIESOFTHESCHECHTMAN{VARCHENKOBILINEARFORMS175[Kohno1986]T.Kohno,\Homologyofalocalsystemonthecomplementofhyper-planes",Proc.JapanAcad.Ser.AMath.Sci.:4(1986),144{147.[Kontsevich1993]M.Kontsevich,\Formal(non)commutativesymplecticgeometry",pp.173{187inTheGel'fandMathematicalSeminars,1992,editedbyL.Corwinetal.,Birkhauser,Boston,1993.[Milnor1968]J.Milnor,Singularpointsofcomplexhypersurfaces,AnnalsofMathe-maticsStudies,PrincetonUniv.Press,Princeton,NJ,1968.[OrlikandRandell1993]P.OrlikandR.Randell,\TheMilnorberofagenericarrangement",Ark.Mat.:1(1993),71{81.[OrlikandTerao1992]P.OrlikandH.Terao,Arrangementsofhyperplanes,Grund-lehrenderMathematischenWissenschaften,Springer,Berlin,1992.[Randell1982]R.Randell,\Thefundamentalgroupofthecomplementofaunionofcomplexhyperplanes",Invent.Math.:1(1982),103{108.[RobinsonandWhitehouse1996]A.RobinsonandS.Whitehouse,\Thetreerepre-sentationofJ.PureAppl.Algebra:1-3(1996),245{253.[Sagan1991]B.E.Sagan,Thesymmetricgroup:Representations,combinatorialalgorithms,andsymmetricfunctions,Wadsworth&Brooks/Cole,PacicGrove,CA,1991.[Salvetti1987]M.Salvetti,\TopologyofthecomplementofrealhyperplanesinInvent.Math.:3(1987),603{618.[SchechtmanandVarchenko1991]V.V.SchechtmanandA.N.Varchenko,\Quantumgroupsandhomologyoflocalsystems",pp.182{197inAlgebraicgeometryandanalyticgeometry(Tokyo,1990),editedbyA.Fujikietal.,Springer,Tokyo,1991.[Stanley1986]R.P.Stanley,Enumerativecombinatorics,Wadsworth&Brooks/Cole,Monterey,CA,1986.[Varchenko1993]A.Varchenko,\Bilinearformofrealcongurationofhyperplanes",Adv.Math.:1(1993),110{144.[Varchenko1995]A.Varchenko,Multidimensionalhypergeometricfunctionsandrep-resentationtheoryofLiealgebrasandquantumgroups,AdvancedSeriesinMathe-maticalPhysics,WorldScientic,RiverEdge,NJ,1995.[Zagier1992]D.Zagier,\Realizabilityofamodelininnitestatistics",Comm.Math.Phys.:1(1992),199{210. NewPerspectivesinGeometricCombinatoricsMSRIPublicationsVolume,1999CombinatorialDierentialTopologyandGeometryROBINFORMANAbstract.Avarietyofquestionsincombinatoricsleadonetothetaskofanalyzingthetopologyofasimplicialcomplex,oramoregeneralcellcomplex.However,therearefewgeneraltechniquestoaidinthisinvestiga-tion.Ontheotherhand,thesubjectsofdierentialtopologyandgeometryaredevotedtopreciselythissortofproblem,exceptthatthetopologicalspacesinquestionaresmoothmanifolds.Inthispaperweshowhowtwostandardtechniquesfromthestudyofsmoothmanifolds,MorsetheoryandBochner'smethod,canbeadaptedtoaidintheinvestigationofcombina-torialspaces.IntroductionAvarietyofquestionsincombinatoricsleadonetothetaskofanalyzingasimplicialcomplex,oramoregeneralcellcomplex.Forexample,astandardapproachtoinvestigatingthestructureofapartiallyorderedsetistoinsteadstudythetopologyoftheassociatedordercomplex.However,therearefewgeneraltechniquestoaidinthisinvestigation.Ontheotherhand,thesubjectsofdierentialtopologyanddierentialgeometryaredevotedtopreciselythissortofproblem,exceptthatthetopologicalspacesinquestionaresmoothmanifolds,ratherthancombinatorialcomplexes.Theseareclassicalsubjects,andnumerousverygeneralandpowerfultechniqueshavebeendevelopedandstudiedovertherecentdecades.Asmoothmanifoldis,looselyspeaking,atopologicalspaceonwhichonehasawell-denednotionofaderivative.Onecanthenusecalculustostudythespace.Ihaverecentlyfoundwaysofadaptingsometechniquesfromdierentialtopologyanddierentialgeometrytothestudyofcombinatorialspaces.Per-hapssurprisingly,manyofthestandardingredientsofdierentialtopologyanddierentialgeometryhavecombinatorialanalogues.Thecombinatorialtheories ThisworkwaspartiallysupportedbytheNationalScienceFoundationandtheNationalSe-curityAgency. COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY179theory,relatingthetopologyofthespacetothecriticalpointsofthefunction,aretrue.Wealsopresentdiscreteanaloguesofsuch(seemingly)intrinsicallysmoothnotionsasthegradientvectoreldandthecorrespondinggradient owassociatedtoaMorsefunction.Usingthese,wedeneaMorsecomplex,adierentialcomplexbuiltoutofthecriticalpointsofourdiscreteMorsefunction,whichhasthesamehomologyastheunderlyingspace.Inthesmoothsetting,theMorsecomplexhasbeensubjecttomuchrecentstudy,andhasplayedacrucialroleinsomerecentapplications(see[Klingenberg1982;Floer1988],forexample).Mostofthissectionwillbeaninformalexpositionofthecontentsof[Forman1995;1998d].Thiscombinatorialtheoryisnotreallyanewtheory,butratheranextractionofthecombinatorialessenceofthesmooththeory.Asevidenceforthis,wewillindicatelaterthatitispossibletodeducemuchofthesmooththeoryfromthiscombinatorialtheory,and,conversely,someoftheresultswepresentcanbeprovedby\smoothing"thecombinatorialMorsefunctionandapplyingthecorrespondingsmooththeory(althoughthatisnottheapproachwewilltakeinthispaper).Aswementionedabove,thisdiscreteMorsetheorycanbedenedforanyCWcomplex,buttoavoidveryminorcomplicationswewillrestrictattention,inthispaper,tosimplicialcomplexes.See[Forman1998d]foramoregeneraltreatment.Letbeanynitesimplicialcomplex.Weemphasizethatneednotbeatriangulatedmanifold,norhaveanyotherspecialproperty.Denotebythesetof(nonempty)simplicesof,andthesimplicesofdimensionWritehasdimension,and�(or)ifisafaceofAdiscreteMorsefunctiononwillactuallybeafunctionon.Thatis,weassignasinglerealnumbertoeachsimplexin.Roughlyspeaking,afunctionisaMorsefunctionifusuallyassignshighervaluestohigherdimensionalsimplices,withatmostoneexception,locally,ateachsimplex.Moreprecisely:Definition1.1.AfunctionisadiscreteMorsefunctionif,forevery(1)#+1)1and(2)#Asimpleexamplewillservetoillustratethedenition.Considerthetwocom-plexesshowninFigure1.Hereweindicatefunctionsbywritingnexttoeachcellthevalueofthefunctiononthatcell.ThefunctionontheleftisnotadiscreteMorsefunctionastheedge(0)violatesrule(2),sinceithas2lower-dimensional\neighbors"onwhichtakesonhighervalues,andthevertex(3) COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY181 Figure2.Thefunctiondecreasesasonemovesfromthe1-simplextoeitherboundarycomponent,andincreasesineachtransversedirection.tothebarycenterofeachsimplex.Onecanthensmoothlyinterpolatebetweenthesevaluestodeneasmoothfunctiononallof.Ifthisisdonecarefully,\smoothing"adiscreteMorsefunctionnearacritical-simplexresultsinasmoothMorsefunctionwithacriticalpointofindexOurnextstepistoshowthatthetopologyofageneralsimplicialcomplexisintimatelyrelatedtothecriticalpointsofadiscreteMorsefunctiononBeforestatingthemaintheorems,weneedonemoredenition.SupposeisadiscreteMorsefunctiononasimplicialcomplex.Foranydenethelevelsubcomplex)byyf()c[:Thatis,)isthesubcomplexofconsistingofallsimpliceswithaswellasalloftheirfaces.Theorem1.3.Supposetheintervala;bcontainsnocriticalvaluesofThenisadeformationretractofMoreoversimpliciallycollapsesontoTodenesimplicialcollapse,supposeisasimplicialcomplex,isasimplexofthatisnotafaceofanysimplex,andisa(1)-face Figure3.Simplicialcollapseofonto COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY183EverysimplicialcomplexsupportsadiscreteMorsefunction.Namely,foranycomplexandanysimplex,set)=dim().ThenisadiscreteMorsefunction,andeverysimplexofiscriticalforTheresultswehavestatedaboveapplytoanysimplicialcomplex.Inthecasethatisacombinatorialmanifold[Glaser1972,p.19]onecanoftensaymore.Forexample,onecanstateacombinatorialanalogueofthestandardSpheretheoremofsmoothMorsetheory;see[Milnor1962,Theorem4.1].Corollary1.7.Supposeisacombinatorial-manifoldwithoutboundarywithaMorsefunctionwithexactlytwocriticalpointsThenisacombinatorial-sphereTheproofofthiscorollaryrestsheavilyonWhitehead'swonderfultheorem[1939]ontheuniquenessofregularneighborhoods,whichimplies,asaspecialcase,thatacombinatorial-manifoldwithboundarywhichsimpliciallycollapsesontoavertexisacombinatorial-ball.Supposeisacombinatorial-manifoldwithadiscreteMorsefunctionwithexactlytwocriticalpoints.Thenthemaximummustoccuratan-simplexwhichisacriticalsimplexofindex.Thus,restrictsontoaMorsefunctionwithexactlyonecriticalsimplex.Theminimumofmustoccuratavertexwhichisacriticalsimplexofindex0,andthisistheonlycriticalsimplexofrestrictedto.Theorem1.3impliesthatsimpliciallycollapsestothatvertex.NowweapplyWhitehead'stheoremtoconcludethatisacombinatorialn-ball.Itfollowsthattisacombinatorialn-sphere.Ratherthanprovethemaintheoremshere,wewillillustratethemainideaswithanexample;seeFigure5.Rigorousproofsappearin[Forman1998d]. 1381011Figure5.ExampleofaMorsefunction.Here(0)isacriticalsimplexofindex0,(9)isacriticalsimplexofindex1,andtherearenoothercriticalsimplices.Thus,itfollowsfromtheabovecorollarythatishomotopyequivalenttoaCWcomplexbuiltfromasingle0-cell,andasingle1-cell.TheonlyCWcomplexwhichcanbebuiltfromtwosuchcellsisacircle.Therefore,wecanconcludethatishomotopyequivalenttoacircle,asisevidentfromthepicture.ConsiderthesequenceoflevelsubcomplexesshowninFigure6.Wecanseewhythetheoremsaretrue.Ifisnotcritical,thenoneoftwocasesmustoccur. COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY185 Figure7.ThegradientvectoreldoftheMorsefunction.beprecise,whenwesay\gradientvectoreld"wearereallyreferringtothenegativeofthegradientvectoreld.)Itisbettertothinkofthisgradientvectoreld,whichwenowcall,asamapoforientedsimplices.Thatis,ifisaboundaryvertexofanedgewith),wewanttothinkof)asadiscretetangentvectorleavingi.e.,withgiventheorientationindicatedbythearrowinFigure8. Figure8.Edgeorientation.Moregenerally,if+1)areorientedsimplicesandsatisfythenwesetwiththesignchosensothat;@Vwhereisthecanonicalinnerproductonorientedchainswithrespecttowhichthesimplicesareorthonormal(inotherwords,;@Vistheincidencenumberofwithrespectto)).Dene)tobe0forallsimpliceswhichthereisnosuch.Nowcanbeextendedlinearlytoamapwhere,foreach)isthespaceofinteger-chainsonInthecaseofsmoothmanifolds,thegradientvectorelddenesadynamicalsystem,namelythe owalongthevectoreld.ViewingtheMorsefunctionfromthepointofviewofthisdynamicalsystemleadstoimportantnewinsights[Smale1961b].Toproceedfurtherinourcombinatorialsetting,wenowdenea(discrete-time) owalongthegradientvectoreld.Deneamapthediscrete-time ow,by=1+V@:Weillustratebyanexample.ConsiderthecomplexshowninFigure9,withtheindicatedgradientvectoreld.Letbethetopedgeorientedfromleft COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY187usethefactthatforsome.If=1,thenworks.Thegeneralcaseisnotmuchharder.Ourgoalinintroducingthiscomplexwastofurtherourunderstandingofthecriticalsimplices.Withthisinmind,wenowshowthattheMorsecomplexcanbealternatelydenedintermsofthecriticalsimplices.Foreach,let)denotethespanofthecritical-simplices.Wethenobviouslyhavedimcriticalsimplicesofdimension(1.1)Foreach,wecanapplythestabilizationmaptogetamapfromTheorem1.9.ForeachthemapisanisomorphismThus,theMorsecomplexcanbedenedequivalentlyasthecomplex�!


whereisthedierentialinducedbytheaboveisomorphism(thatis,).Inparticular(1.2)Itisinterestingtonotethattogether(1.1)and(1.2)implytheMorseinequalities,sothisgivesanalternateproofofCorollary1.6.From(1.2),weseethattheMorsecomplexcontainsamorecompletedescriptionoftherelationshipbetweenthecriticalsimplicesof,andthehomologyof.Allthatremainsistogetaexplicitdescriptionofthedierential.Thisrequirestheintroductionofthenotionofacombinatorialgradientpath.Letand~-simplices.Agradientpathfrom~isasequenceofsimplices+1)+1);:::;+1)suchthat)foreach;:::;r.Equivalently,werequire,and.InFigure11weshowasinglegradientpathfromtheboundaryofacritical2-simplextoacriticaledge,wherethearrowsindicatethegradientvectoreld Figure11.Agradientpathfrom COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY189Suchpairingswerestudiedin[Stanley1993;Duval1994]asatoolforinvesti-gatingthepossible-vectorsforasimplicialcomplex.In[Forman1998b]weinvestigatedthedynamicalpropertiesofthediscrete-time owcorrespondingtoageneraldiscretevectoreld.isacombinatorialvectoreld,wedenea-pathtobeanypathasinFigure11(wherenowthearrowsindicatethemap).Moreprecisely,a-pathisasequenceofsimplices+1)+1);:::;+1)suchthatforeach;:::;r)and.Wesaysuchapathisanontrivialclosedpathif0and.ThereisasimplecharacterizationofthosediscretevectoreldswhicharethegradientofadiscreteMorsefunction.Theorem1.12.AdiscretevectoreldisthegradientvectoreldofadiscreteMorsefunctionifandonlyiftherearenonontrivialclosed-pathsThistheoremhasaverynicepurelycombinatorialdescription,usingwhichwecanrecasttheMorsetheoryinanappealingform.WebeginwiththeHassediagramof,thatis,thepartiallyorderedsetofsimplicesoforderedbythefacerelation.ConsidertheHassediagramasadirectedgraph.Theverticesofthegraphareinone-to-onecorrespondencewiththesimplicesof,andthereisadirectededgefromifandonlyifisacodimension-onefaceof.Nowletbeacombinatorialvectoreld.Wemodifythedirectedgraphasfollows.thenreversetheorientationoftheedgebetweenand(sothatitnowgoesfrom).A-pathisjustadirectedpathinthismodiedgraph.FromTheorem1.12,isagradientvectoreldifandonlyiftherearenocloseddirectedpaths.Thatis,inthiscombinatoriallanguage,adiscretevectoreldisapartialmatchingoftheHassediagram,andadiscretevectoreldisagradientvectoreldifthepartialmatchingisacyclicintheabovesense.Wecannowrestatesomeofourearliertheoremsinthislanguage.ThereisaveryminorcomplicationinthatoneusuallyincludestheemptysetasanelementoftheHassediagram(consideredasasimplexofdimension1)whilewehavenotconsideredtheemptysetpreviously.Theorem1.13.LetbeanacyclicpartialmatchingoftheHassediagramofofthesortdescribedaboveAssumethattheemptysetismatchedwithavertexLetdenotethenumberofunpaired-simplicesThenishomotopyequivalenttoaCWcomplexwithverticesforeachcellsofdimensionAnimportantspecialcaseofTheorem1.13iswhenisacompletematching,thatis,everysimplexispairedwithanother(possibleempty)simplex.Theorem1.14.LetbeacompleteacyclicmatchingoftheHassediagramofthencollapsesontoavertexsothatinparticulariscontractible COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY191ornot�is-connecteduntilweaskquestions.Weobservethatinducesacompletepairingofthesetofgraphson.Namely,pair�and�ifwecannotdistinguishbetweenthemuntilallquestionshavebeenasked.Bydenition,anot--connectedgraphisanevaderofifandonlyifitispairedwitha-connectedgraph,andviceversa.Themainresultof[Forman1998c]isthatthisisanacyclicpairing,andhence,byTheorem1.13,thegradientvectoreldofaMorsefunction.Wecanrestrictthisvectoreldto
,togetthegradientofaMorsefunctionon
.Thecriticalsimplicesofthisrestrictedgradientvectoreldarethesimplicesof
whichwerepairedwithsimplicesnotin
.Thesearepreciselythenot--connectedgraphswhichwerepairedwitha-connectedgraph,i.e.,theevadersof.WecannowusetheweakMorseinequalities(Corollary1.6)alongwithTheorem1.15toconclude:Theorem1.16.Supposeourgoalistodeterminewhetherornotanunknowngraphis-connectedForanyquestionalgorithmthereareatleast2)!evadersofwhicharenot-connected2)!evadersofwhichare-connectedBeforeendingthissection,wemakesomenalremarksaboutthecombinatorialMorsetheory,anditsrelationtothesmooththeory.OneofthemainproblemsinMorsetheoryisndingaMorsefunction,foragivenspace,withthefewestpossiblecriticalpoints.Ingeneralthisisaverydicultproblem,since,inparticular,itcontainsthePoincareconjecture|spherescanberecognizedasthosespaceswhichhaveaMorsefunctionwithpreciselytwocriticalpoints.(See[Milnor1965]foracompletelyMorsetheoreticpresentationofSmale'sproof[1961a]ofthehigher-dimensionalPoincareconjecture.)AsanapplicationofTheorem1.12,weeasilyprovea\cancellation"theorem,whichenablesone,undercertainconditions,tosimplifyaMorsefunction(thisresultisadiscreteanalogueofin[Milnor1965,Theorem5.4],the\FirstCancellationTheorem").Thatis,ifand+1)aretwocriticalsimplices,andifthereisexactlyonegradientpathfrom,thenandcanbecancelled.Moreprecisely:Theorem1.17.SupposeisadiscreteMorsefunctiononsuchthat+1)arecriticalandthereisexactlyonegradientpathfromThenthereisanotherMorsefunctionwiththesamecriticalsimplicesexceptthatarenolongercriticalMoreoverthegradientvectoreldassociatedisequaltothegradientvectoreldassociatedtoexceptalongtheuniquegradientpathfromInthesmoothcase,theproof,eitheraspresentedoriginallybyMorse[1965]oraspresentedin[Milnor1965],israthertechnical.Inourdiscretecasetheproofissimple.If,inFigure11,theindicatedgradientpathistheonlygradientpathfrom,thenwecanreversethegradientvectoreldalongthispath,replacingthegurebythevectoreldinFigure12. COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY193thatthecurvatureislocalizedtothelower-dimensionalcells.Thecurvatureisthendenedintermsofhowtheneighborhoodofacelldiersfromwhatonewouldseeina atcomplex.Thissectionisalsorelatedtoeortstode-rivelocalcombinatorialformulasforthecharacteristicclassesofcombinatorialmanifolds;see[BanchoandMcCrory1979;Gabrielovetal.1974;GelfandandMacPherson1992;HalperinandToledo1972;Levitt1989;LevittandRourke1978;MacPherson1993;Stone1981;Yu1983].Inthecaseofsmoothmani-folds,suchlocalformulasareexpressedintermsofcurvature,so,presumably,localcombinatorialformulasforcharacteristicclassescanbethoughtofascom-binatorialformulasforcurvature;thisconnectionismadeexplicitin[Cheeger1983].Inthemathematicalphysicsliterature,similarquestionshavebeenstud-iedunderthenameoflatticegaugetheory;see[PhillipsandStone1990],forexample.Themainresultof[LuoandStong1993]isalsosimilarinspirittotheresultswewillpresent,inthatthecombinatoricsoftheedgesandfacesofacombinatorial3-manifoldarerelatedtothetopologyoftheunderlyingmani-fold.Oneshouldalsocomparewith[Alexandrov1951;Gromov1987],andthereferencestherein,inwhichonedenesnotionsofcurvatureoncombinatorialspacesbycomparingdistancesbetweenpointswiththedistancesonewouldndonsmoothRiemannianmanifoldswithconstantcurvature.Aswillsoonbecomeclear,ourapproachiscompletelydierentfromthosefollowedintheseworks,andisbasedinsteadonananalysisofthecombinatorialLaplaceoperator(thisoperatoralsoappearsin[Yu1983],butplaysaverydierentrolethere).ThecombinatorialLaplaceoperatorhasalonghistory.Inthecaseofaone-dimensionalsimplicialcomplex,thisoperatorappeared,implicitly,inKircho'sworkonelectricalnetworks[1847].See[Chung1997;Forman1993]formorerecentworkonthiscase.ThecombinatorialLaplaceoperatorongeneralcomplexeswasintroducedbyEckmann[1945].ThecloserelationshipbetweenthecombinatorialandRiemann-ianLaplaceoperatorshasbeenexploredinanumberofpapers,mostnotablybyDodziukandhiscollaborators;see,forexample,[Dodziuk1974;1981;Dodz-iukandKarp1988;DodziukandPatodi1976].TheworkbyGarland[1972;1973;1975],isverycloselyrelatedtoours,inthathealsousesthecombina-torialLaplaceoperatortodenecombinatorialcurvatures,whicharethenusedtodeduceglobaltopologicalpropertiesofthecomplex.Thereadercanconsult[Friedman1996;FriedmanandHandron1997;Kooketal.1998]forsomefasci-natingrecentapplicationsofthecombinatorialLaplaciantosomeproblemsincombinatorics.Letusbeginwithsomeexamplesofthetheorywedevelop.Althoughthetheorycanbeappliedtoverygeneralcellcomplexes,inthisintroductionwewillrestrictattentiontoCWcomplexessatisfyingacombinatorialconditionmodelledonthenotionofconvexity.Thereadermaythinkonlyofsimplicialcomplexes,butitisinsightfultoallowmoregeneralconvexcells. COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY195Ofparticularinterestisthecase=1.Inthiscasewegivethecurvaturefunctionaspecialname.Foranyedge(1-cell),denetheRiccicurvatureRic(Moreexplicitly,Ric(2-cells�feparallelneighborsofForexample,inFigure13Ric()=2+24=0Thus,thestandardlatticecelldecompositionoftheplaneisa\Ricci at"cellcomplex.Infact,thereadercaneasilycheckthatforallvanishesonthestandardlatticecelldecompositionofEuclideanspaceofanydimension.Attherightweshowa2-dimensionalspherewith\cubic"celldecomposition.Inthiscasetheedgehas2parallelneighbors(theedgesand)sothatRic()=2+22=2 ThepurposeofFigure14istodemonstratethatthedenitionsandtheoremsdiscussedinthispapercan,withafewexceptions,beappliedtoverygeneralcomplexes.Thecomplexneednotbeamanifold,orhaveanyotherspecialstructure. Figure14.TheRiccicurvature|indicatedhereforeachedge|canbecom-putedforarbitrarycomplexes.IntheRiemanniancaseBochnershowedthatonedoesnotneedthe-thcurvaturefunctiontobestrictlypositive,aswerequiredinTheorem2.4,inordertodrawsometopologicalconclusions.Infact,nonnegativityalsohasstrongimplications.Inthecombinatorialsetting,wecanproveacorrespondingresultinthecasep=1.Theorem2.5.Letbeaconnectedquasi-convexcomplex(i)SupposethatRic(foralledgesIfthereisavertexsuchthatRic(forallthen COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY197Stone[Stone1976]denesadierentnotionofcurvature,inasomewhatmorerestrictedsetting,fromwhichaversionofMyers'theoremisdeduced.EventhoughtheproofofTheorem2.6isverysimilartoStone'sproof(andtheproofin[Myers1941]),therelationshipbetweenthetwonotionsofcombinatorialcur-vatureisnotcleartothisauthoratpresent.Thesetheoremsshowthattherearestrongtopologicalrestrictionsforatopo-logicalspacetohaveaquasi-convexcelldecompositionwithRic(0(orforeachedge.Itisnaturaltoconsidertheotherextreme,andtoinvestigatethepossibilityofdecompositionswithRic(0foreveryedge.In[Gao1985;GaoandYau1986;Lohkamp1994a;1994b]itwasshownthateverysmoothmanifoldofdimension3hasaRiemannianmetricwitheverywherenegativeRiccicurvature.Thesame,andmore,istrueinthecombinatorialsetting.Theorem2.7.Letbeanotnecessarilycompactcombinatorial-manifoldThenthereisanitesubdivisionsuchthatRic(foralledgesItisinterestingtonotethatthecorrespondingRiemanniantheoremisonlytrueformanifoldsofdimension3.TheGauss{BonnetTheoremprovidesanobstructionforasurfacetohaveametricwithnegativeRiccicurvature(whichistosayGaussiancurvatureif=2).Theorem2.7impliesthatthereisnosimpleGauss{BonnettheoremforcombinatorialRiccicurvature.See[Forman5]forawaytorecoveraversionofGauss{Bonnetinthissetting.WewillnowgiveanideaoftheproofsofTheorems2.4and2.5.ReadersfamilarwithBochner'soriginalproofwillnoticethatwearesimplyfollowinghisproof(nowknownas\Bochner'smethod")inacombinatorialsetting.LetbeaniteCWcomplex.Considerthecellularchaincomplex�!


Endoweach)withapositivedeniteinnerproductsuchthatthecellsareorthogonal.Thisinvolveschoosingapositive\weight"foreachcellandsetting;(Theformulaspresentedinthispaperresultfromsetting=1foreverycell.)Let)denotetheadjointofandthecorrespondingLaplacian.ThecombinatorialHodgetheoremfollowsfrombasiclinearalgebra.Tomotivatewhatfollows,wenowpresentaverybriefoverviewofBochner'soriginalproofinthesmoothcategory.WewillfreelyusestandardnotationandterminologyofRiemanniangeometry.Thereadermayfeelfreetoskip COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY199Thelinearmapdenedbythematrixisnonnegative(withrespecttothestandardinnerproducton).Thisisaspecialcaseofthefactthatanysym-metricmatrix()suchthatforeachisnonnegative.Thisisoneofthoseveryusefulfactsinmatrixtheorywhichisconstantlybeingre-discovered.Forthehistoryofthisresult(notstatedinquitethesameway)andextensions,see[Taussky1949].Thedesiredconclusionisalsoimpliedbytheclassicaleigenvalueestimateknownas\theGershgorincirclemethod"[Gersh-gorin1931],whichsaysthattheeigenvalueslieincirclescenteredatwithradiusThenonnegativityofimpliesthatinthissimpleexamplewecanalreadymaketheBochner-likestatementthatifeachofthe3numbers)ispositivethentheoriginal33matrixhasatrivialkernel.ToapplythisdecompositiontothecombinatorialLaplaceoperator,wemustrstrepresenttheLaplaceoperatorasasymmetricmatrix,andthisrequireschoosinganorderedorthonormalbasisforthespaceofchains.Wecandenesuchabasisbychoosinganorientationforeachcellof,andthenchoosinganorderingofthecells.Makingadierentchoicefortheorientationofacellhastheeectofmultiplyingthecorrespondingrowandcolumnofthesymmetricmatrix1.Permutingtheorderingofthecellshastheeectofapplyingthesamepermutationtoboththerowsandthecolumnsofthematrix.Therefore,foradecompositionofsymmetricmatricestoinduceawell-deneddecompositionofthecombinatorialLaplaceoperator,thedecompositionmustbeequivariantundermultiplyingarowandcolumnby-1,andapplyingasinglepermutationtoboththerowsandthecolumns.Itiseasytoseethatthedecompositionshownin(2.2)hasthisproperty,andhenceinducesadecomposition.InanalogywiththeBochner{Weitzenbockformula,wecallthecombinatorialBochnerLaplacian,and-thcombinatorialcurvaturefunction.Itfollowsimmediatelythatif0then0,sothat)=0.Forany-cell,wedeneOurdenitionofRiccicurvature,Ric()wasmotivatedbyformula(2.1).Wenotethat(2.1)isatheoremintheRiemanniansetting,sinceRiccicurvaturewasoriginallydenedinacompletelydierentmanner,butitisadenitioninthecombinatorialcase.ProvingTheorem2.4isnowreducedtoexplicitlycalculatingasdenedabove,andshowingthatthisisequivalenttotheformulagiveninDenition2.3.See[Forman1998a]forthedetailsofthisstraight-forwardcalculation.IntheRiemanniancase(Ihopethereaderisnotgettingdizzyfromallofthisbouncingbackandforthbetweencategories),Theorem2.5followsfromtheBochner{Weitzenbockformulawithjustabitmorework.Namely,ifisa COMBINATORIALDIFFERENTIALTOPOLOGYANDGEOMETRY201weknewthat,thenwouldbedeterminedbyand.Puttingtheseideastogether,itisnottoohardtogettothegeneralresultthatifisconnected,and,thenforanyvertexiscompletelydeterminedby.Since=Kerthedesiredtheoremfollows.ToseehowTheorem2.5(i)follows,supposeRic(0foreachedge.Then,sinceisalsoanonnegativeoperator,=KerKerRic=(KerKerRic=(KerKerRicNowsuppose.ThenKer(Ric).SinceRic(0foreachedge,wemusthave=0foreachedge.Nowweobservethatsinceiscompletelydeterminedby,andhencewemusthave=0.Thisdemonstratesthat)=0.Theresultswehavepresentedhere,aswellasthoseinthereferencescitedearlier,indicatethatthereshouldexistmorecompletetheoriesofcombinatorialdierentialtopologyandgeometry.However,atthispointthecombinatorialtheoriescaptureonlyasmallportionofthesebeautifulandfar-reachingclassicalsubjects.Thereremainsmuchtobedone.AcknowledgementsTheauthorthanksthestaoftheMathematicalSciencesResearchInstitute,aswellastheorganizersofthespecialyearinCombinatoricsandTopology,fortheirhospitalitywhilesomeofthisworkwascompleted.Theauthoralsothankstherefereesandeditor,whosethoughtfulcommentshavedrasticallyimprovedthispaper.References[Alexandrov1951]A.D.Alexandrov,\Atheoremontrianglesinametricspaceandsomeofitsapplications",TrudyMat.Inst.Steklov(1951),5{23.InRussian.[AllendoerferandWeil1943]C.B.AllendoerferandA.Weil,\TheGauss{BonnettheoremforRiemannianpolyhedra",Trans.Amer.Math.Soc.(1943),101{129.[Babsonetal.1999]E.Babson,A.Bjorner,S.Linusson,J.Shareshian,andV.Welker,\Complexesofnot-connectedgraphs",Topology:2(1999),271{299.[Bancho1967]T.Bancho,\Criticalpointsandcurvatureforembeddedpolyhedra",J.DierentialGeometry(1967),245{256.[Bancho1983]T.F.Bancho,\Criticalpointsandcurvatureforembeddedpolyhedra,II",pp.34{55inDierentialgeometry(CollegePark,MD,1981/1982),editedbyR.Brooksetal.,Progr.Math.,Birkhauser,Boston,1983. 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NewPerspectivesinGeometricCombinatoricsMSRIPublicationsVolume,1999MacdonaldPolynomialsandGeometryMARKHAIMANAbstract.Weexplainsomeremarkableconnectionsbetweenthetwo-parametersymmetricpolynomialsdiscoveredin1988byMacdonald,andthegeometryofcertainalgebraicvarieties,notablytheHilbertschemeHilb)ofpointsintheplane,andthevarietyofpairsofcommutingmatrices.Contents1.Introduction2072.SymmetricFunctionsandMacdonaldPolynomials2103.The!Conjecture2184.TheHilbertSchemeand5.FrobeniusSeries2296.TheIdeals7.DiagonalHarmonics2448.TheCommutingVariety250References2521.IntroductionThisarticleisanexplicationofsomeremarkableconnectionsbetweenthetwo-parametersymmetricpolynomialsdiscoveredbyMacdonald[1988]andthegeometryofcertainalgebraicvarieties,notablytheHilbertschemeHilb)ofpointsintheplaneandthevarietyofpairsofcommutingmatrices(\commutingvariety",forshort).Theconjecturesondiagonalharmonicsintro-ducedin[Haiman1994;GarsiaandHaiman1996a]alsorelatetothisgeometricsetting. MathematicsSubjectClassication.Primary05-02;Secondary14-02,05E05,13H10,14M05.Keywordsandphrases.Macdonaldpolynomials,Hilbertscheme,commutingvariety,sheafcohomology,Cohen{Macaulay,Gorenstein.SupportedinpartbyNSFMathematicalSciencesgrantDMS-9400934. MACDONALDPOLYNOMIALSANDGEOMETRY209Moregenerally,thegeometryoftheblowupdependsonmodule-theoreticpropertiesofthepowers.Weareledtoextend(1.1)andconjecturethat(1.2)thatis,thepowersofarethesymbolicpowersoftheidealofthecoinci-dencelocus.Infact,weconjecturethatthevariables;:::;xformaregularsequencefortheex;y]-module,forall.Asweshow,thisconjectureim-plies(1.2).Itfurtherimpliesthatthe'sformaregularsequenceon(thatis,onitsstructuresheaf,viewedasasheafoffx;y]-algebras).Assumingthisregular-sequenceconjecture,weareabletogiveaninductivesheaf-cohomologicalargumenttoshowthatisCohen{Macaulay,andthusthe!andMacdonaldpositivityconjecturesfollow.Animportantpointtoremarkonhereisthatthe!conjectureandmanyoftherelatedgeometricconjectureshaveevidentanalogousstatementsinmorethantwosetsofvariablesX;Y;Z;:::.Forthemostpart,theseanalogsfailtohold.However,theaboveconjecturesontheidealsareanexception,asweexpectthemtoholdinanynumberofsetsofvariables(thelastconjecturethenbeingthatanyoneofthesetsofvariablesformsaregularsequence).Ofcoursethereasoningleadingfromtheretothe!conjecturemakesessentialuseofhavingonlytwosets.TheisospectralHilbertschemealsoprovidesthegeometricsettingforthestudyofdiagonalharmonics,thesubjectofaseriesofconjectures[GarsiaandHaiman1996a;Haiman1994].Thespaceofdiagonalharmonicsmaybeiden-tiedwiththequotientringngx;y]bytheidealgeneratedbyallinvariantpolynomialswithzeroconstantterm.Itisconjectured,amongotherthings,thatthedimensionofasavectorspaceis(+1).Furtherconjec-turesin[Haiman1994]describeaspectsofitsstructureasagradedmoduleincombinatorialterms.In[GarsiaandHaiman1996a]weconjecturedacompleteformulaforthedoublygradedcharacterof,intermsofMacdonaldpolyno-mials,andprovedthatthismasterformulaimpliesalltheearliercombinatorialconjectures.Ingeometrictermsisthecoordinateringofthescheme-theoreticberovertheoriginunderthenaturalmap(fromordered-tuplesofpointsintheplanetounordered-tuples.Now,thereisabersquareHilb The!conjecturehasacquiredaminorhistoryofexcitingbutunsuccessfulideasforsimpleproofs,bytheauthorandothers.Agoodrealitycheckonacontemplatedproofistoaskwheretheargumentbreaksdown|asitmust|inthreesetsofvariables. MACDONALDPOLYNOMIALSANDGEOMETRY211Weworkthroughoutwithsymmetricfunctionsininnitelymanyindetermi-nates,withcoecientsintheeldq;t)ofrationalfunctionsoftwovariablesand.Thevariousclassicalbasesoftheringofsymmetricfunc-tionsareindexedbyintegerpartitions,anddenotedasfollows:themonomialsymmetricfunctionsby,thepower-sumsby,theelementarysymmetricfunctionsby,thecompletehomogeneoussymmetricfunctionsby,andtheSchurfunctionsby.Ineachbasis,asrangesoverpartitionsofagiveninteger,weobtainabasisforthesymmetricpolynomialshomogeneousofdegreeThestandardpartialorderingonpartitionsofisthedominanceorder,de-nedbyforall.ThetriangularityoftransitionmatricesbetweencertainbasesofsymmetricfunctionswithrespecttodominanceplaysacrucialroleinthedenitionanddevelopmentofMac-donaldpolynomials,aswellasinthereasoningwewilluselatertodeducetheMacdonaldpositivityconjecturefromthe!conjecture.Wenowturntotheimportantdeviceofplethysticsubstitution.Thefactthatthepower-sumsformabasismeans,equivalently,thattheringofsymmet-ricfunctionscanbeidentiedwiththeringofpolynomialsinthepower-sums.Inparticular,the'smaybespecializedarbitrarilytoelementsofanyalgebraoverthecoecienteld,andthespecializationextendsuniquelytoanalgebrahomomorphismonallsymmetricfunctions.NowletbeaformalLaurentserieswithrationalcoecientsinindetermi-nates,whichmayincludeourparametersand.WedeneneA]tobetheresultofreplacingeachindeterminate.ExtendingthespecializationnA]toarbitrarysymmetricfunctions,weobtaintheplethysticsubstitutioninto,denotededA].IfAismerelyasumofindeterminates,,thenweseethattA]=pk(a1;a2;:::;a),andhenceforeverywehaveveA]=f(a1;a2;:::;aThisiswhyweviewtheoperationasakindofsubstitution.Similarly,ifhasaseriesexpansionasasumofmonomials,,A]isevaluatedonthesemonomials;forexample,,=(1�t)]=;t;tOurconventionwillbethatinaplethysticexpressionstandsforthesumoftheoriginalindeterminates,sothattX]isthesameassX=(1�t)]=;:::;tx;tx;:::;tandsoforth.Amongthevirtuesofthisnotationisthatthesubstitutionof)forasabovehasanexplicitinverse,namelytheplethysticsubsti-tutionof)forTheonecautionthatmustbeobservedwithplethysticnotationisthatinde-terminatesmustalwaysbetreatedasformalsymbols,neverasvariablenumericquantities.Forinstance,ifishomogeneousofdegreethenitistrue(and MACDONALDPOLYNOMIALSANDGEOMETRY213validforallf;g.Inotherwords,theplethysticsubstitutionofself-adjoint.Macdonalddeneshispolynomialsbyrstintroducingaq;t-analogoftheHallinnerproduct,whichinplethysticnotationissimplyf;gq;t Inviewof(2.7),thisdenitionissymmetricinand.Ifandareq;t-dualbases,thenand areHalldual,sotheCauchyidentitygives or X]v[Y]:(2.8)Notethatthenonplethysticexpressionfor ,asin[Macdonald1995],istherathermysteriousproduct where(Asaparticularcaseof(2.8),weseethattheq;t-dualbasistothemonomialsisthebasisoftransformedcompletesymmetricfunctions (denotedin[Macdonald1995]).TheMacdonaldpolynomialsq;t)maybedenedbyrequiringthattheyareorthogonalwithrespecttoq;t,andlowerunitriangularwithrespecttothemonomials,thatis,q;tq;t(2.9)forsomecoecients.Here,however,weshalldenethemdirectlyaseigen-functionsoftheplethysticoperatororX�(1�q)=z] [(2.10)wheretheverticalbarindicateswearetotaketheconstanttermwithrespectBeforefurtherexaminingtheoperator
wedeneanimportantquantitythatappearsintheeigenvaluesoftheoperator,andwillturnouttohavegeometricsignicancelateron.(ThetheoryofMacdonaldpolynomialsisrifewithnumerology.Quantitiessuchasandvariousq;t-hookproductscropupagainandagain;see[GarsiaandHaiman1998;1996a]formanyexamples.Inourgeometriccontext,thesequantitieswillturnouttohavenaturalinterpretations.)Firstrecallthatthediagramofapartitionisthearrayoflatticepointsi;jj(2.11) MACDONALDPOLYNOMIALSANDGEOMETRY215From(2.14),usingtheidentity),weseethatforanyfunction andtherefore,sinceceX�xi(1�q)]= Notethatthesubstitutionofinsidetheplethysmispermissible,sincewearesubstitutingoneindeterminateforanother.RecalltheJacobiformula)=detFromthisweseethatthecoecientofintheSchurfunctionexpansionofanysymmetricfunction)isthecoecientof),where;:::;0).Inparticularthecoecientin
givenby(2.15)whereisthecoecientof,so=1.Nowineachsummandabove,theleadingtermindominanceorderisclearly,establishingthetriangularity.Thistermarisesfromtheterm,multipliedbytheterm).Inthe-thsummandtheindicatedsubstitutionsmultiplyitby.ThuswendWiththeunderstandingthatiszeroforexceedingthenumberofparts,wereadilyverifythattheaboveexpressionisindependentof�nlasitmustbe,andreducestoq;tCorollary2.3.Theoperatorhasdistincteigenvaluesq;tanditscorrespondingeigenfunctionisalinearcombinationofthemonomialsymmetricfunctionswithnonzerocoecientofDefinition.Macdonaldpolynomialq;t)istheeigenfunctionoftheoperator
q;t MACDONALDPOLYNOMIALSANDGEOMETRY217whereq;tq;tInparticularsinceitisknownthat[Macdonald1995]),isnormalizedsothatitscoecientofNowtheoperator
issymmetricinand,whilet;qq;t).HenceweobtainProposition2.5.Forallwehaveq;tt;qconse-quentlyq;tt;qAsmentionedearlier,thefunctionscanbecharacterizedbycertaintriangu-larityrelations.Proposition2.6.ThetransformedMacdonaldpolynomialssatisfyandareuniquelycharacterizedbytheseconditions(1))X(1�q);q;tq;t(2))X(1�t);q;tq;t(3)Proof.NotethattX(1�t);q;tt�X(1�t�1);q;t]isascalarmulti-pleoff�X;q;t]andthusofq;t).Sincebelongstothespaceq;t,andconjugationreversesthedominanceorder,be-longstoq;t,whichis(2)above.FromthesymmetrygivenbyProposition2.5wethenobtain(1).ThenormalizationfromProposition2.4is(3).Foruniqueness,suppose)isanothersolutionof(1)and(2).Then(1)impliesthattX(1�q)]2Q(q;ttX(1�q)]:andhencethatq;t.Similarly,(2)impliesthatq;tTogetherthesemeanthatisascalarmultipleof,and(3)xesthescalarfactoras1.Corollary2.7.Forallwehaveq;tconsequentlyq;tProof.Oneverieseasilythat)satises(1)and(2)ofProposition2.6,andhenceisascalarmultipleof.Toxthescalaras1requiresthat.Butthisisknown[Macdonald1995],asitisequivalenttoToconclude,wewillrecovertheorthogonalityofthe'swithrespecttoq;tasinMacdonald'soriginaldenition.Replacing,wearetoshowthatq;t q;t,orequivalentlythatq;t q;t MACDONALDPOLYNOMIALSANDGEOMETRY219Itisknown[Macdonald1995]that1)=(1),thedegreeofthecharacterorthenumberofstandardYoungtableauxofshape.Hence,accordingtoConjecture3.2,wemusthavemult(ch())=(1),sothatwhenweignorethegrading,aordstheregularrepresentationof,andhencehasdimension!.ThusConjecture3.2impliesConjecture3.1.OneofthechiefthingswewillachieveinthegeometricsettingofSections4and5istoprovetheconverseimplication.Itwillbehelpfultoreformulatetheconjecturesintwoways.Therstistointroduceamoreconvenientnotationfor(3.2).RecallthattheFrobeniusmapfromcharacterstosymmetricfunctionshomogeneousofdegreeisdenedby (3.3)where)isthepartitionwhosepartsarethelengthsofthecyclesofthepermutation,andwheretheindeterminatesarenottobeconfusedwiththecoordinates.Fortheirreduciblecharacterswehavethesymmetricfunctionidentity(),fromwhichfollows,foranycharacter,mult(Now,byanalogytotheHilbertseries,wedenetheFrobeniusseriesofadoublygradedrepresentationtobeq;tr;sch(r;sThisgiven,Conjecture3.2takesthesimpleform(3.4)InSection5weextendthenotionofFrobeniusseriestoactionsonmodulesoverageometricregularlocalringwithanequivarianttwo-dimensionaltorusaction,providingthebasictooltolinkConjecture3.2withthegeometry.Thesecondreformulationweneedisofthedenitionofitself|thedef-initionintermsofderivativesissimple,butgeometricallymisleading,andweneedaderivative-freeversion.Thisisgivenbythenextpropositions.Wewillneedthefollowingdenitionhereandlater.Definition.alternationoperatoroverthesymmetricgroupisAltSincethepolynomials
formabasisofallthe-alternatingpolynomialsinnx;y],itmakessensetospeakofthecoecientof
inAlt.Indeed,itismerelythecoecientofthemonomial MACDONALDPOLYNOMIALSANDGEOMETRY221(compareCorollary2.7).Proof.IntheproofofProposition3.4therearetwoisomorphismsoffx;y]=Jwith.Therstistheinclusionoffx;y],followedbyprojectionmod,whichisanisomorphismofgradedrepresentations.Thesecondisthe.As
-alternatingandhomogeneousofdegrees)),thismapreversesdegreesandtensorscharactersbythesigncharacter.RecallingthattheFrobeniusmapsatises(),weseethatreversingdegreesandtensoringwiththesigncharacteronyieldsaspacewithFrobeniusseries).CombiningthetwoisomorphismsshowsthisisequaltoRemark:ThealgebraiccorrelateofthissymmetryofisthattheringGorenstein.Moregenerally,foranyhomogeneousidealalx],thequotienttx]=Jisnite-dimensional(asavectorspace)andGorensteinifandonlyifistheidealofdierentialoperatorsannihilatingsomehomogeneouspolynomial.Proposition3.7.WehavetheidentityProof.Theleft-handsideistheFrobeniusseriesofthesubspaceex]\D.TheGarnirpolynomial)isdenedastheproductoftheVandermondedeterminantsintherstvariables,thenextvariables,andsoon.Itisnothardtoseethatthespaceex]\Disspannedbyallderivativesofanditsimagesunderpermutationofthevariables.Thisspacehasbeenwellstudied[BergeronandGarsia1992;DeConciniandProcesi1981;GarsiaandProcesi1992;HottaandSpringer1977;Kraft1981;Springer1978],anditsFrobeniusseries(inonevariable)isknowntobethetransformedHall{LittlewoodpolynomiallX=(1�t�1);t�1].Since),theresultfollows.Itisalsopossibletogiveanelementaryproof(see[GarsiaandHaiman1996b])thatthe!conjectureimplies).Since1)de-terminesthecharacterandthusthedimensionof,onemustofcourseassume!conjectureforthis.Butaswearegoingtoprovethatthe!conjectureimplies,theproofforthe=1specializationwouldberedundanthere.4.TheHilbertSchemeandLet=Speccx;y]betheaneplaneover.Bydenition,closedsub-schemescorrespondtoidealssx;y].Thesubschemeis0dimensional,oflength,iffx;yhasdimensionasavectorspace.Thegenericexampleofsuchasubschemeisasetofpointsin,withthere-ducedsubschemestructure.Inthiscaseisaradicalidealanddx;ycanbeidentiedwiththeringofcomplex-valuedfunctionsontheniteset,whichisclearly-dimensional. MACDONALDPOLYNOMIALSANDGEOMETRY223 atandniteofdegreeover,thereisauniquemorphismHilbsuchthatistheberproductHilb.Theuniversalproperty,asusual,characterizesHilb)andtheuniversalfamilyuptocanonicalisomorphism.Thefollowingresult,whichisuniquetoandfailsinmorethantwodimen-sions,isoneofthetruemiraclesofthemathematicaluniverse.Theorem4.1[Fogarty1968].TheHilbertschemeHilbisirreducibleandnonsingularofdimensionGivenHilb),theoperatorsandofmultiplicationbyandrespectively,arecommutingendomorphismsofthevectorspaceex;yS;PwhichpreservethedirectfactorsS;P.ForandarenilpotentonS;P,sothatS;Pisthecharacteristicsubspaceassociatedwiththejointeigenvalue()of(X;Y).Thusthepoints(;:::;)of,eachincludedwithitsmultiplicityin,arethejointspectrumof(X;Y).Inparticular,thepolarizedpowersumsh;ksatisfytheidentityh;k)=tr(4.4)NotethattracemapassociatedtothenitemorphismHilb)sendstheregularfunction(comingfrom)totr,sothelatterisaregularfunctiononHilbNowthesymmetricgroupactsonthevariety=Speccx1;y1;:::;xofordered-tuplesofpointsintheplane,andwemayidentifySpecoftheringofinvariantsasthevarietyofunordered-tuples,or-elementmultisets,=Speccx;y]Sn.ByatheoremofWeyl[1946],thepolarizedpower-sumsh;kgeneratethisringofinvariants.Itfollowsfromthisandtheprecedingparagraphthatthemap:Hilbsendingtothemultisetofpointsofthecorrespondingsubscheme,withtheirmultiplicitiesin,isamorphism.Itextendstoamorphism^:Hilb,withbeingtherestrictionto^).SinceHilb)isaprojectivevariety,isaprojectivemorphism,calledtheChowmorphism.Notethatrestrictstoanisomorphismoftheopensetswherethepointsaredistinct,andsoisbirational. MACDONALDPOLYNOMIALSANDGEOMETRY225inducedby(4.6).Proof.Sinceisreduced,asectionofbelongstotheidealsheafofifandonlyif,regardedasaregularfunctionon,itvanishesonanydenseopensubset.Henceitisenoughtocheckthepropositiongenerically,overthelocuswhereconsistsofdistinctpoints.Supposeisasectionthatvanishes.ThensodoandAlt,foranysection.Butsinceitisalternating,AltalsovanishesatanypointI;P;:::;Pforwhichtwoofthepointscoincide.Henceitvanishesonandthusonallof,whichmeansitiszeroin.Theconditionthatbelongstothekernelof(4.7)ispreciselythatAlt=0forallConversely,supposedoesnotvanishon,andchooseapointI;P;:::;Poutsidethevanishinglocus),withthealldistinct.Aftermultiplyingbyasuitablewecanarrangethatvanishesatallpointsoforbitof,except.ThenAltdoesnotvanishat,soAlt=0.Nowconsiderthesituationoveroneofthedistinguishedpoints.Theber)ofthevectorbundleex;y,andthatofofx;y]=�I(x1;y1)+


+I(xn;yn)
:Noticethateveryalternatingpolynomial
asin(3.1)vanishesmoduloexceptfor
.Inotherwords,the1-dimensionalspace)isspannedbytheimageof
,andthelinearfunctionalAlt:,composedwiththenaturalprojectionnx;y]!B n(I),isjustthemapsendingtothecoecientof
inAlt.TogetherwithProposition3.3,thisprovesthenextresult:Proposition4.3.Theidealofpolynomialsthatasdierentialoperatorsannihilateisthekernelofthemapmapx;y]!B n(I)!(B n)(I) VnB(I)inducedbythemap(4.7).ComparingthiswithProposition4.2wemightwellexpecttheberoftheidealsheafofovertobe,andthescheme-theoreticberofovertobeSpec.Wecannotyetdrawthisconclusion,however,sincethemapisonlyasheafhomomorphism,notnecessarilyahomomorphismofvectorbundles,andthustheberofitskernelatneednotequalthekernelofitsber.Whatwecansayisisthattheberofthekernelfactorsthroughthekerneloftheber,whichgivesthefollowingresult.Proposition4.4.Theimageoftheidealcontainstheimageofthebermapwhereistheberoftheidealsheaf MACDONALDPOLYNOMIALSANDGEOMETRY227ofidealswhicharestableandforwhichhx;y]=JhasagivencharacterisclosedinHilb,soforeveryeryx;y]=JaordstheregularrepresentationofNowthereisanaturalmapfromtoHilb),whichmaybedescribedasfollows.Sinceex;y]=Jaordstheregularrepresentation,itsonlyinvariantsaretheconstants.Thismeansthatmodulowehaveh;kh;ksomeconstanth;k,forallh;k.ByWeyl'stheorem,the-invariantsinnx;y],fortheactionofthrough,aregeneratedbyandthepolarizedpower-sumsh;k;:::;x).Modulo,thelatterarecongruenttoh;k,sotheinvariantsofofx;y]=Jaregeneratedbyand.Inotherwords,,x1;y1]=(J\C[x1;y1])=((x;y]=J)Sn�1.Itfollowsthattx1;y1]belongstoHilb),afteridentifyingwithx;yTheaboveconstructiondenesthemapHilb),whichalsohasthefollowgeometricdescription.Letbetheuniversalfamilyover.Ithasanatural-action,inwhicheveryberaordstheregularrepresentation.ThenW=Sis atandniteofdegreeover,andbytheabovecalculationcanbeidentiedwithafamilyofsubschemesof.ThemapHilb)istheonegivenbytheuniversalpropertyofHilb),forthefamilyW=SIftheequivalentconditionsofTheorem4.5hold,thenisa atfamilyofsubschemesof(,ofdegree!overHilb).TheuniversalpropertyofHilbthenyieldsamapHilbHilb,whoseimagelies.Generically,forsetsdistinctpointsinandtheircorrespondingregularorbitsin(,thesetwomapsaremutuallyinverse.Hence,assuming!conjecture,theyareinverseeverywhere,andthenaturalmapHilb)isanisomorphism.Conversely,ifHilb)isanisomorphism,thenitsinversedenesafamilyHilbthatis atoverHilb)andcoincideswithgenerically.Butthenisreducedandhenceequalto,sois at,andthe!conjectureholds.ItwouldevensucetoshowthatHilb)isinjective.Foritisproperandbirational,hencesurjective,andabijectivemorphismontoanonsingularvariety(oranynormalvariety)isanisomorphism,byZariski'stheorem.Tosummarize,wehaveProposition4.6.TheequivalentconditionsofTheorem4.5,forallpartitionsarealsoequivalentto(4)ThenaturalmapHilbisinjective(5)ThismapisanisomorphismProposition4.6impliesthatthe!conjectureisequivalenttoaninstanceofaconjectureofNakamura[1996],citedin[Reid1997],connectedwiththeMcKaycorrespondence,aswenowexplain.Letbeanitesubgroupof),where.ForanynitelineargroupactiononV=G=SpecisCohen{Macaulay,anditscanonical MACDONALDPOLYNOMIALSANDGEOMETRY2295.FrobeniusSeriesLetdenotethetwodimensionalalgebraictorusgroup,i.e.,themultiplicativegroup.Itactsalgebraicallyonbytherulet;qx;y)=(tx;qyfor(t;qx;yTheuniversalconstructionoftheHilbertschemeisfunctorialwithrespecttoautomorphismsof,soalsoactsonHilb).Thisactionsendsasub-schemeto(t;q,soonidealssx;y]itisgivenbythepullbackthroughtheringendomorphism(t;q),thatis,t;qx=t;y=qSimilarlyactsonthetheschemesand,andthevariousmapsbetweentheseschemesare-equivariant.Observethatapolynomial)isdoublyhomogeneousofdegree(r;s)ifandonlyifitisaneigenfunctionfortheactionwitheigenvalue.HencethebivariateHilbertseriesofanite-dimensionaldoublygradedspaceofpolynomi-alsissimplythecharacterofthe-action,asafunctionofand.SimilarconsiderationsapplytotheFrobeniusserieswhenacts,asitisjustagener-atingfunctionfortheHilbertseriesofthethevarious-isotypicsubspaces.Inthegeometricsituationwehavetodealwithactionsonlocalringsandsheavesofmodulesthatmaybeneithergradednornite-dimensional.Forthisweneedrecoursetoa\formal"Hilbertseriesthatcapturesthenaivecharacterinthenitedimensionalcase,andextendstothemoregeneralsetting.Definition.Letbethelocalringofaschemeofnitetypeoveraclosedpointwithmaximalideal.Assumeisnonsingularatandthatisanisolatedxedpointforanalgebraicactionof.Letanitelygenerated-modulewithanequivariantaction.ThentheformalHilbertseriesisgivenbyq;ttr(Tor det(witht;q(5.1)Toseethatthedenitionissound,observerstthatthemodulesTorandthecotangentspacearenitedimensionalrepresentationsof,sothetraceanddeterminantintheformulamakesense.Sinceisregular,thesyzygytheoremimpliesthatthesuminthenumeratorisnite.Sinceisanisolatedxedpoint,theactiononthecotangentspacedoesnothave1asaneigenvalue,sothedenominatordoesnotvanishidentically,butisaproductoffactorsoftheform(1)withr;snotbothzero.Notethatq;t)isarationalfunctionofandProposition5.1.(1)isanexactsequencethen MACDONALDPOLYNOMIALSANDGEOMETRY231Proposition5.3.TheformalFrobeniusserieshasthefollowingproperties(1)Ifthecharacterhasmultiplicityzerointhen=0.(2)Letbeanite-dimensionaldoublygradedmoduleThentensoringwehavewhereistheinternalproductofsym-metricfunctions(3)Supposeisan-equivariant-modulewhereisanite-algebrawithanactionSuppose;:::;xisan-regularsequencethatactsontheelementst;qandthatactsontheseelementsbypermutingthemThenq;ttX(1�t);q;tProof.(1)LetbetheReynoldsoperator,thecentralidempotentinthegroupalgebraofthatactsineachrepresentationastheprojectionontheisotypiccomponentwithcharacter.Ifisanitelygenerated-modulewithactionthenithasacanonicalisotypicdecompositionandeachisalsoanitelygeneratedmodulewithaction.Thedecompositionmeansthattheidentityfunctorisnaturallyisomorphictothedirectsumofthefunctors.Inparticularthefunctorsareexact,andcommutewithTor,sincetheactiondoes.Now,comparingthedenitionsoftheformalHilbertandFrobeniusseries,andusingthefactthat,weseethat (1)ByNakayama'sLemma,if=0,then=0.(2)Recallthattheinternalproductisdenedby,andsatisestheidentity)=(forallcharacters.Fromthisitisclearthat(2)holdsinthecasewherehasnitelength,byProposition5.2.ForthegeneralcasewehaveTorTor),whichreducestheidentitytothecorrespondingonewiththenite-lengthmodulesTor)inplaceof(3)Letdenotethespacespannedbyelementsoftheregularsequenceasanmoduleitaordsthepermutationrepresentationoftensoredbytheone-dimensionalrepresentationofwithcharacter.Sinceisareg-ularsequencewehavethefollowingexactsequence,theKoszulresolutionof!


!(5.2)wherethetensorproductsareover.ThemapsintheKoszulcomplexareequivariantifwetaketheactionontobemultiplicationby MACDONALDPOLYNOMIALSANDGEOMETRY233Viathemap,thecoordinates;:::;xon(deneglobalregularfunctionsonandthuselementsofthelocalringLemma5.5.isCohen{Macaulayatthen;:::;yisaregularse-quenceProof.Wearetoshowthat;:::;ycutoutacompleteintersectionin.Sincedim(wemustshowthatdim.Now)consistsofthosepoints(I;P;:::;Pforwhichallthepointslieonthe-axis,whichisthesameassayingthat)liesonthe-axis.EllingsrudandStromme[EllingsrudandStrmme1987],studyingthecoho-mologyofHilb),constructedacelldecompositionthatcontainswithinitacelldecompositionofthesubsetHilb)consistingofpointsforwhich)liesonthe-axis.EverycellintheirdecompositionofhasdimensionSincethelocusisniteoveritsdimensionisaswell.Thelocalringisnotonlyniteover,itisfree,andbytheproofofTheorem4.5,whereisthemaximalidealof.ByNakayama'slemma,isfreelygeneratedasanmodulebyanysubspacecomplementarytotheideal.Wemaychoosetobestable(infact,wecanchoosetobethespaceofderivatives),andthenwillhavethesameFrobeniusseriesas.Thisshowsthatq;tq;tq;t(5.3)Incidentally,thequantityq;t)hasatantalizingexplicitvalue.In[Haiman1998]weconstructedanexplicitsystemofdoublyhomogeneousregularlocalparametersfor.Fromtheirdegreesoneobtainsq;t wherethearms)andlegs)areasin(2.16).AfterthereplacementtherstfactorinthedenominatorisexactlythenormalizingfactorforthedenitionofMacdonald'sintegralforms.ThiscoincidenceisatypicalexampleofthelinksbetweenthenumerologyassociatedwithMacdonaldpolynomialsandgeometricallysignicantquantitiesattachedtotheHilbertscheme.Nowconsiderthering).Justasisgeneratedasanmodulebyanysubspacerepresenting)isgenerated(nolongerfreely,however)byrepresentativesof).Thislastspacecanbeidentiedwiththecomponentof,orof,homogeneousofdegreezeroin.AsmentionedintheproofofProposition3.7,thisspaceiswell-understood,anditsFrobeniusseriesiswhereisaHall{Littlewoodpolynomialand)denotestheclassicalone-variableKostkacoecient.Inparticular,since)=0unless,the MACDONALDPOLYNOMIALSANDGEOMETRY235Proof.Theproofisbyinductionon.Notethatfor=1and=2wetriviallyhave,andwehavetheremainingcasesthrough1byinduction.Firstconsiderthesituationlocallyatapointtx;y]wherethe(arenotallequal.Withoutlossofgeneralitywecanassumethatnoneof;:::;)isequaltoanyof(;:::;).Inthelocalring,thedierencesandareinvertiblewheneverisintherstgroupandinthesecond.Hencereduceslocallytotheproductoftheidealsintherstindicesandthelastseparately.Lessobvious,butstilltrue,isthat,andhence,decomposessimilarly.Toseethis,letbeageneratorof;:::;x;:::;xalternatingintherstandlastindices,i.e.,thesubgroupactsonbythesigncharacter.Letbeanypolynomialthatbelongstothelocalizationateverypointintheorbitof,butdoesnotvanish.Now=Altbelongsto.Thetermsinthealternationcorrespondingtoelementsthatdonotstabilizebelongto,byconstructionofSinceisalreadyalternatingwithrespecttothestabilizerof,theremainingtermssumto,andthesumhereisinvertiblein.Thisshows,andso.Thereverseinclusionisclear.Usingtheinductionhypothesis,weconcludethatlocallyoutsidethelocuswhereallpointscoincide.Nowsinceandbelongtheidealof,ourhypothesisimpliesdepth2,andthelocalcohomologyexactsequenceforthesheafofidealsassociatedtogives)=0(6.3)where.Thus).Thelatteristheidealofallpolynomialswhoserestrictionstobelonglocallyto,sowehaveshown.Aswehadtobeginwith,wemusthaveThereisofcoursenothingspecialaboutthechoiceofandintheaboveProposition;it'sjustanexplicitwaytoguaranteethatdepthThiswouldalsofollowifdepth1,whichmeanstheidealofcontainsanelementthatisanon-zero-divisormodulo.Note,bytheway,thattheproofofProposition6.2worksequallywellwithmorethantwosetsofvariables.Someexplorationswehavedoneforsmallvaluesofusingthecomputeralge-brasystemMacaulay[BayerandStillman1989]suggestthefollowingconjecture.Conjecture6.3.denotestheidealgeneratedbythealternantsinnx;y;:::;foranynumberofsetsofvariablesisthelocuswhereallthepoints;:::;z),(;:::;zcoincidethenisamaximal-regularsequenceintheidealofforallparticular,depth MACDONALDPOLYNOMIALSANDGEOMETRY237Lemma6.6.LetI;P;:::;PbeapointofLetthedistinctpoints;:::;P;:::;Qwithmultiplicities;:::;rTheninthereisaneigborhoodofisomorphictoanopensetintheproductProof.Withoutlossofgeneralitywecanassume,andsoon.Forourneighborhoodofwecantakethepreimageinoftheopensetofpointswheretheonlycoincidencesthatoccurhavei;jwithinoneoftheseconsecutiveblocks.ThentheresultisclearfromProposition6.4,togetherwiththeproductdecomposition,validon,oftheideal;:::;r;:::;rfromtheproofofProposition6.2.Forsomedimensionargumentsbelowwewillneedthefollowingresults.Lemma6.7.ThereisadecompositionofHilbintolocallyclosedanecellssuchthateverypointofcontainsintheclosureofitsorbitdimwhereisthenumberofpartsofLemma6.8.Thereisadecompositionofthezero-ber(0)HilbintolocallyclosedanecellssuchthateverypointofcontainsintheclosureofitsorbitdimProof.See[EllingsrudandStrmme1987].Lemma6.9.LetbetheclosedlocusofidealsHilbforwhichsomepointofhasmultiplicityatleastThenhascodimensionhasonlyoneirreduciblecomponentofmaximaldimensionProof.Itisknown[Briancon1977]thatthezero-ber(0)isirre-ducibleofdimension1.Thelocuswhereallthepointscoincideisjusttheproductof(forthechoiceoforigin)by,soitisirreducibleofdimension+1.ByLemma6.6,itfollowsthatthe(locallyclosed)locuswherethemultiplicitiesare;:::;rhasdimension+1)=andcodimension.Ifonemultiplicityisatleast,thiscodimensionisatleast1,withequalityonlyformultiplicities;:::;1.AgainbyLemma6.6,thelocusinwhere,andtheotherpointsaredistinctfromandeachother,isirreducible.ItsurjectsonthelocusinHilb)wherethemultiplicitiesare;:::;1,sothelatterisirreducibleaswell.AsasteptowardtheCohen{MacaulaypropertyweneednormalityresultsforandDefinition.AnidealHilb)iscurvilinearifthelocalringsS;Phaveembeddingdimension1,i.e.,theirmaximalidealsareprincipal.Thisisequivalentto)beingasubschemeofasmoothcurveinwhencethename. MACDONALDPOLYNOMIALSANDGEOMETRY239Proof.Recallthatanidealinanormaldomainissaidtobeintegrallyclosedifeveryelementsatisfying(6.6)alreadybelongsto.TheaboveconditionforisequivalenttosayingthatbelongstotheintegralclosureofftJ]innt],soalltheidealsareintegrallyclosedifandonlyifftJ]isnormal.Forourand,Conjecture6.3implies,byProposition6.2.Itiswell-knownthatthepowersofanidealgeneratedbyaregularsequenceareintegrallyclosed,anditisobviousthatanintersectionofintegrallyclosedidealsisintegrallyclosed,soisintegrallyclosed.ThisshowsthatttJ]isnormal,so=ProjjtJ]is,bydenition,arith-meticallynormalinthegivenprojectiveembedding.Inparticularitisnormal.Nowwecometothegeometricconstructionthatsuppliestheinductivemachin-ery.Definition.nestedHilbertschemeisthesubvarietyofpairsHilbHilbProposition6.13[Cheah1998;Tikhomirov1992].ThenestedHilbertschemeisirreducibleofdimensionandnonsingularIf()isapointofthenthecorrespondingsubschemeisasubschemeof),sothemultiset)contains)alongwithoneadditionalpoint,orelsewiththemultiplicityofoneoftheoriginalpointsincreasedby1.Soifthespectrumofis(;:::;)thenthatofis(;:::;)foradistinguishedpointNowboththeinvariantsh;k;:::;x)andtheinvari-antsh;k;:::;x)areregularfunctionson,hencesoare;:::;x;:::;x)andsimilarly.Thismeanswehaveamor-phism=Speccxn;yn];mappingapairtoitsdistinguishedpoint.Ofcourse(),andbysuitablechoiceof,given,thedistinguishedpointcanbeanypointof).HencethecombinedmapHilb)factorsHilb)throughasurjectivemorphism(6.7)totheuniversalschemeoverHilb).Wherethepointsaredistinct,thismapislocallyanisomorphism,soitisbirational.TheabovemapandthemapHilb)areprojective.Infact,givenisdeterminedbyitssinglegeneratormod,soisa MACDONALDPOLYNOMIALSANDGEOMETRY241eachpointbyequations(6.5).Onthelocuswhereistheidealofareducedsubscheme;:::;,therstequation�


mustbethepolynomial),sotheparametersaretheelementarysymmetricfunctions).Fromthesecondequation,arethecoecientsoftheinterpolatingpolynomial)thatsatises)whenthe'sarealldistinct.Itiswell-knownthattheelementarysymmetricfunctionssatisfy^


^^


^,where)istheVandermondedeterminant.Inparticular,sincethe)and)areglobalregularfunctionsonHilband)isinvariant,^


^^


^^


^


)makessenseasarational-formonHilbMoreover,theequations)saythatthevector(;:::;y)istheproductof(;:::;a)bytheVandermondematrix,so^


^^


^),andhence^


^^


^Inparticular,therational2-formisregularandhasnozeroesonButisinvariantundertheactionof,soitfollowsthatisregularandnowherevanishingonevery.Sincewehavealreadyseenthatthecomplementofhascodimensiongreaterthan1,itfollowsthatisregulareverywhereandvanishesnowhere,whichshowsthatLemma6.17.Thecanonicalsheafofregular-formsonisisomor-phictowhereisthelinebundledenedbytheexactsequence0(6.8)inducedonthetautologicalbundlesoverbythecontainmentProof.Wearetoshowthatthelinebundleistrivial,anditsucestodothisonanopensetwhosecomplementhascodimension2.ByLemma6.15,wecanusetheopensetwhereiscurvilinearandthemaprestrictstoanisomorphism,whichmeanswecanverifyitonthecurvilinearlocusinByLemma6.16anddualityforthenite, atmorphismHilb),wehave,whereisthecanonicalsheafontheHilbertscheme.Sowehavetoshowthatasa-module,sinceNowlet'sexamineasasubbundleofonthecurvilinearlocusin.Toavoidconfusion,sincewealreadyhaveregularfunctionsx;y,wewriteforthevariablesof,sotheberofat(I;P)issx0;y0]=I(x0;y0).Then MACDONALDPOLYNOMIALSANDGEOMETRY243isaproperclosedsubvarietyoftheirreduciblevariety.Sinceis atoveritcannothaveacomponentcontainedinthecoincidencelocus,sothenoncoincidencelocusisdensein.Thisshowsisirreducibleandthemorphismisbirational.wehave,bypullbackfrom,thetwotautologicalbundlesand.Setk;l.ByLemma6.17thecanonicalsheafon1)inthisnotation.ByLemmas6.16and6.18therelativecanonicalsheafofover,whichispulledbackfromthatofoverHilb),is0).Hencethecanonicalsheaf1).Inparticular,itisapullbackfromNowwearegoingtoprovethatforthederivedfunctorofthepushforwardwehave.Since1)thisalsoproves1).Bydualityfortheprojectivemorphismweconcludethatthesheaf1)isthedualizingcomplexon,soisCohen{Macaulay.(ThisalsoshowsisGorensteinwith1),sowecouldhavemadeLemma6.18partoftheinduction.)Sinceisproperandbirational,andisnormalbyLemma6.12,wehave.Wehavetoprovethat=0forall0.Nowthebersarealsobersofthemap,andthustheberdimensionsareboundedby,byLemma6.14.Inparticular,sinceweareassuming4,thisimpliesdn2(exerciseforthereader).ItfollowsthatFor2weusethefollowinglemma.Lemma6.19.Letbeamorphismandlet;:::;xbeglobalregularfunctionsonandsoalsoonSupposethatisan-regularsequenceateverypointofbothinandinLetThenimpliesforProof.Thehypothesisandconclusionarebothlocalwithrespectto,sowecanassumeisane.Thenwearetoshow)=0for01.Let)(inbothand,byabuseofnotation).Theregularsequenceconditionimplies)=0forik,onbothand.Thehypothesisimpliesthat).Thenfromthelocalcohomologyexactsequences


!!


and


!!


weobtain)=0,for0Conjecture6.1impliesthat(;:::;x)isaregularsequenceonntJ],andhence,byProposition6.4,on.ToapplytheLemmausingthis MACDONALDPOLYNOMIALSANDGEOMETRY245InparticularthediagonalharmonicsaresolutionsoftheLaplaceequationsotheyareharmonicpolynomialsintheclassicalsense.Itiseasytoseethatthepolynomials
ofSection3arediagonalharmonics,andhencethespacesaresubspacesofLettx;y]betheidealgeneratedbythepolarizedpowersumsh;k0.Wemaydescribeinderivative-freetermsasfollows.Proposition7.1[Haiman1994].Thequotientringgx;y]=InisisomorphicasadoublygradedmoduletoForgeometricpurposeswewillworkinsteadwiththeringgx;y]=In;Inbeingagaingeneratedbythepolarizedpowersums,whichofcoursehasthesameFrobeniusseriesassx;y]=InorDn.ComputationshavesuggestedaseriesofsurprisingcombinatorialconjecturesconcerningtheHilbertandFrobeniusseriesoftherings.Asthesearetreatedatlengthin[Haiman1994],weherementiononlythreethataresimpletostate.Conjecture7.2.Thedimensionofasavectorspace1),equalto+1)Moreoverq;q)=(1+Conjecture7.3.Thespecializationenumeratesspanningtreesonthevertexset;:::;neachcountedwithweightwhereisthenumberofinversionsininversionisapairijforwhichvertexliesontheuniquepathinfromvertextovertexConjecture7.4.AsanmoduleisisomorphictothesigncharactertensoredwiththepermutationrepresentationofontheniteAbeliangroup+1)whereactsbypermutingthefactors+1);:::;isthesubgroupof-invariantelementsTheaboveconjecturesarecorollariestoapairofmoregeneralconjecturesgiv-ingtheFrobeniusseriesspecializations1)and).In[GarsiaandHaiman1996a]weshowedthatthesespecializationsareinturncorollariestothefollowingmasterformula.Conjecture7.5.TheFrobeniusseriesofisgivenbyq;tq;tq;tq;t (7.1) MACDONALDPOLYNOMIALSANDGEOMETRY247AsaschemeniteoverHilb),wehave=Spec,andsinceweareassumingthe!conjecture,islocallyfreeofrank!,thatis,itisthesheafofsectionsofavectorbundle,theimageofthehomomorphism(4.7).Then)=Specasaschemeover(0).Hencewecanidentifytheglobalsections)withProposition7.7[Haiman1998].Theschemetheoreticzerober(0)intheHilbertschemeisreducedCohen{Macaulayandhasa-equivariantresolutionbylocallyfreesheavesonHilb!


!(7.2)whereisasummandofthetautologicalbundledenotethetrivialbundletensoredbythe-dimensionalrepresentationofwithcharacterrespectivelyToproceedfurtherwewillneedtoassumethevalidityofthefollowingconjecture.Conjecture7.8.Forallwehave)=0,andfor=0,thecanonicalmapmapx01;y01;:::;x)(7.3)issurjectiveToclarify,recallthattx0;y0]=I,whereex0;y0]reallymeansthetrivialbundleex0;y0],andweuseprimestoavoidconfusionwiththevari-ables.Themapin(7.3)isinducedbythemapssx0;y0]!B,withtheidenticationssx0;y0] k=C[x01;y01;:::;x]anddx0;y0] k)=C[x0;y0] H0(Xn;O)=C[x0;y0;x;y].Notethattx;y]becausethemapisproperandbirational,and(isobviouslynormal.Notealsothattheexteriorpowerisasummandof,sotheconjectureextendstotensorsofexteriorpowersaswell.WedonotusethefullstrengthofConjecture7.8below,onlythevanishingpropertyforbundlesandthesurjectivitypropertyforandProposition7.9.AssumethatConjecture7.8holdsThenthecanonicalho-momorphismisanisomorphismProof.Sinceweareassumingis atoverHilb),thepullbackfunctoronsheavesisexact.Applyingtotheresolution(7.2)wegetaresolution!


!(7.4) MACDONALDPOLYNOMIALSANDGEOMETRY249Thisleavesusonlytoconsidertherstmap(7.6).Thesummandisdenedtobethekernelofthetracemap.Henceif)representsasectionin),thenisthezerosectionin).Nowforeachthereisahomomorphismofsheavesofalgebrasmapping()to().Thisissobecause,andthehomomorphismcorrespondstotheprojectiononthe-thfactor.Applyingthesehomomorphismstothesumabove,weseethat)=0innx;y];forall.By(7.7)thisimpliesthatforeach.Summingoverandusing(7.7)againwendthatThemapX;B)ismultiplicationin,whichsends)to).Modulo,thetracemapisthesameasevaluationat()=(00),againby(7.7).Hencetheimageof)in)isgivenmodulo),andsincethelatterbelongstotheproofiscomplete.Theorem7.10.AssumingtheconjectureandConjecture7.8holdtheFrobe-niusseriesofisgivenbythemasterformula(7.1)inConjecture7.5.Proof.In[Haiman1998]wederivedanAtiyah{BotttypeLefschetzformula-equivariantvectorbundleson,usingtheresolution(7.2)andexplicitlocalparametersforHilb)atthe-xedpoints.Thisformulatakestheformq;tq;tq;tq;t (7.8)Actually,thisformulawasderivedforHilbertseries,butwhenisabundlemodulesitgeneralizesimmediatelytoFrobeniusseries.Theq;t-Catalannumbersstudiedin[Haiman1998]correspondtothelinebundle(1).Ifthe!conjectureandConjecture(7.8)hold,thenbyProposition7.9,theFrobeniusseriesofisequalto,where.Moreover,usingtheresolution(7.4),weseethatConjecture7.5implies)=0for0,orequivalently,sinceisnite,)=0.ThereforetheEulercharacteristicontheleft-handsideof(7.8)reducestoq;t MACDONALDPOLYNOMIALSANDGEOMETRY251kernelisanidealHilb).Theninducesanisomorphismmx;y,underwhichtheunitcoordinatebasis;:::;ecorrespondstoabasis1;:::;v;vx;y.ItiseasytoseethatthesetwoconstructionsaremutuallyinverseandsodeneasmoothbrationHilbwithber,whereisthestabilizerofparametrizesorderedbasesofwhoserstvectorisgiven).Inparticularthisshowsthatnonsingular.Definition.isospectralcommutingvarietyisthevarietyoftuplesX;Y;suchthat(;:::;)isthejointspectrumofandinsomeorder.Inotherwords,wehavetheidentitydet((1+(8.1)wherer;sareindeterminates.Notethatifandcommutethereisasuchthatandarebothuppertriangular,byLie'stheorem.Inparticulartheyhaveajointspectrumasdenedabove,givenbythediagonalentriesofthetriangularform.Notealsothatthereisanactionof,permutingthepairs(Underthisactionwehave,sincetheinvariantsh;k)areequaltotrandsoreducetofunctionsonLetdenotetheopensubsetoflyingover.FromthedenitionoftheisospectralHilbertschemeitfollowsimmediatelythatwehave(set-theoretically)abersquare������HilbSincethebottomarrowisasmoothmorphism,soisthetoparrowinthescheme-theoreticbersquare.Hencethescheme-theoreticberproductisreducedandthereforeequaltotheset-theoreticberproduct.ThisprovesProposition8.1.TheopensetisCohen{MacaulayandhenceGorensteinifandonlyifConjecture8.2.TheisospectralcommutingvarietyisGorensteinNotethatthisimpliestheconjecturethatthecommutingvarietyisCohen{Macaulay,aswellasthe!conjecture.Weshouldpointoutherethattheidealofiscertainlygeneratedbytheidealof(conjecturally)togetherwithequations(8.1).Thisfailsevenfor=2. MACDONALDPOLYNOMIALSANDGEOMETRY253[Grothendieck1961]A.Grothendieck,\Techniquesdeconstructionettheord'existenceengeometriealgebrique,IV:LesschemasdeHilbert",pp.249{276(exposeno.221)ineminaireBourbaki1960/1961(exposes205{222),IHP,Paris,1961.ReprintedbyBenjamin,NewYork,1966,andSoc.Math.France,Paris,1995.[Haiman1994]M.D.Haiman,\Conjecturesonthequotientringbydiagonalinvari-ants",J.AlgebraicCombin.:1(1994),17{76.[Haiman1998]M.Haiman,\t;q-CatalannumbersandtheHilbertscheme",DiscreteMath.:1-3(1998),201{224.[HottaandSpringer1977]R.HottaandT.A.Springer,\AspecializationtheoremforcertainWeylgrouprepresentationsandanapplicationtotheGreenpolynomialsofunitarygroups",Invent.Math.:2(1977),113{127.[Iarrobino1972]A.Iarrobino,\Reducibilityofthefamiliesof0-dimensionalschemesonavariety",Invent.Math.(1972),72{77.[KirillovandNoumi1998]A.N.KirillovandM.Noumi,\AneHeckealgebrasandraisingoperatorsforMacdonaldpolynomials",DukeMath.J.:1(1998),1{39.[Knop1997]F.Knop,\IntegralityoftwovariableKostkafunctions",J.ReineAngew.Math.(1997),177{189.[Kraft1981]H.Kraft,\ConjugacyclassesandWeylgrouprepresentations",pp.191{205inYoungtableauxandSchurfunctionsinalgebraandgeometry(Torun,Poland,1980),Asterisque,Soc.Math.France,Paris,1981.[Macdonald1988]I.G.Macdonald,\Anewclassofsymmetricfunctions",pp.131{171ActesdueminaireLotharingien,I.R.M.A.Publ.372/S{20,Strasbourg,1988.[Macdonald1995]I.G.Macdonald,SymmetricfunctionsandHallpolynomials,2nded.,OxfordUniv.Press,1995.[MotzkinandTaussky1952]T.S.MotzkinandO.Taussky,\PairsofmatriceswithpropertyL",Trans.Amer.Math.Soc.(1952),108{114.[Nakamura1996]I.Nakamura,\Simplesingularities,McKaycorrespondence,andHilbertschemesof-orbits",preprint,HokkaidoUniversity,1996.[Reid1997]M.Reid,\McKaycorrespondence",preprint,1997.Availableathttp://xxx.lanl.gov/abs/math.AG/9702016.[Richardson1979]R.W.Richardson,\CommutingvarietiesofsemisimpleLiealgebrasandalgebraicgroups",CompositioMath.:3(1979),311{327.[Sahi1996]S.Sahi,\Interpolation,integrality,andageneralizationofMacdonald'spolynomials",Internat.Math.Res.Notices(1996),457{471.[Springer1978]T.A.Springer,\AconstructionofrepresentationsofWeylgroups",Invent.Math.:3(1978),279{293.[Tikhomirov1992]A.S.Tikhomirov,\OnHilbertschemesand agvarietiesofpointsonalgebraicsurfaces",preprint,1992.[Weyl1946]H.Weyl,Theclassicalgroups:theirinvariantsandrepresentations,2nded.,PrincetonUniversityPress,Princeton,NJ,1946. NewPerspectivesinGeometricCombinatoricsMSRIPublicationsVolume,1999EnumerationofMatchings:ProblemsandProgressJAMESPROPPDedicatedtothememoryofDavidKlarner(1940{1999)Abstract.Thisdocumentisbuiltaroundalistofthirty-twoproblemsinenumerationofmatchings,thersttwentyofwhichwerepresentedinalectureatMSRIinthefallof1996.Ibeginwithacapsulehistoryofthetopicofenumerationofmatchings.Thetwentyoriginalproblems,withcommentary,comprisethebulkofthearticle.Igiveanaccountoftheprogressthathasbeenmadeontheseproblemsasofthiswriting,andincludepointerstoboththeprintedandon-lineliterature;roughlyhalfoftheoriginaltwentyproblemsweresolvedbyparticipantsintheMSRIWorkshoponCombinatorics,theirstudents,andothers,between1996and1999.Thearticleconcludeswithadozennewopenproblems.1.IntroductionHowmanyperfectmatchingsdoesagivengraphhave?Thatis,inhowmanywayscanonechooseasubsetoftheedgesofsothateachvertexofbelongstooneandonlyonechosenedge?(SeeFigure1(a)foranexampleofaperfectmatchingofagraph.)Forgeneralgraphs,itiscomputationallyhardtoobtaintheanswer[Valiant1979],andevenwhenwehavetheanswer,itisnotsoclearthatweareanythewiserforknowingthisnumber.However,formanyinnitefamiliesofspecialgraphsthenumberofperfectmatchingsisgivenbycompellinglysimpleformulas.Overthepasttenyearsagreatmanyfamiliesofthiskindhavebeendiscovered,andwhilethereisnosingleuniedresultthatencompassesallofthem,manyofthesefamiliesresembleoneanother,bothintermsoftheformoftheresultsandintermsofthemethodsthathavebeenusefulinprovingthem. ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS257Atapproximatelythesametime,scientistsweretryingtounderstandthebehaviorofliquids.Asanextensionofamorebasicmodelforliquidscontainingonlymoleculesofonetype,FowlerandRushbrooke[1937]devisedalattice-basedmodelforliquidscontainingtwotypesofmolecules,onelargeandonesmall.Inthecasewherethelargemoleculewasroughlytwicethesizeofthesmallmolecule,itmadesensetomodelthesmallmoleculesasoccupyingsitesofathree-dimensionalgridandthelargemoleculesasoccupyingpairsofadjacentsites.Inmodernparlance,thisisamonomer-dimermodel.Inlateryears,thetwo-dimensionalversionofthemodelwasfoundtohaveapplicabilitytothestudyofmoleculesadsorbedonlms;iftheadsorptionsitesareassumedtoformalattice,andanadsorbedmoleculeisassumedtooccupytwosuchsites,thenonecanimaginectitiousmoleculesthatoccupyalltheunoccupiedsites(oneeach).MajorprogresswasmadewhenTemperleyandFisher[1961]andKasteleyn[1961]independentlyfoundwaystocountpuredimercongurationsonsub-graphsoftheinnitesquaregrid,withnomonomerspresent.Althoughthephysicalsignicanceofthisspecialcasewas(andremains)unclear,thisresult,alongwithOnsager'searlierexactsolutionofthetwo-dimensionalIsingmodel[Onsager1944],pavedthewayforotheradvancessuchasLieb'sexactsolutionofthesix-vertexmodel[Lieb1967],culminatinginaneweldattheintersec-tionofphysicsandmathematics:exactlysolvedstatisticalmechanicsmodelsintwo-dimensionallattices.(Intriguingly,virtuallynoneofthethree-andhigher-dimensionalanaloguesofthesemodelshavesuccumbedtoresearchers'eortsatobtainingexactsolutions.)Forbackgroundonlatticemodelsinstatisticalmechanics,seethebookbyBaxter[1982].Aninnitetwo-dimensionalgridhasmanynitesubgraphs;inchoosingwhichonestostudy,physicistswereguidedbytheideathattheshapeofboundaryshouldbechosensoastominimizetheeectoftheboundary|thatis,tomaximizethenumberofcongurations,atleastintheasymptoticsense.Forexample,Kasteleyn,inhisstudyofthedimermodelonthesquaregrid,countedthematchingsofthe-by-rectangle(seethedouble-productformulaatthebeginningofSection5)andofthe-by-rectangulartorus,andshowedthatthetwonumbersgrowatthesamerateasm;ngotoinnity,namelyaknownconstant.(Analytically,,whereisCatalan'sconstant 9+1 1 1 �


;numerically,isapproximately134.)Kasteleyn[1961]wrote:\Theeectofboundaryconditionsis,however,notentirelytrivialandwillbediscussedinmoredetailinasubsequentpaper."(SeethearticleofCohn,KenyonandPropp[Cohnetal.1998a]forarigorousmathematicaltreatmentofboundaryconditions.)Kasteleynneverwrotesuchafollowuppaper,butotherphysicistsdidgivesomeattentiontotheissueofboundaryshape,mostnotablyGrensing,CarlsenandZapp[Grensingetal.1980].Theseauthorsconsideredaone-parameterfamilyofgraphsofthekindshowninFigure1(a),andtheyassertedthateverygraphinthisfamilyhas2 ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS259 (a)(b)Figure2.Amatchinganditsassociatedtiling.Similarly,variantsofMacMahon'sprobleminwhichtheplanepartitionissubjectedtovarioussymmetryconstraints(consideredbyMacdonald,Stanley,andothers[Stanley1986a;1986b])correspondtotheproblemofenumeratingmatchingspossessingcorrespondingkindsofsymmetry.Kuperberg[1994]usedthiscorrespondenceinsolvingoneofStanley'sopenproblems,andthiscreatedfurtherinterestinmatchingsamongcombinatorialists.OneofKuperberg'schieftoolswasanoldresultofKasteleyn,whichshowedthatforanyplanargraph,thenumberofmatchingsofisequaltothePfaanofacertainmatrixofzerosandonesassociatedwith.Aspecialcaseofthisresult,enunciatedbyPercus[1969],canbeusedwhenisbipartite;inthiscase,onecanuseadeterminantinsteadofaPfaan.Percus'determinantisamodiedversionofthebipartiteadjacencymatrixofthegraph,inwhichrowscorrespondto\white"verticesandcolumnscorrespondto\black"vertices(underacoloringschemewherebywhiteverticeshaveonlyblackneighborsandviceversa);the(i;j)-thentryis1ifthe-thwhitevertexand-thblackvertexareadjacent,and0otherwise.Formoredetailsonhowthesignsoftheentriesarechosen,seetheexpositionsofKasteleyn[1967]andPercus[1969].Percus'theorem,incorporatedintocomputersoftware,makesiteasytocountthematchingsofmanyplanargraphsandlookforpatternsinthenumbersthatarise.Twosuchprogramsare,writtenbyGregKuperberg,DavidWilsonandmyself,and,writtenbyDavidWilson.Mostofthepat-ternsdescribedbelowwerediscoveredwiththeaidofthissoftware,whichisavailablefromhttp://math.wisc.edu/propp/software.html.Bothprogramstreatsubgraphsoftheinnitesquaregrid;thismightseemrestrictive,butitturnsoutthatcountingthematchingsofanarbitrarybipartiteplanargraphcanbetintothisframework,withabitoftweaking.Themathematicallyinterestingpartofeachprogramistheroutineforchoosingthesignsofthenonzeroentries.Therearemanychoicesthatwouldwork,butWilson'ssign-ruleisfarandawaythesimplest:Ifanedgeishorizontal,wegiveitweight+1,andifanedgeisvertical,joiningavertexinonerowtoavertexintherowbelowit,wegivethe ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS261hasturnedouttobearichavenueofcombinatorialinquiry,andmanymorebeautifulpatternsundoubtedlyawaitdiscovery.UpdatesonthestatusoftheseproblemscanbefoundontheWebathttp://math.wisc.edu/propp/update.ps.gz.2.LozengesWebeginwithproblemsrelatedtolozengetilingsofhexagons.Alozengearhombusofside-length1whoseinternalanglesmeasure60and120degrees;allthehexagonswewillconsiderwilltacitlyhaveintegerside-lengthsandin-ternalanglesof120degrees.Everysuchhexagoncanbedissectedintounitequilateraltrianglesinauniqueway,andonecanusethisdissectiontodeneagraphwhoseverticescorrespondtothetrianglesandwhoseedgescorrespondtopairsoftrianglesthatshareanedge;thisisthe\nitehoneycombgraph"dualtothedissection.Itiseasytoseethatthetilingsofbylozengesareinone-to-onecorrespondencewiththematchingsofa;b;csemiregularhexagonisthehexagonwhosesidelengthsare,incyclicalorder,a;b;c;a;b;c.Lozengetilingsofthisregionareincorrespondencewithplanepartitionswithatmostrows,atmostcolumns,andnopartexceeding.SuchhexagonsarerepresentedinVAX-formatbydiagramslikeAVAVAVAVAAVAVAVAVAVAAVAVAVAVAVAVAAVAVAVAVAVAVAVAVAVAVAVAVAVAVAVVAVAVAVAVAVAVVAVAVAVAVAVVAVAVAVAVwhereA'sandV'srepresentupward-pointinganddownward-pointingtriangles,respectively.Inthisarticlewewillusetrianglesinstead: MacMahon[1915{16]showedthatthenumberofsuchplanepartitionsis (ThisformofMacMahon'sformulaisduetoMacdonald;ashort,self-containedproofisgivenbyCohnetal.[1998b,Section2].) ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS263Problem2.Enumeratethelozengetilingsoftheregionobtainedfromthe+1,+1,+1hexagonbyremovingthecentraltriangle.Progress.MihaiCiucuhassolvedthemoregeneralproblemofcountingtherhombustilingsofan(+1,+1,+1)-hexagonwiththecentraltriangleremoved[Ciucu1998].IraGesselprovedthisresultindependentlyusingthenon-intersectinglattice-pathsmethod[HelfgottandGessel1999].SoichiOkadaandChristianKrattenthalerhavesolvedtheevenmoregeneralproblemofcountingtherhombustilingsofan(+1,+1,+1)-hexagonwiththecentraltriangleremoved[OkadaandKrattenthaler1998].Onecanalsotakea2+3,2+3,2+3hexagonandmakeitlozenge-tilablebyremovingatrianglefromthemiddleofeachofitsthreelongsides,asshown: Hereoneobtainsanequallytantalizingsequenceoffactorizations:Problem3.Enumeratethelozengetilingsoftheregionobtainedfromthe2+3,2+3,2+3hexagonbyremovingatrianglefromthemiddleofeachofitslongsides.Progress.TheresiaEisenkolblsolvedthisproblem.Whatshedoesinfactistocomputethenumberofallrhombustilingsofahexagonwithsides+3,+3,+3,whereanarbitrarytriangleisremovedfromeachofthe\long"sides ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS265oflength.FortheproofoftheirformulatheycomputeHankeldeterminantsfeaturingBernoullinumbers,whichtheydobyusingfactsaboutcontinuedfrac-tions,orthogonalpolynomials,and,inparticular,continuousHahnpolynomials.ThespecialcasesolvesthesecondpartofProblem4.ImentionedearlierthatKasteleyn'smethod,asinterpretedbyPercus,al-lowsonetowritethenumberofmatchingsofabipartiteplanargraphasthedeterminantofasignedversionofthebipartiteadjacencymatrix.Inthecaseoflozengetilingsofhexagonsandtheassociatedmatchings,itturnsoutthatthereisnoneedtomodifysignsofentries;theordinarybipartiteadjacencyma-trixwilldo.GregKuperberg[1998]hasnoticedthatwhenrow-reductionandcolumn-reductionaresystematicallyappliedtotheKasteleyn{Percusmatrixofa;b;csemiregularhexagon,onecanobtainthe-by-Carlitzmatrix[Car-litzandStanley1975]whose(i;j)-thentryis.(ThismatrixcanalsoberecognizedastheGessel{Viennotmatrixthatarisesfrominterpretingeachtilingasafamilyofnonintersectinglatticepaths[GesselandViennot1985].)Suchreductionsdonotaectthedeterminant,sowehaveapleasingwayofun-derstandingtherelationshipbetweentheKasteleyn{PercusmatrixmethodandtheGessel{Viennotlattice-pathmethod.Infact,suchreductionsdonotaectcokernelofthematrix(anabeliangroupwhoseorderisthedeterminant).Ontheotherhand,thecokerneloftheKasteleyn{Percusmatrixforthea;b;chexagonisclearlyinvariantunderpermuting,and.ThisgivesrisetothreedierentCarlitzmatricesthatnontriviallyhavethesamecokernel.Forexample,=1,thenonegetsan-by-matrixanda-by-matrixthatbothhavethesamecokernel,whosestructurecanbedetermined\byinspection"ifonenoticesthatthethirdCarlitzmatrixofthetrioisjusta1-by-1matrixwhosesoleentryis(plusorminus)abinomialcoecient.Inthisspecialcase,thecokernelisjustacyclicgroup.GregKuperbergposesthischallenge:Problem5.DeterminethecokerneloftheCarlitzmatrix,orequivalentlyoftheKasteleyn{Percusmatrixofthea;b;chexagon,andifpossiblendawaytointerpretthecokernelintermsofthetilings.ThiscombinesQuestions1and2ofKuperberg[1998].Ashepointsoutinthatarticle,inthecase=2,onegetsthenoncyclicgroupasthecokernel.Aswasremarkedabove,onenicethingabouttheKasteleyn{Percusmatricesofhoneycombgraphsisthatitisnotnecessarytomakeanyoftheentriesnega-tive.Forgeneralgraphs,however,thereisnocanonicalwayofdening,inthesensethattheremaybemanywaysofmodifyingthesignsofcertainentriesofthebipartiteadjacencymatrixofagraphsothatallnonzerocontributionstothedeterminanthavethesamesign.Thus,oneshouldnotexpecttheeigenvaluesoftopossesscombinatorialsignicance.However,thespectrumoftimesitsadjointisindependentofwhichKasteleyn{Percusmatrixonechooses(as ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS267candoasimilar(buteveneasier)calculationyourselfforthecase=1,toseethatthe\momentsofinertia"oftheverticaledge-probabilitiesaroundthehorizontalandverticalaxesare0and1,respectively.Usingtostudythecaseforlargervaluesof,Indthatthemomentofinertiaaboutthehorizontalaxisgoeslike100;:::andthemomentofinertiaabouttheverticalaxisgoeslike;::::Itiseasytoshowthattheformermomentsofinertiaaregiveningeneralbythepolynomial(6(infact,thenumberofverticallozengesthathaveanyparticular-coordinatedoesnotdependonthetilingchosen).Thelattermomentsofinertiaaresubtler;theyarenotgivenbyapolynomialofdegree4,thoughitisnoteworthythatthe-thtermisanintegerdivisibleby,atleastfortherstfewvaluesofProblem7.Findthe\momentsofinertia"forthemassonedgesarisingfromedge-probabilitiesforrandommatchingsofthea;b;choneycombgraph.3.DominoesNowletusturnfromlozenge-tilingproblemstodomino-tilingproblems.Adominoisa1-by-2or2-by-1rectangle.Althoughlozengetilings(intheguiseofconstrainedplanepartitions)werestudiedrst,itwasreallythestudyofdominotilingsinAztecdiamondsthatgavecurrentworkonenumerationofmatchingsitscurrentimpetus.HereistheAztecdiamondoforder5: Atilingofsucharegionbydominosisequivalenttoamatchingofacertain(dual)subgraphoftheinnitesquaregraph.Thisgridisbipartite,anditisconvenienttocoloritsverticesalternatelyblackandwhite;equivalently,itisconvenienttocolorthe1-by-1squaresalternatelyblackandwhite,sothateverydominocontainsone1-by-1squareofeachcolor.Elkies,Kuperberg,Larsen,andProppshowedin[Elkiesetal.1992]thatthenumberofdominotilingsofsucharegionis2+1)(where2isthenumberofrows),andGessel,Ionescu,andProppprovedin[Gesseletal.1999]anexactformula(originallyconjectured ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS269factor73mighthaveledustoexpect.(OneofthefactorsinHelfgott'sproductformulaisasingle-indexedsum;73arisesas12860+5.)Onecanalsolookat\Aztecrectangles"fromwhichsquareshavebeenre-movedsoastorestorethebalancebetweenblackandwhitesquares(anecessaryconditionfortileability).Forinstance,onecanremovethecentralsquarefrom-by-Aztecrectangleinwhichanddierby1,withthelargerofa;bodd: Problem9.Findaformulaforthenumberofdominotilingsofa2-by-(2+1)Aztecrectanglewithitscentralsquareremoved.Progress.ThishadalreadybeensolvedwhenIposedtheproblem;itisaspecialcaseofaresultofCiucu[1997,Theorem4.1].EricKuosolvedtheproblemindependently.Whatabout(21)-by-2rectangles?Fortheseregions,removingthecen-tralsquaredoesnotmaketheregiontilable.However,ifoneremovesanyoneofthefoursquaresadjacenttothemiddlesquare,oneobtainsaregionthatistilable,andmoreover,forthisregionthenumberoftilingsappearstobeaniceroundnumber.Problem10.Findaformulaforthenumberofdominotilingsofa(21)-by-Aztecrectanglewithasquareadjoiningthecentralsquareremoved.Progress.Thisproblemwassolvedindependentlythreetimes:byHaraldHelfgottandIraGessel[1999],byChristianKrattenthaler[1997],andbyEricKuo(privatecommunication).GesselandHelfgottsolveamoregeneralprob-lemthanProblem10.Krattenthaler'spreprintgivesseveralresultsconcerningtheenumerationofmatchingsofAztecrectangleswhere(asuitablenumberof)collinearverticesareremoved,ofwhichProblem10isjustaspecialcase.ThereissomeoverlapbetweentheresultsofHelfgottandGesselandtheresultsofKrattenthaler.Atthispoint,somereadersmaybewonderingwhy-by-rectangleshavenotplayedabiggerpartinthestory.Indeed,oneofthesurprisingfactsoflifeinthestudyofenumerationofmatchingsisthatAztecdiamondsandtheirkinhavebeenmuchmorefertilegroundforexactcombinatoricsthattheseeminglymorenaturalrectangles.Thereare,however,afewcasesIknowofinwhichsomethingratherniceturnsup.OneistheproblemofIraGesselthatappearsasProblem20inthisdocument.AnotheristheworkdonebyJockusch[1994]and,later,Ciucu[1997]onwhythenumberofdominotilingsofthesquareisalwayseitheraperfectsquareortwiceaperfectsquare.Inthespiritofthework ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS271ifnotalloftheirentriesnonzero.)Nonetheless,someexploratory\numerology"leavesroomforhopethatthisisdo-able.ConsidertheKasteleyn{PercusmatrixfortheAztecdiamondoforderinwhicheveryverticaldominowithitswhitesquareontop(relativetosomexedcheckerboardcoloring)hasitssigninverted|thatis,thecorresponding1inthebipartiteadjacencymatrixisreplacedbyProblem12.Showthatthesumoftheentriesofthematrixinverseof 1)(+3)+2.(Thisformulaworksfor=1through=8.)Progress.HaraldHelfgotthassolvedasimilarproblemusingthemainresultofhisthesis[1998],anditislikelythattheresultassertedinProblem12canbeprovedsimilarly.(AslighttechnicalhurdlearisesfromthefactthatHelf-gott'sthesisusesadierentsign-conventionfortheKasteleyn{Percusmatrix,whichresultsindierentsigns,andadierentsum,fortheinversematrix;how-ever,Helfgott'smethodsarequitegeneral,sothereisnoconceptualobstacletoapplyingthemtoProblem12.)Ishouldmentionthatmyoriginalreasonforexaminingthesumoftheen-triesoftheinverseKasteleyn{Percusmatrixwastoseewhethertheremightbeformulasgoverningtheindividualentriesthemselves.Helfgott'sworkprovidessuchformulas.Also,inthisconnection,GregKuperbergandDouglasZarehavesomehigh-techruminationsontheinversesofKasteleyn{Percusmatrices,andthereisachancethatrepresentation-theorymethodswillgiveadierentwayofprovingtheresult.NowweturntoaclassofregionsIcall\pillows".Herearea\0mod4"pillowof\order5"anda\2mod4"pillowof\order7": Itturnsout(empirically)thatthenumberoftilingsofthe0-mod-4pillowoforderisaperfectsquaretimesthecoecientofintheTaylorexpansionof(5+3).Thisfactcametolightinseveralsteps.Firstitwasnoticedthatthenumberoftilingshasacomparativelysmallsquare-freepart.Thenitwasnoticedthatinthederivedsequenceofsquare-freeparts,manytermswereroughlythreetimestheprecedingterm.Thenitwasnoticedthat,byjudiciouslyincludingsomeofthesquarefactors,onecouldobtainasequenceinwhicheachtermwasroughlythreetimesthepreceding ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS273Moregenerally,allthepolynomialsinthatariseinthisfashionappearto\factor"inthesenseoffunctionalcomposition.Herearethefactoredformsofthepolynomialsfor10: 2
228x2x�x2241 2x+7 8�x+3 2
2221 4+7 3+ 1442+ 18�x2241 3+ 5762+ 2304+ 1024�x+5 Ingeneraltherightmostpolynomialis(,andtheleftmostpolynomialiseitheraperfectsquare,twiceafourthpower,orhalfafourthpower,dependingmod8.Apatternforthemiddlepolynomialhoweveriselusive.Problem14.FindageneralformulaforthenumberofdominotilingsofAztecwindows.Progress.ConstantinChiscanufoundapolynomialboundonthenumberofdominotilingsoftheAztecwindowofinnerorderandouterorder[Chiscanu1997].DouglasZareusedthetransfer-matrixmethodtoshowthatthenumberoftilingsisnotjustboundedbyapolynomial,butgivenbyapolynomial,foreachxed[Zare1997{98].4.MiscellaneousNowwecometosomeproblemsinvolvingtilingthattneitherthedomino-tilingnorthelozenge-tilingframework.Herethemoregeneralpictureisthatwehavesomeperiodicdissectionoftheplanebypolygons,suchthatanevennumberofpolygonsmeetateachvertex,allowingustocolorthepolygonsalternatelyblackorwhite.Wethenmakeasuitablechoiceofaniteregioncomposedofequalnumbersofblackandwhitepolygons,andwelookatthenumberof\diform"tilingsoftheregion,whereadiformistheunionoftwopolygonalcellsthatshareanedge.Inthecaseofdominotilings,theunderlyingdissectionoftheinniteplaneisthetilingbysquares,4aroundeachvertex;inthecaseoflozengetilings,theunderlyingdissectionoftheinniteplaneisthetilingbyequilateraltriangles,6aroundeachvertex.Othersortsofperiodicdissectionshavealreadyplayedaroleinthetheoryofenumerationofmatchings.Forinstance,thereisatilingoftheplanebyisoscelesrighttrianglesassociatedwithadiscretere ectiongroupintheplane;inthiscase,therightchoiceof(seeFigure3)givesusaregionthatcanbetiledinwayswhenisevenandin5or2wayswhenisodd[Yang1991]. ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS275 Figure5.Adragonoforder10(tiled).ofhexagons[Cohnetal.1998b]anddominotilingsofAztecdiamonds[Cohnetal.1996].Onewaytogetanewdissectionoftheplanefromanoldoneistoreneit.Forinstance,startingfromthedissectionoftheplaneintosquares,onecandrawinevery-thsouthwest-to-northeastdiagonal.Whenis1,thisisjustadistortionofthedissectionoftheplaneintoequilateraltriangles.Whenis2,thisisadissectionthatleadstoniteregionsforwhichthenumberofdiformtilingsisaknownpowerof2,thankstoatheoremofChrisDouglas[1996].Butwhatabout=3andhigher?Forinstance,wehavetheroughlyhexagonalregionshowninFigure6;certainboundaryverticeshavebeenmarkedwithadotsoastobringoutthelarge-scale2hexagonalstructuremoreclearly. Figure6.AregionforProblem16. ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS277andafterthesubstitution,onecancheckthatthesumoftheweightsofthematchingsofequalsthesumoftheweightsofthematchingsofdividedbyItisrequiredthatthefourinnermostverticeshavenoneighborsotherthanthefourverticesshown;thisconstraintisindicatedbycirclingthem.Noncircledverticesmayhaveanynumberofneighbors. Figure8.RickKenyon'ssubstitution.ThesubstitutionshowninFigure8(astraightforwardgeneralizationofacleversubstitutionduetoRickKenyon)hasalsobeenofuse.Herethenewweightsarenotentirelydeterminedbytheold,buthaveasingledegreeoffree-dom;therelevantformulascanbewrittenasaegcdf b;D E;F E1 withfree.Asbefore,thecircledverticesmustnothaveanyneighborsotherthantheonesshown.Inthiscase,thesumoftheweightsinthebefore-graphisexactlyequaltothesumoftheweightsintheafter-graph;thereisnoneedforacorrectionfactorlikethe1)thatarisesinurbanrenewal.Theextremelypowerful\wye-delta"substitutionofColbourn,Provan,andVertigan[Colbournetal.1995]shouldalsobementioned.Uptillnowwehavebeendealingexclusivelywithbipartiteplanargraphs.Wenowturntothelesswell-explorednonbipartitecase.Forinstance,onecanlookatthetrianglegraphoforder,showninFigure9inthecase=4.(Hereisthenumberofverticesinthelongestrow.)Let)denotethenumberofmatchingsofthetrianglegraphoforderWhenis1or2mod4,thegraphhasanoddnumberofverticesand)is0;henceletusonlyconsiderthecasesinwhichis0or3mod4.Herearetherstfewvaluesof),expressedinfactoredform:2,23,261,229,2461,233721523657, Figure9.Thetrianglegraph. ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS279NowI'lldenewhatIcall\dimertableaux."Takean2by2rectangleandsplititintotwopartsbyapathfromthelowerleftcornertotheupperrightcorner.Forexample(with=6and=10) Thenllintheupperleftpartwithentriesfrom1,2,...,1sothatforadjacententries wehaveij1andforadjacententries wehave+1,andllinthelower-rightpartitionwithentriesfrom1:::;mwiththereverseinequalities( implies+1and implies1).Weweightanintheupper-leftpartbyandainthelower-rightpartbyTheorem1.ThesumoftheweightsofthedimercoveringsisequaltothesumoftheweightsofthedimertableauxMyproofisnotveryenlightening;itessentiallyinvolvesshowingthatbothofthesearecountedbythesameformula.Problem20.Istherean\explanation"forthisequality?Inparticular,isthereareasonablebijectiveproof?Notes:(1)Thecase=2iseasy:the210dimercoveringabovecorrespondstothe5dimertableau 2 x5 x7 y1 (there'sonlyonepossibility!).(2)Ifweset=0wheniseven(sothateverytwo-by-twosquareofthedimercoveringmaybechosenindependently),thentheequalityisequivalenttotheidentitycompare[Macdonald1995,p.37].ThisidentitycanbeprovedbyavariantofSchensted'scorrespondence,soabijectiveproofofthegeneralequalitywouldbeessentiallyageneralizationofSchensted.SeveralpeoplehavelookedattheproblemofaSchenstedgeneralizationcorrespondingtothecaseinwhich=0wheniseven.(3)Theanalogousresultsinwhichisoddareincludedinthecaseinwhichandarebotheven.Forexample,ifwetake=4andset=0,thenthefourthrowofadimercoveringmustconsistof2horizontaldominoes,whichcontribute totheweight,soweareessentiallylookingatdimercoveringswiththreerows. ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS281inthe-by-grid;thisgivesus2\mutilated"versionsofthegraph.Wecansetuprecurrencesthatlinkmatchingsofmutilatedgraphsofwidthwithmatchingsofmutilatedgraphsofwidthand1,andstandardalgebraicmethodsallowustoturnthissystemofjointmutualrecurrencesoflowdegreeintoasinglerecurrenceofhighdegreegoverningtheparticularsequenceofinterest,whichenumeratesmatchingsofunmutilatedrectangles.Therecurrenceobtainedinthiswayisnot,however,bestpossible,asonecanseeeveninthesimplecase=2.Problem23(Stanley).Proveordisprovethattheminimumdegreeofalinearrecurrencegoverningthesequence;:::is2+1)Progress.ObservationsmadebyStanley[1985,p.87]implythattheconjec-tureistruewhen+1isanoddprime.Theideaofmutilatingagraphbyremovingsomeverticesalongitsboundaryleadsustothenextproblem.Ithasbeenobservedforsmallvaluesofthatifoneremovesequalnumbersofblackandwhiteverticesfromtheboundaryof-by-2squaregrid,thenumberofmatchingsofthemutilatedgraphislessthanthenumberofmatchingsoftheoriginalgraph.Infact,itappearstobetruethatonecandeletesubsetoftheverticesofthesquaregridandobtainaninducedgraphwithstrictlyfewermatchingsthantheoriginal.Itisworthpointingoutthatnoteverygraphsharesthispropertywiththesquaregrid.Forinstance,ifistheAztecdiamondgraphoforder5andthegraphobtainedfrombydeletingthemiddleverticesalongthenorthwestandnortheastborders,thenhas32768matchingswhilehas59493.Problem24.Proveordisprovethateverysubgraphofthe2-by-2gridgraphhasstrictlyfewermatchings.NextwecometoavariantontheAztecdungeonregionshowninFigure4.Figure10showsan\hexagonaldungeon"withsides24.MattBlum'sinvestigationoftheseshapeshasledhimtodiscovermanypatterns;themoststrikingofthesepatternsformsthebasisofthenextproblem.Problem25.Showthatthehexagonaldungeonwithsidesa;b;a;a;bhasexactlydiformtilings,forallUnmatchablebipartitegraphscansometimesgiverisetointerestingquasi-matchingproblems,eitherbywayof(seeProblem6)orbysystematicadditionordeletionofverticesoredges.Theformersortofproblemsimplyasksforthedeterminantof(wherewemayassumethathasmorecolumnsthanrows).Whentheunderlyinggraphhasequalnumbersofblackandwhite ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS283 Figure11.Ahexagonwithextraedges.Progress.InthecaseofAztecrectangles,MattBlumhasfoundgeneralfor-mulasfordet()whenis1,2,or3.Forfool'sdiamonds,weget(Onemightalsolookat\fool'srectangles".)Anotherthingonecandowithanunmatchablegraphisaddextraedges.Evenwhenthisruinsthebipartitenessofthegraph,therecanstillbeinterestingcombinatorics.Forinstance,considerthe24hexagon-graph;ithasanevennumberofvertices,butithasasurplusofblackverticesoverwhitevertices.Wethereforeintroduceedgesbetweeneveryblackvertexandthesixnearestblackvertices.(Thatis,ineachhexagonofthehoneycomb,wedrawatriangleconnectingthethreeblackvertices,asinFigure11.)Thenthegraphhas5187=319matchings.Problem28.Countthematchingsofthea;b;c;d;e;fhexagon-graphinwhichextraedgeshavebeendrawnconnectingverticesofthemajoritycolor.Whatworksforhoneycombgraphsworks(orseemstowork)forsquare-gridgraphsaswell.Ifoneaddsedgesjoiningeachvertexofmajoritycolortothefournearestlike-coloredverticesinthe+2AztecrectanglegraphasinFigure12,onegetsagraphforwhichthenumberofmatchingsgrowslike27,211,231,etc.Ifonedoesthesamefortheholey21by2Aztecrectanglefromwhichthecentralvertexhasbeenremoved,asinFigure13,onegetsthenumbers27,217,231,etc. ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS285 Figure14.Anisoscelesrighttrianglegraphwithextraedges.modulo4.Herearethedatafortherstfewcases,courtesyofMattBlum:numberofmatchingsfactorization33346271065386276211459495633128073431282152217970805948013161181229925380302422460895677875631911706649125094394956776331549900237171420120191008032503971486078526402881534305952328473ThefollowingproblemdescribessomeofBlum'sconjectures:Problem30.Showthatfortheisoscelesrighttrianglegraphwithextraedges,thenumberofmatchingsisalwaysamultipleof3.Furthermore,showthattheexactpowerof2dividingthenumberofmatchingsis2whenis0modulo4,and2(=1)whenis3modulo4.Thispropertyofdivisibilityby3popsupinanotherproblemofasimilar avor.ConsiderthegraphshowninFigure15,whichisjustliketheoneshowninFigure9,exceptthathalfofthetriangularcellshaveanextravertexinthem,connectedtothethreenearestvertices.(NotealsotheresemblancetoFigure11.)Problem31.Showthatfortheequilateraltrianglegraphwithextraverticesandedges,thenumberofmatchingsisalwaysamultipleof3. ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS287verticesoredges.Mightthisinvariancenonethelesshavesomedeepersigni-cance?Cohn[1999]hasfoundanotherexampleofgratuitoussymmetryrelatedtotilings.AcknowledgementsThisresearchwasconductedwiththesupportoftheNationalScienceFoun-dation,theNationalSecurityAgency,andtheM.I.T.Classof1922CareerDe-velopmentfund.IamdeeplyindebtedtothepastandpresentmembersoftheTilingsResearchGroupfortheirmanyformsofassistance:PramodAchar,KarenAcquista,JosieAmmer,FedericoArdila,RobBlau,MattBlum,CarlBosley,RuthBritto-Pacumio,ConstantinChiscanu,HenryCohn,ChrisDou-glas,EdwardEarly,NicholasEriksson,DavidFarris,LukaszFidkowski,MarisaGioioso,DavidGupta,HaraldHelfgott,SharonHollander,DanIonescu,SameeraIyengar,JuliaKhodor,NeelakantanKrishnaswami,EricKuo,YvonneLai,ChingLaw,AndrewMenard,AlyceMoy,Anne-MarieOreskovich,BenRaphael,VisTaraz,JordanWeitz,BenWieland,LaurenWilliams,DavidWilson,JessicaWong,JasonWoolever,andLaurenceYogman.IalsoacknowledgethehelpfulcommentsonthismanuscriptgivenbyHenryCohnandRichardStanley,andtheinformationprovidedbyJerryDias,MichaelFisherandHorstSachsconcerningtheconnectionsbetweenmatchingtheoryandthephysicalsciences.ReferencesManyofthepapersinthebibliographyareincludedase-printsinthemathematicsarchive;thisisindicatedbytheinclusionofaidentier.Suche-printscancurrrentlybeaccessedacrosstheWorldWideWebbyappendingtheidentiertotheURLhttp://front.math.ucdavis.edu/orhttp://xxx.lanl.gov/abs/.[Andrews1976]G.E.Andrews,Thetheoryofpartitions,vol.2,EncyclopediaofMathematicsanditsApplications,Addison-Wesley,Reading,MA,1976.ReprintedbyCambridgeUniversityPress,NewYork,1998.[Baxter1982]R.J.Baxter,Exactlysolvedmodelsinstatisticalmechanics,AcademicPress,London,1982.Reprinted1989.[CarlitzandStanley1975]L.CarlitzandR.P.Stanley,\Branchingsandpartitions",Proc.Amer.Math.Soc.:1(1975),246{249.[Chiscanu1997]C.Chiscanu,\CommentsonPropp'sproblem14",1997.Availableathttp://math.wisc.edu/propp/chiscanu.ps.gz.[Ciucu1997]M.Ciucu,\Enumerationofperfectmatchingsingraphswithre ectivesymmetry",J.Combin.TheorySer.A:1(1997),67{97.[Ciucu1998]M.Ciucu,\Enumerationoflozengetilingsofpuncturedhexagons",Combin.TheorySer.A:2(1998),268{272. ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS289[GesselandViennot1985]I.GesselandG.Viennot,\Binomialdeterminants,paths,andhooklengthformulae",Adv.Math.(1985),300{321.[Gesseletal.1999]I.Gessel,A.Ionescu,andJ.Propp,Inpreparation.[GordonandDavison1952]M.GordonandW.H.T.Davison,\Theoryofresonancetopologyoffullyaromatichydrocarbons,I",J.ofChem.Phys.(1952),428{435.[Grensingetal.1980]D.Grensing,I.Carlsen,andH.-C.Zapp,\Someexactresultsforthedimerproblemonplanelatticeswithnon-standardboundaries",Phil.Mag.(1980),777{781.[Helfgott1998]H.Helfgott,Edgeeectsonlocalstatisticsinlatticedimers:astudyoftheAztecdiamondnitecase,Bachelor'sthesis,BrandeisUniversity,1998.Availableathttp://www.math.wisc.edu/propp/helfgott.ps.[HelfgottandGessel1999]H.HelfgottandI.Gessel,\Tilingsofdiamondsandhexwithdefects",Electron.J.Combin.(1999),R16.[Jockusch1994]W.Jockusch,\Perfectmatchingsandperfectsquares",J.Combin.TheorySer.A(1994),100{115.[JohnandSachs1985]P.JohnandH.Sachs,\WegesystemeundLinearfaktoreninhexagonalenundquadratischenSystemen",pp.85{101inGrapheninForschungundUnterricht,BarbaraFranzbeckerVerlag,BadSalzdetfurth,1985.[Kasteleyn1961]P.W.Kasteleyn,\ThestatisticsofdimersonalatticeI:Thenumberofdimerarrangementsonaquadraticlattice",Physica(1961),1209{1225.[Kasteleyn1967]P.W.Kasteleyn,\Graphtheoryandcrystalphysics",inGraphtheoryandtheoreticalphysics,editedbyF.Harary,AcademicPress,1967.[Kenyon1997]R.Kenyon,\Localstatisticsoflatticedimers",Ann.Inst.H.PoincarProbab.Statist.(1997),591{618.[Krattenthaler1997]C.Krattenthaler,\Schurfunctionidentitiesandthenum-berofperfectmatchingsofholeyAztecrectangles",1997.Availableathttp://radon.mat.univie.ac.at/People/kratt/artikel/holeyazt.htmlormath.CO/9712204.[Kuperberg1994]G.Kuperberg,\Symmetriesofplanepartitionsandthepermanent-determinantmethod",J.Combin.TheorySer.A:1(1994),115{151.[Kuperberg1998]G.Kuperberg,\Anexplorationofthepermanent-determinantmethod",Electron.J.Combin.(1998),R46.[Lieb1967]E.H.Lieb,\Residualentropyofsquareice",Phys.Rev.(1967),[LovaszandPlummer1986]L.LovaszandM.D.Plummer,Matchingtheory,Elsevier,AmsterdamandNewYork,1986.[Macdonald1995]I.G.Macdonald,SymmetricfunctionsandHallpolynomials,2nded.,OxfordUniv.Press,1995.[MacMahon1915{16]P.A.MacMahon,Combinatoryanalysis(2v.),CambridgeUniversityPress,1915{16.ReprintedbyChelsea,NewYork,1960.[Millsetal.1983]W.H.Mills,D.P.Robbins,andH.RumseyJr.,\Alternatingsignmatricesanddescendingplanepartitions",J.Combin.TheorySer.A(1983), ENUMERATIONOFMATCHINGS:PROBLEMSANDPROGRESS291[Yang1991]B.-Y.Yang,TwoenumerationproblemsabouttheAztecdiamonds,Ph.D.thesis,Mass.Inst.Tech.,1991.[Zare1997{98]D.Zare,emailfromMay20,1997andMarch2,1998.Availableathttp://math.wisc.edu/propp/zare.JamesProppDepartmentofMathematicsUniversityofWisconsinMadison,WI53706UnitedStatespropp@math.wisc.edu NewPerspectivesinGeometricCombinatoricsMSRIPublicationsVolume,1999TheGeneralizedBauesProblemVICTORREINERAbstract.WesurveythegeneralizedBauesproblemofBilleraandSturm-fels.Theproblemisoneofdiscretegeometryandtopology,andasksaboutthetopologyofthesetofsubdivisionsofacertainkindofaconvexpoly-tope.Alongwithadiscussionofmostoftheknownresults,wesurveythemotivationfortheproblemanditsrelationtotriangulations,zonotopaltilings,monotonepathsinlinearprogramming,orientedmatroidGrass-mannians,singularities,andhomotopytheory.Includedareseveralopenquestionsandproblems.1.IntroductionThegeneralizedBauesproblem,orGBPforshort,isaquestionarisingintheworkofBilleraandSturmfels[1992,p.545]onberpolytopes;seealso[Billeraetal.1994,3].Thequestionaskswhethercertainpartiallyorderedsetswhoseelementsaresubdivisionsofpolytopes,endowedwithacertaintopology[Bjorner1995],havethehomotopytypeofspheres.Casesareknown[RambauandZiegler1996]wherethisfailstobetrue,butthegeneralquestionofwhenitistrueorfalseremainsanexcitingsubjectofcurrentresearch.ThegoalofthissurveyistoreviewthemotivationforberpolytopesandtheGBP,anddiscussrecentprogressontheGBPandtheopenquestionsremain-ing.Somerecommendedsummarysourcesonthissubjectaretheintroductorychaptersinthedoctoraltheses[Rambau1996;Richter-Gebert1992],Lecture9in[Ziegler1995],andthepaper[Sturmfels1991].Thearticles[Billeraetal.1990;1993],thoughnotdiscussedinthetext,arenonethelessalsorelevanttotheGBP.BeforedivingintothegeneralsettingofberpolytopesandtheGBP,itisworthwhiletoponderthreemotivatingclassesofexamples. PartiallysupportedbySloanFoundationandUniversityofMinnesotaMcKnightLandGrantFellowships. THEGENERALIZEDBAUESPROBLEM295 Figure2.AlltriangulationsandbistellaroperationsforasetpointsinThepointsofformtheverticesoftwohomotheticandconcentricequilateraltriangles.Adaptedfrom[deLoera1995b].ofsuch ips.Thereisawell-knownbijectionbetweentriangulationsofanandnonassociativebracketingsofaproduct,andunderthisiden-ticationbistellaroperationscorrespondto\rebracketings".Fromthispointofview,thegraphoftriangulationsofan-gonanddiagonal ipswasperhapsrststudiedinthe1950'sbyTamari[1951]andlaterincollaborationwithothers[Tamari1962;FriedmanandTamari1967;HuangandTamari1972;HuguetandTamari1978].Theseauthorsdistinguishedadirectiononeachrebracketinganddenedaposetonthetriangulationshavingthesedirectededgesasitscoverrelations.TheywereabletoshowthatthisTamariposetisalattice[FriedmanandTamari1967;HuangandTamari1972].ItsHassediagramisdepictedinFigure3for=6,forachoiceofaparticularconvex6-gonwhoseverticeslieonasemicircle.Theseauthorsseemalsotohavebeenaware(withoutproof)thatthisgraphappearstobethe1-skeletonofacellular(4)-sphere,andprovedresultsabouthowits\facial"structureinteractswiththeTamarilatticestructure.Meanwhile,similarissuesofassociativityappearedintheearly1960'sinStashe'swork[1963]onhomotopyassociativity.Stashevindicatedthisapparentsphericity 296VICTORREINER Figure3.Triangulationsofaconvex6-gon:theassociahedron.byshowing(essentially)thatthesetofallpolygonalsubdivisionsofanindexesthecellsinaregularcellcomplex[Bjorner1995,(12.3)]homeomorphictothe(4)-sphere.Notethatinthiswayofthinking,adiagonal ipbistellaroperationcorrespondstoapolygonalsubdivisionwhosemaximalcellsarealltrianglesexceptforonequadrangle(containingthe ippingdiagonal),andlessrenedsubdivisionsofthe-goncorrespondtohigherdimensionalcellsinthesphere.Inanunpublishedwork(see[Kapranov1993,p.120]),Milnorproducedasetofvertexcoordinatesfortheverticesofthis(4)-spherewhichembeditastheboundarycomplexofan(3)-dimensionalpolytope.Unfortunately,theexistenceofthispolytopalembeddingseemstohavebeenunknowninthecombinatorialgeometrycommunity,andwasrediscoveredinthemid1980'safterPerlesposedtheproblemofwhetherthiscomplexwaspolytopal;see[Lee1989].Independently,Haiman[1984]andLee[1989]constructedthispolytope,whichHaimandubbedtheassociahedron.In[KapranovandVoevodsky1991;Gel'fandetal.1994]itissometimescalledtheStashepolytope.KapranovandSaito[1997]documentitsoccurrenceinothersurprisinggeometriccontexts.Theassociahedronalsomakesitsappearanceincomputerscience,wheretri-angulationsofan-gonshowupintheequivalentguiseofbinarytrees,andbistellaroperationscorrespondtoanoperationonbinarytreescalledrotationHereSleator,TarjanandThurston[Sleatoretal.1988]wereabletodetermine THEGENERALIZEDBAUESPROBLEM299 Figure6.Theincoherenttriangulationsandtheirneighbors:twoequivalentpictures.Weremarkthatinthecasewhereisthesetofverticesofaconvex-gon,everysubdivisioniscoherent,andhencethesecondarypolytope()istheassociahedronencounteredearlier.Thefactthatthesubgraphofcoherenttriangulationsandbistellaroperationsishighlyconnectedandformsthe1-skeletonofacellular(evenpolytopal)sphereraisesthefollowingbasicquestion:Question1.2.Isthegraphofalltriangulationsofandtheirbistellaropera-tionsconnectedAglanceatFigure2illustratesthateveninsmallcaseswherethereareinco-herenttriangulations,thegraphstillappearstobeconnected.WecanprovidesomemotivationfortheGeneralizedBauesProblembyperformingthefollowingmentalexercisewhilestaringatFigure2.Firstpicturetheplanarsubgraphofcoherenttriangulations,byignoringthetwoverticescorrespondingtotheinco-herenttriangulationsinFigure2(callthemand).Whenoneimaginesthisplanarsubgraphasatwo-dimensionalsphericalcellcomplex,thatistheboundaryofthethree-dimensionalsecondarypolytope(),theunionoftheneighborsofandformtheverticesofahexagonalcell,correspondingtotheunboundedregionintheplanarembedding;seeFigure6.Now\in ate"thishexagonalcellonthe2-sphereintoacubical3-dimensionalcellwiththeextratwoverticescorrespondingto.Thisgivesa3-dimensionalcellcomplexwhichisstillhomotopyequivalent(butnothomeomorphic)toa2-sphere.Roughlyspeaking,theBauesquestioninthiscontextaskswhetherthisbe-haviorisgeneral:Dotheincoherenttriangulationsandsubdivisionsofattachthemselvestothesphericalboundaryof()insuchawayastonotchangeitshomotopytype?Zonotopaltilings.ConsiderFigure7,similarto[BilleraandSturmfels1992,Figure1],depictingthetilingsofacentrallysymmetricoctagonhavingunitside THEGENERALIZEDBAUESPROBLEM301 Figure9.Thegraphoftilingsofadecagon,seeninfrominnity(left)andfromanearbypoint.Compare[Ziegler1993,Figure3].Theseoperationshavebeengivenvariousnamesintheliterature,dependinguponthecontextinwhichthetilingsarise.Inthecrystallinephysicsliterature[Destainvilleetal.1997;MosseriandBailly1993],wherethesetoftilingsisamodelforthepossiblestatesofacrystallinesolid,thesemovesarecalledelementary ipslocalizedphasons.Ratherthanconsideringtilingsofa2-gon,anequivalent(anduseful)viewpointcomesfromconsiderationofarrangementsofpseudolines(see[Bjorneretal.1993,Chapter6]fordenition,backgroundandreferences).Anarrangementofanepseudolinesintheplanelabelled;:::;ncounterclockwisegivesrisetoarhombictilingofacentrallysymmetric-gonwhichis\dual"tothelinearrangementinthesenseofplanarmaps;seeFigure10.Inthepseudolinepicture,themovedepictedinFigure8correspondstomovingonepseudolinelocallyacrossthenearbycrossingpointoftwootherpseudolines; 12233445566 Figure10.Acongurationofanepseudolinesanditsassociatedtiling. THEGENERALIZEDBAUESPROBLEM303 V Z Figure11.Acoherenttiling,inducedbyliftingintothegeneratingseg-mentsofthe-dimensionalzonotope,thenprojectingtheupperfacetsoftheresulting-dimensionalzonotopebackintotheplane.ThisraisestheanalogousquestiontoQuestion1.2:Question1.3.Isthegraphofallcubicaltilingsofazonotopeandtheircube ipsconnectedOnecanalsoviewcoherenceoftwo-dimensionaltilingsintermsofpseudolinesandstraightlines.Acoherenttilingisonewhosepseudolinearrangementisisomorphictoa(straight)linearrangementinwhicheachlinehasslopeperpen-diculartotheslopeoftheedgeinthepolygontowhichitcorresponds(thatis,totheedgeofthepolygonlabelledwiththesamenumberinFigure10).SomeofthisviewpointisexplainedintheinstructionsforthedelightfulpuzzleHexa-Grid[MRIn.d.],whichsuppliesfoamrubberversionsoftherhombictilesoccurringinFigure10,andaskstheconsumertoassemblethemintoatilingofazonotopal12-gon!Instudyingtilingsandzonotopalsubdivisionsofhigherdimensionalzono-topes,theorientedmatroidpointofviewhasbecomeindispensable;see[Bjorneretal.1993,2.2;Ziegler1996].TheBohne{DressTheorem[Bohne1992;Richter-GebertandZiegler1994]statesthatzonotopalsubdivisionsofbijectwiththesingle-elementliftingsoftherealizedorientedmatroidassociatedwiththegeneratingsegments,orusingorientedmatroidduality,tothesingle-elementextensionsofthedualorientedmatroid(see[Bjorneretal.1993,7.1]).Fromthispointofview,thesubsetofcoherentzonotopalsubdivisionsofcor-respondstothecoherentliftings[BilleraandSturmfels1992,5].Ifoneviewsrealizedorientedmatroidsandtheirliftingsintermsofsphereandpseu-dospherearrangements,thenthenotionofacoherentliftingwasexploredintheworkofBayerandBrandt[1997]ondiscriminantalarrangements,generalizing THEGENERALIZEDBAUESPROBLEM305 Figure13.Monotonepaths(thicklines)inacyclic-polytopewithvevertices.isisomorphictothe1-skeletonofthewell-knownpermutohedron[Ziegler1995,Example0.10];seealso[Milgram1966].Whathappensforotherpolytopesandfunctionals?Figure13showsthegraphof-monotonepathsina(cyclic)3-polytopewithveverticesinwhichistheconvexhullofthepointst;t,andAlthoughthisgraphisconnected,itisperhapsdisappointingthatitisnotcircularasinthecaseofFigure12.Onceagain,geometrycomestotherescueinsinglingoutawell-behavedsubsetof-monotonepaths.Saythatan-monotonepathcoherentifthereexistssomelinearfunctionalwhichinducesinthefollowingway:eachpointof(notnecessarilyavertex)isthe-maximalpointamongallthosepointsofwiththesamevalue,orinotherwords,istheunionoverallpointsofthe-maximalpointsinthebers).Withthisdenition,themonotonepathinthemiddleofthegraphinFigure13isincoherent.Toseethis,assumethereissomefunctionalinducingthismonotonepath,andidentifyasthedotproductwithsomexedvector.Thenthisvectormustpointroughlytowardthefront(thevisibleside)ofthepolytopeinordertoinducetherightportionofthepath,butalsopointtowardtheback(theinvisibleside)inordertoinducetheleftportionofthepath;contradiction.TheremainingsixpathsinFigure13areeasilyseentobecoherent(byimaginingappropriatefunctionals)andthesubgraphonthecorrespondingsixverticesisindeedcircular.Ingeneral,itfollowsasaspecialcaseofBilleraandSturmfels'berpoly-topeconstruction[BilleraandSturmfels1992,7]thatthegraphofcoherent-monotonepathsinapolytopeisthe1-skeletonofapolytopecalledthemonotonepathpolytope.Higherdimensionalfacesofthemonotonepathpoly-topecorrespondtoobjectscalledcoherentcellularstringswithrespecttocellularstringwithrespecttoisasequence(;:::;F)ofboundaryfacesofwiththefollowingproperties: THEGENERALIZEDBAUESPROBLEM3072.FiberPolytopesandtheBauesProblemThetheoryofberpolytopes[BilleraandSturmfels1992]providesacom-monframeworkinwhichtodiscusstriangulations,tilings,andmonotonepaths,andalsoacommonnotionofcoherencefortheseobjects.Theberpolytope),isapolytopenaturallyassociatedtoanylinearprojectionofpoly-topes.Letbea-dimensionalpolytopein-dimensionalpolytopeinandalinearmapwithpolytopalsubdi-visionisapolytopalcomplexwhichsubdivides.Apolytopalsubdivisioninduced(i)itisoftheformforsomespeciedcollectionoffacesofhavingall)distinct,and(ii))implies)),andinparticularItispossiblethatdierentcollectionsoffacesofprojecttothesamesub-division,sowedistinguishthesesubdivisionsbylabellingthemwiththefamily.Wepartiallyorderthe-inducedsubdivisionsofifandonlyif.Theresultingpartiallyorderedsetisdenotedby)andcalledtheBauesposet.Theminimalelementsinthisposetarethetightsubdivisions,thatisthoseforwhichand)havethesamedimensionforallWenextexplainhow-inducedsubdivisionsofgeneralizetriangulations,tilings,andmonotonepaths.Thisisperhapseasiesttoseeformonotonepathsandcellularstrings.Givenapolytopeandfunctional,letbethe1-dimensionalpolytope)in.Thenacellularstring(;:::;F)onwithrespecttogivesrisetoafamilysatisfyingthedenitionfora-inducedsub-divisionofasfollows:consistofthe'salongwiththeir-minimizingand-maximizingfaces.Tight-inducedsubdivisionsofcorrespondtomonotonepathsonFortriangulationsandtilings,thereisaconcealedprojectionofpolytopeslurkinginthebackground.Givenapointsetwithcardinality,letdenoteitsconvexhull.Thereisanaturalsurjectionfromasimplex
havingvertices,whichsendseachvertexofthesimplextooneofthepointsof.Onecanthencheckthatthe-inducedsubdivisionsofdenedabovecorrespondtothefollowingnotionofasubdivisionof,whichreplacesthenaivedenitiongiveninSection1.Asubdivisionofisacollectionofpairs,wherearesubsetsofeachistheconvexhullofandis-dimensional,theunionofthecovers,andforany;,theintersectionisaface(possiblyempty)ofeach,and THEGENERALIZEDBAUESPROBLEM309coherentsubdivisionsofapointsetandforcoherentzonotopalsubdivisionsofazonotopeLetcoh)denotetheinducedsubposetoftheBauesposetonthesetof-coherentsubdivisionsof.ThefollowingbeautifulresultofBilleraandSturmfelswhichexplainsallofourprettypolytopalpicturesisthefollowing:Theorem2.1[BilleraandSturmfels1992,Theorem3.1].Letbea-polytope-polytopealinearsurjectionThentheposetcohisthefaceposetofa-polytopeInparticularthetight-coherentsubdivisionsofcorrespondtotheverticesThe()-polytope()iscalledtheberpolytopeofthesurjectionItgeneralizesthesecondarypolytopesberzonotopes,andmonotonepathpolytopesencounteredinSection1.Astrikingfeatureof()isthatitcanalsobeconstructedasthe\Minkowskiaverage"overpoints(inawell-denedsense;see[BilleraandSturmfels1992,2])ofallofthepolytopalbers).Foranalgebro-geometricinterpretationoftheberpolytope(intermsofChowquotientsoftoricvarieties;see[Kapranovetal.1991;Hu1999].AsaconsequenceofTheorem2.1,ifoneremovesthetopelement1fromcoh),correspondingtotheimproper-coherentsubdivisiononeobtainsthefaceposetofapolytopal(1)-sphere,thatistheboundaryof().ThegeneralizedBauesproblemasksroughlyhowclosethewholeBauesposet1istopologicallytothissphere.Beforephrasingtheproblemprecisely,wemustrstgivetheposet1atopology.Thestandardwaytodothisistoconsideritsordercomplex,theabstractsimplicialcomplexofchainsintheposet[Bjorner1995,(9.3)].Fromhereon,wewillabusenotationandusethenameofanyposetalsotorefertothetopologicalspacewhichisthegeometricrealizationofitsordercomplex.WecannowstatetheGeneralizedBauesProblem,inatleasttwoforms,onestrongerthantheother.Bothoftheseformsappear,implicitlyorexplicitly,eitherintherstmentionoftheproblembyBilleraandSturmfels[1992,p.545]orinthelaterformulationof[Billeraetal.1994,3].Question2.2(WeakGBP).homotopyequivalenttoa-sphereQuestion2.3(StrongGBP).Istheinclusioncohastrongdeformationretraction THEGENERALIZEDBAUESPROBLEM311sucharegularcellcomplexdoesnotexistingeneral;relativelysmallexamplesshowthatlowerintervalsin1neednotbehomeomorphictospheres,whichisthenecessaryconditionforaposettobetheposetoffacesofaregularcellcomplex[Bjorner1995,(12.5)].OnewaytoobtainsuchanexampleistoaddaseventhpointtothepointcongurationinFigure2,inanylocationinthesameplane.Thentheuniquepropersubdivisionoffa0gwhichleavestheconvexhullofcompletelyunrenedliesatthetopofalowerintervalthatisnothomeomorphictoasphere.3.RelationstoOtherProblemsHavingstatedtheGBP,wecannowexplainhowitrelatestosomeofourpreviousquestions,andtootherproblemsindiscretegeometryandtopologicalcombinatorics.Connectivityquestions.Questions1.2,1.3,and1.4areclearlyrelatedtotheGBP,andappearatrstglancetobeweaker,inthattheyonlyaskforconnectivityofacertaingraphratherthanhomotopysphericityofacomplex.However,apositiveanswertothestrongGBPdoesnotquiteimplyapositiveanswertoeitherofthesequestions.Thereareatleasttwosubtletiesassociatedwiththisconclusion,whichwewillnowattempttomakeprecise.Foranelementofaniteposet,letitsrankbethelengthoftheshortestsaturatedchainbelowitintheposet,sothatminimalelementshaverank0.Letdenotetheelementsatrank0andrank1respectivelyintheBauesposet),andletbethegraphontheunionnBobtainedbyrestrictingtheHassediagramfor)tothisunionofitsbottomtworanks.Givenapointset,letdenoteitsgraphoftriangulationsandbistellaroperations.Similarly,forazonotope,letbeitsgraphofcubicaltilingsandcube ips,andforapolytopewithalinearfunctional,letP;fbethegraphofmonotonepathsandpolygonmoves.TherstsubtletyweencounteristherelationbetweenthegraphsP;fandthegraph.Itistemptingtosaythatthebarycentricsubdivisionof,orP;fisthesameasfortheappropriatemap,sincethevertexsetsofeachofthesebarycentricsubdivisionsformsasubsetoftheverticesoftheappropriate.However,ittakessomeworktoshowthatthesegraphscoincide.Inthecaseofpointsets,aslightgeneralizationofthepullingconstructiondescribedbyLee[1991,2]canbeusedtoshowthateverysubdivisioncanberenedtoatriangulation,soelementsofrank0intheBauesposetcoincidewiththetriangulations.Furthermore,theresultsandideasof[Santos1999]canbeusedtoshowthattheelementsofrank1coincidewiththebistellaroperations.Bothoftheseassertionsareeasywhenisingeneralposition,butotherwisebecomesubtle. THEGENERALIZEDBAUESPROBLEM313Lemma3.1.LetbeaniteposetwithatopelementandassumethathasthepropertythateverystrictprincipalorderidealeitherisconnectedemptyorconsistsoftwoincomparableposetelementsThenthegraphobtainedbyrestrictingtoitselementsatrankconnectedOfcoursetheGBPonlyimpliesthehypothesesofthislemmaaresatisedwith)forthestrictprincipalorderideal.ButthereissomehopethatifonecouldprovetheweakGBPinsomecase,thenonecanalsoprovehomotopysphericityfortherestoftheorderideals,andhencecanusethelemma.Underthegenericityassumptionswhichwerementionedabovefortriangulationsandmonotonepaths,onecancheckthattheseprincipalorderidealsareCartesianproductsofBauesposetsforsmallerpolytopes,andhencetheirconnectivityfollowsfrompositiveanswerstotheGBPforthesesmallerpolytopes.Inparticular,thepositiveanswerforthestrongGBPformonotonepaths[Billeraetal.1994](tobediscussedinthenextsection)impliesapositiveanswertoallofQuestion1.4undertheassumptionthatthefunctionalgeneric.Withoutsuchgenericityassumptions,thestructureoftheseprincipalorderidealsmaybemorecomplicated.AspecicstudyoftheseprincipalorderidealsinthecaseoftriangulationsofapointsetwasinitiatedbySantos[1999].Flipdeciency.WhilewearediscussingQuestions1.2,1.3,and1.4,itisap-propriatetomentionquestionsaboutthenumberofbistellarneighborsofatriangulation,thenumberofcube ipneighborsofatiling,andthenumberofpolygon-moveneighborsofamonotonepath.Inthegeneralsettingofeverytight-coherentsubdivisionofrepresentsavertexofthe()-polytope),andthereforewillhaveatleastneighboringtight-coherentsubdivisionslyingalongtheedgesofthepolytope.Ontheotherhand,-inducedsubdivisionswhicharenot-coherentmayhavefewerneighbors,inwhichcasewewillsaythatthesubdivisioninquestionhas ipdeciency.Ifthesubdivi-sionhasnoneighborswesaythatitisisolated,whichofcoursegivesanegativeanswertotheGBPif1inthatcase.NotethattheexamplewithallcoherentcellularstringsinFigure16showsthatformonotonepaths,wemusteitherbecarefultorestrictourselvestothecaseofagenericfunctional,orelseredenewhatismeantbya\polygon-move"intalkingabout ip-deciency.Flipdeciencyhasbeenverywell-exploredforcubicaltilingsofzonotopesintheguiseofcountingsimplicialregionsofhyperplanearrangementsormutationsinorientedmatroids;see[Richter-Gebert1992,Introduction3]foranicesum-mary.Fortriangulations, ip-deciencyhasbeenexploredonlymorerecently;see[deLoeraetal.1999;Santos1997b].Formonotonepaths,thequestionof ipdeciencyappearsnottohavebeenconsideredmuchatall.ArelatedquestionconcernsthelevelofconnectivityofthegraphsP;foftriangulationsandbistellarmoves,tilingsandcube ips,monotonepathsandpolygonmovesrespectively.Foreachofthesegraphs,theinducedsubgraph THEGENERALIZEDBAUESPROBLEM315Conjecture3.3.isarealizableorientedmatroidofrankthenishomotopyequivalenttotheGrassmannianof-planesinBabson[1993]showedthatConjecture3.3istruefor2,andintheBooleancase,thatMacP(3)ishomotopyequivalenttotheappropriateGrasmmannian.See[MnevandZiegler1993].TherelationtoextensionspacesandtheBauesproblemisthattheextensionposet)isadoublecoverof)inthesensethatthereisatwo-to-oneorder-preservingmap).Asaconsequence,onecanviewtheconjecturethat)ishomotopyequivalenttotheGrassmannianof(1)-planesin(or(1)-dimensionalrealprojectivespace)asaprojectivizedversionoftheExtensionSpaceConjecture.ThisalsoimpliesthatthepositiveresultsofSturmfelsandZiegler[1993]ontheExtensionSpaceConjecture3.2givesomespecialcasesofConjecture3.3.4.PositiveResultsInthissectionwereviewresultswhichgiveapositiveanswertotheweakorstrongGBP.Themethodsusedtendtosegregateintothethreeparadigmsdescribedbelow,wherewehaveindicatedthereferenceswhoseproofsexemplifytheseparadigms:Retraction:AproofofthestrongGBP,byexhibitinganexplicithomotopyretracting)ontocoh).See[Billeraetal.1994,Theorem2.3;RambauandZiegler1996,Theorem1.4;Athanasiadisetal.1997,Theorem1.2].Homotopies:AproofoftheweakGBPbyashortchainofhomotopyequiva-lencesfrom)tosomeposetknowntohavesphericalhomotopytype.See[Bjorner1992,Theorem2;Edelmanetal.1997,Theorem1.2].Deletion-Contraction:AninductiveproofoftheweakGBPusing(sometimesimplicitly)thenotionofdeletion-contractionfrommatroidtheory.See[Billeraetal.1994,Theorem1.2;SturmfelsandZiegler1993,Theorem1.2;EdelmanandReiner1998,Theorem3;RambauandSantos1997,Theorem1.1].Recallthegeneralset-up:weconsideralinearsurjectionofpolytopeswithP;Qbeing-dimensional,respectively,andwithhavingvertices.Wedivideourdiscussionofpositiveresultsintothefollowingcategories:=1(monotonepaths),=2(lowcodimension),=cube(zonotopaltilings),=1or=simplex(triangulations),cyclicpolytopes.Thecase:Monotonepaths.TheoriginalpaperofBillera,KapranovandSturmfelsthatposedtheGBP[Billeraetal.1994]provesboththeweakand THEGENERALIZEDBAUESPROBLEM317TheextensionspaceconjecturewasinvestigatedbySturmfelsandZiegler[1993],whoprovedmostofthestrongestpositiveresultsatpresent.TheyshowedthataninductivelydenedtechnicalhypothesiscalledstrongEuclideannesstheorientedmatroidimpliesthattheextensionspaceconjectureholds,usingtheDeletion-Contractionparadigm.TheythenshowedthatanorientedmatroidelementswithrankisstronglyEuclideanundervarioushypotheses:if3,or2,orwhenisthealternatingorientedmatroidn;rthatcomesfromacyclicarrangementofvectors[Bjorneretal.1993,9.4].Sinceorientedmatroiddualityexchangesandkeepsxed,andsincethealternatingorientedmatroidssatisfy(n;rn;n,theirresultsimplytheweakGBPwhenisa-cubeandisa-dimensionalzonotopeunderthefollowingconditions:3,or2,orisacycliczonotope.ItwasalsoshownbyBailey[1997]thatthehypothesisofstrongEuclideannessholdsforwhenistheorientedmatroidassociatedtoa-dimensionalzonotopehaving+1genericgeneratingsegments,butwitharbitrarymultiplecopiesofeachsegment.HencetheweakGBPalsoholdsfortilingsofsuchzonotopes.Weremarkthatfor=2,thecubicaltilingsofthesezonotopes(hexagons)wereenumeratedbyMacMahon[1915{16,vol.2,X]in1899.BeforeclosingourdiscussionoftheBauesproblemfortilings,wewouldliketomentionanimportantresultofSantoswhichshowsthattheGBPforzonotopaltilingsisaspecialcaseoftheGBPfortriangulations.Toanyrealizedorientedmatroidonecanassociateapolytope()knownasitsLawrencepolytope[BayerandSturmfels1990;BilleraandMunson1984;Santos1997a,Chapter4;Bjorneretal.1993,9.3],usingthetechniqueofGaletransforms.Thisconstruction,duetoJimLawrence(unpublished;see[Ziegler1995,p.183])givesanencodingofalltheinformationoftheorientedmatroidintothefacelatticeofthepolytope(),andisusefulfortransferringmatroidconstructionsandexamplesintotheworldofpolytopes.Theorem4.1[Santos1997a,Theorem4.14;Huberetal.1998].LetbeazonotopewithassociatedorientedmatroidThereisanaturalbijectionbetweenthesubdivisionsofandthezonotopalsubdivisionsof=single-elementliftingsofwhichinducesanisomorphismbetweentheassociatedBauesposetsConsequently,anegativeanswertotheGBPforzonotopaltilingsproducesanegativeanswerfortriangulations.WeremarkthattheLawrenceconstructionappliesmoregenerallytoorientedmatroidswhicharenotnecessarilyrealiz-able,yieldingamatroidpolytope)[Bjorneretal.1993,9.1]ratherthanapolytope.Santos'resultalsoappliesinthissituation,whereonedenesthetriangulationofamatroidpolytopeviahisdenitionofatriangulationofan THEGENERALIZEDBAUESPROBLEM319resultsaretight,inasense,sinceanunpublishedexampleofdeLoera,SantosandUrrutiagivesatriangulationofacongurationof8pointsinwithonepointinterior,havingonly3bistellarneighbors.Theyalsoexhibitin[deLoeraetal.1999],atriangulationof9pointsiningeneralpositionwithonpointinterior,havingonly4bistellarneighbors,andatriangulationof10pointsinconvexgeneralpositioninhavingonly4bistellarneighbors.Therearerelativelyfewfamiliesofpolytopesinhigherdimensionswhosetriangulationshavebeenwell-studied,otherthanthecyclicpolytopeswhichwillbediscussedinthenextheading.Wementionafewoftheseotherfamilieshere.Triangulationsofthe-cubewhichusefewmaximalsimplicesaredesirableforthepurposesofxedpointalgorithms[Todd1976;Ziegler1995,Problem5.10].Thereforeonewouldbeinterestedinalgorithmswhichenumeratethetriangula-tions,suchastheprogramPUNTOS[deLoera1995a],whichenumeratesallthetriangulationslyinginthesameconnectedcomponentofthegraphofbistellaroperationsasthecoherenttriangulations.Unfortunately,deLoera[deLoera1995b,Theorem2.3.20;1996]hasshownthatincoherenttriangulationsofthe-cubeexistfor4(includingsomewith ipdeciency)soitisnotknownwhetheronecanproducealltriangulationsofthecubebythismethod.Wemomentarilydigresstopointouta(perhaps)surprisingfactaboutthetriangulationsofapointsetwhichareextremalwithrespecttothenumberofmaximalsimplices|theyneednotbecoherent!SuchanexamplecomesfromworkofOhsugiandHibi[1997],andwasfurtheranalyzedbydeLoera,FirlaandZiegler;see[FirlaandZiegler1997].Thisexampleisapointcongurationhaving15pointsinlyinginconvexposition(infact,havingallcoordinates0or1),forwhichthethemaximalnumberofmaximalsimplicesinaregulartriangulationissmallerthanforanarbitrarytriangulation.Thisexamplealsohasthesamepropertyfortriangulationswiththeminimalnumberofmaximalsimplices.Cartesianproductsofsimplices
wereconjecturedtohaveonlyco-herenttriangulations(see[Ziegler1995,Problem5.3]).Thisistruewhenisequalto1,asthesecondarypolytopeinthiscaseisknowntobetheper-mutohedron[Gel'fandetal.1994,p.243].However,deLoera[1995b,Theorem2.2.17;1996]showedthatthereareincoherenttriangulationswheneverm;nandSturmfels[1996,Theorem10.15]showedthattheyexistwhen=2and5.Aclosestudyofthesecondarypolytope(
)anditsfacetswasinitiatedbyBilleraandBabson[1998],whosepointofdeparturewasthefactthatatypicalberofthemap
+1)(+1)isatransportationpolytope,i.e.,thepolytopeofnonnegative(+1)+1)matriceswithsomeprescribedrowandcolumnsums.ThePh.D.thesisofR.Hastings[1998]con-tainssomeinterestingwaystoviewarbitrarytriangulationsof
,andafewdierentwaystoviewincoherencefortriangulationsofpointsetsingeneral.Anotherinterestingfamilyofpolytopesarethe(k;nhypersimplicesk;ndenedin[GelfandandMacPherson1992]astheconvexhullofallsumsofdis- THEGENERALIZEDBAUESPROBLEM321In[Athanasiadisetal.1997],theauthorsdeterminewhentheberpolytopen;dn;d))iscanonicalineitherofthefollowingtwoways:all-inducedsubdivisionsofn;d)are-coherent(thishappensonlywhen2,or=1and=2,exceptingafewsporadiccases),ornotall-inducedsubdivisionsare-coherent,butthesubsetof-coherentsubdivisionsdoesnotdependuponthechoiceofparameters(thishappensexactlyif=1,2,and2).Theremainingresultsaboutcyclicpolytopesdealexclusivelywiththecaseoftriangulationsofn;d)andtheircombinatorics,thatis,=1.Thephilosophyherehasbeentotryandgeneralizeasmanythingsaspossiblefromthecase=2,wherethecyclicpolytope2)isaconvexpolygonasinFigure3.For=2weknowalmosteverythingaboutthetriangulationsandsubdivisions,aswasdescribedinSection1.Allthesesubdivisionsofarecoherent,sotheposetofsubdivisionsisthefaceposetofthesecondarypolytope(),the(3)-dimensionalassociahedron.The1-skeletonoftheassociahedronistheHassediagramfortheTamariposet(seeFigure3),andthisposetturnsouttobealattice(orientedsidewaysinthatgure).Incontrasttothe=2case,noteverytriangulationofn;d)iscoherentingeneral,startingwith3),4),5);see[Athanasiadisetal.1997].Itisalsoperhapsdisappointingthattriangulationsofn;d)canhave ip-deciency[RambauandSantos1997],andthehigherStashe{Tamariposetsarenotlat-ticesfor4[Edelmanetal.1997].Onthebrightside,RambauandSantos[1997]provethateventhoughtriangulationsofn;d)arenotalwayscoherent,theydoenjoythesomewhatweakerpropertyofbeingliftingtriangulations;see[Santos1997a,Denition3.4;Bjorneretal.1993,p.410].Rambau[1997b]alsoprovestheinterestingfactthattriangulationsofn;d)arealwaysshellablesimplicialcomplexes(see[Bjorner1995,11.1]forthedenitionandsignicanceofshellability).KapranovandVoevodsky[1991]suggestedageneralizationoftheTamariposetontriangulationsof2)toapartialorderontriangulationsofn;dwhichtheycalledthehigherStasheorders,andwhichwerestudiedbyEdelmanandReiner[1996]underthenameofhigherStashe-Tamariorders.Actually,thislatterpaperdenestwopossiblesuchorderswhicharerelatedtoeachother,anditisnotquiteclear(thoughpresumablytrue)thatoneoftheseordersisthesameasthatconsideredbyKapranovandVoevodsky.In[EdelmanandReiner1996]itwasprovedfor3thatthesetwopartialorderscoincideandbotharelattices,andalsothatfor5thegraphofbistellaroperationsontriangulationsofn;d)isconnected.ThislastresultwasgreatlyimprovedbyRambau[1997b],whoshowedthatthegraphisconnectedforall.Inthispaper,Rambauintroducestheimportant\sliding"ideamentionedearlier:whenoneslidesthe-thvertexonthemomentcurvedowntowardthe(1)-stver-tex,asubdivisionofn;d)inducesasubdivisionof).Thismapon THEGENERALIZEDBAUESPROBLEM323farfromsettlingtheGBP.Inparticular,Santos'constructions[Santos1997b]showthattheratioofthenumberofbistellar ipsofatriangulationoftothe\expectedlowerbound"1canapproachzero.Wesummarizethemainopencasesofthe(weak)GBPhere:Question5.1.(i)Istheposetofzonotopalsubdivisionsofa-dimensionalzonotopewithgeneratorshomotopyequivalenttoa-dimensionalsphere(ii)Istheposetofsubdivisionsofapointsethomotopyequivalenttoa-dimensionalsphereAswasmentionedearlier,theworkofSantos[1997a]showsthattherstquestionisaspecialcaseofthesecond,andthereforeacounterexamplefortherstwouldalsosettlethesecond,aswellastheExtensionSpaceConjecture3.2andConjecture3.3.6.OpenQuestions,Problems,ConjecturesThemainopenproblemsrelatedtotheGBPareQuestions1.2,1.3,5.1.Inthissection,wecollectotherproblemsandquestions,someofwhichaddressmorespecicallytheexpectedfrontierbetweenthecasesofforwhichtheGBPhaspositiveandnegativeanswer.Insomecases,wegooutonalimbbyoeringourpredictions,butwewarnthereaderthatmanyoftheseopinionsarenotbasedonverymuchdata,andareonlytheopinionofthisauthor.Webeginbyconjecturingthefrontierbetweengoodandbadbehaviorfortriangulations,inspiredbythepositiveandnegativeresultscontainedin[AzaolaandSantos1999;deLoeraetal.1999].Conjecture6.1.LetbeapointcongurationinandinconvexpositionThen(a)ThestrongGBPhaspositiveanswerforsubdivisionsofwithoutthegen-eralpositionassumptionneededin[EdelmanandReiner1998]).(b)Furthermore,thegraphoftriangulationsandbistellaroperationsis(1)-vertex-connected,soinparticulareverytriangulationofhasatleast1bistellarneighbors.(c)Ontheotherhand,thereexistsapointcongurationinconvexpositioninandalsooneinconvexpositionin,eachofwhichhasanisolatedtriangulationwhichrenesnoothersubdivision.Infact,itwouldbenicetohaveasimplerproofoftheweakGBPforevenassuminggeneralposition,orperhapsaproofofthestrongGBPviatheRetractionparadigm.Forzonotopaltilings,onewonderswhethertherearerealizableorientedma-troidsexhibitingthebehaviorofthecounterexamplesofMnevandRichter-Gebert[1993]. THEGENERALIZEDBAUESPROBLEM325knownformulacountingtriangulationsistheCatalannumber enumeratingtriangulationsofthe2).Wealsohavethefollowingmostlytrivialresults:1)has2triangulations,)has1triangulation,)has2triangulations,and)has+3triangulationsbytheresultsof[Lee1991].Santos[1998]recentlymadethefollowingconjecturefor),basedontheknowndata:Conjecture6.5.LetbethenumberoftriangulationsofThentheseconddierencehasthefollowingform(provedbySantos),HepointsoutthatthisconjectureeasilyleadstosimpleclosedformforSpecically,onewouldhave+4)where+8ifiseven,+23)13ifisodd.Aswasmentionedearlier,in[Athanasiadisetal.1998]theauthorsgiveapositiveanswertotheGBPforalloftheprojectionsbetweencyclicpolytopes.Thespecialcaseoftheprojectionn;d2),alongwiththemethodsusedin[Athanasiadisetal.1997,Theorem1.2;Reiner1998]inspirethefollowingconjecture,whichwouldalsopartlyexplaintheimportanceoftheinteriorpointpresentintheRambau{Zieglercounterexample[RambauandZiegler1996].Conjecture6.6.ThestrongGBPhaspositiveanswerforliesandallverticesofprojectundertotheboundaryofTheproofoftheweakGBPfortriangulationsofcyclicpolytopesgivenbyRam-bauandSantos[1997]usestheDeletion-Contractionparadigm.Wenextdiscusssomeotherconjecturalapproachestothisresult,involvingtherelationofcyclicpolytopestocycliczonotopesandalternatingmatroids[Bjorneretal.1993,9.4].Foranypointconguration,thedualpointcongurationGaletransformlivesin[Ziegler1995,Lecture6].Asingleelementex-tensionoftheorientedmatroidcorrespondingtogivesrisetoasubdivisioncalledaliftingsubdivision[Bjorneretal.1993,p.410],andhencegivesamapfromtheextensionposet)totheposetofsubdivisionsof.Inthe THEGENERALIZEDBAUESPROBLEM327Thefactthattheseposetshavehomotopyequivalentproperpartsalreadyfollowsfromthesphericityresultspreviouslymentioned.WhatdoesthishavetodowiththeGBP?Itiseasytoseethatforanyzonotopalsubdivisionofn;d),thesetofallcubicaltilingswhichreneitformsanintervalbothinn;d)andinn;d).Thisgivesaverynaturalorder-preservingmapfromtheBauesposetn;dtotheposetofproperintervalsn;d)orn;d).Similarly,foranysubdi-visionofn;d),thesetoftriangulationswhichreneitformsanintervalbothn;d)andinn;d),givinganorder-preservingmapfromtheBauesposetn;d))totheposetofproperintervalsinn;d)orn;dConjecture6.9.(a)Theimageofthemapfromn;dtotheposetofproperintervalsineithern;dn;disexactlythesetofnoncontractibleopenintervals.(Truefor=1,by[BjornerandWachs9].)Theimageofthemapfromn;dtotheposetofproperinter-valsineithern;dn;disexactlythesetofnoncontractibleintervals(Truefor3,by[Edelmanetal.1997,Lemma6.3].)Thepreviousconjecturewouldhavetwoniceconsequences:(i)Itwouldcompletelydescribethehomotopytypeofallintervals(andhencecomputetheobiusfunction)inbothhigherBruhatordersn;dn;dandinbothhigherStashe{Tamariordersn;dn;d).TheintervalswhicharetheimagesoftheabovemapsarealwaysisomorphictoCartesianproductsofposets)or)forsmallerval-ues,andhencebytheknownsphericityresults,arealsohomotopyspherical.(ii)Itwouldimplythatn;d))ishomotopyequivalenttothesus-pensionoftheproperpartofn;d)orn;d),andsimilarlyn;d))ishomotopyequivalenttothesuspensionoftheproperpartofn;dn;d).ThisfollowsfromthefactobservedbyWalker[Walker1988]thattheposetofproperintervalsinaboundedposetishomeomorphictothesuspensionoftheproperpartof,andthefactthattheposetofpropernon-contractibleintervalsinisadeformationretractoftheposetofallproperintervalsin[Edelmanetal.1997,Lemma6.5].Conjectures6.8and6.9tintoadiagramofconjecturalhomotopyequivalences(amongspaceswhichareallknowntobehomotopyequivalenttoan(1)-sphere)connectingtheBauesposetsfortriangulationsofn;d)andzonotopaltilingsofn;d)toeachotherandtothehigherBruhatandhigherStashe{Tamariorders: THEGENERALIZEDBAUESPROBLEM329AswasthecaseinConjecture6.9,whenevertheabovequestionhasapositiveanswer,themapinquestioninducesahomotopyequivalencebetweenandthesuspensionoftheproperpartof).TheexamplesofMnevandRichter-Gebert[1993]showthat)doesnotalwayshavesphericalhomotopytype,butitisstillpossiblethatsuchahomotopyequivalencemayexistevenincaseswheresphericityfails.OurlastquestionrelatestoStembridge'sphenomenonoccurringinthecontextofcubicaltilingsofzonotopes;see[Stembridge1994].Azonotopeacentrally-symmetricpolytope,andhencetheantipodalmapinducesanaturalinvolutiononitssetofcubicaltilings.Saythatatilingofcentrallysymmetricifitisxedbythisinvolution.Consideralsothegraphcubicaltilingsandcube ipson.Itcanbeshownthatthisgraphwillalwaysbebipartite.WesaythatthephenomenonholdsforifthenumberofcentrallysymmetrictilingsofisthesameasthedierenceincardinalityofthetwosidesofthebipartitionofStembridge[1994]observedthatknownformulascountingsymmetryclassesofplanepartitionsimpliedthe1phenomenonforzonotopalhexagonsintheplane(withmultiplecopiesofthethreelinesegmentswhichgeneratethehexagonasazonotope).FurtherexamplesinvolvingcertainzonotopaloctagonswerefoundbyElnitsky[1997]andBailey[1997].However,onecancheckthatthephenomenondoesnotholdforallzonotopes,astherearealreadyexamplesofzonotopaloctagonsforwhichitfails.Question6.12.ForwhichzonotopesdoesthephenomenonholdAcknowledgmentsTheauthorthanksLauraAnderson,ChristosAthanasiadis,LouBillera,PaulEdelman,SilvioLevy,JesusdeLoera,JorgRambau,JurgenRichter-Gebert,PacoSantos,ColinSpringer,JimStashe,GunterZiegler,andananonymousreferee.Theyprovidedmanyhelpfulconversations,insights,edits,andinsomecasesgavepermissiontousetheirguresorincludetheirconjecturesinthispaper.NoteAddedinProofTherecentpaper[Athanasiadisetal.1999]resolvesConjecture6.3arma-tivelyforsimplepolytopesandfor3-dimensionalpolytopes,butnegativelyingeneral.Specically,itisshownthatfor3,thegraphof-monotonepathsina-polytopewithrespecttoagenericfunctionalisatleast2-connected,butthereexistexamplesforeach3inwhichthegraphcontainsavertexofdegree2. THEGENERALIZEDBAUESPROBLEM331[Billeraetal.1993]L.J.Billera,I.M.Gel'fand,andB.Sturmfels,\Dualityandminorsofsecondarypolyhedra",J.Combin.TheorySer.B:2(1993),258{268.[Billeraetal.1994]L.J.Billera,M.M.Kapranov,andB.Sturmfels,\Cellularstringsonpolytopes",Proc.Amer.Math.Soc.:2(1994),549{555.[Bjorner1992]A.Bjorner,\Essentialchainsandhomotopytypeofposets",Proc.Amer.Math.Soc.:4(1992),1179{1181.[Bjorner1995]A.Bjorner,\Topologicalmethods",pp.1819{1872inHandbookofcombinatorics,v.2,editedbyR.Grahametal.,North-Holland,Amsterdam,1995.[BjornerandWachs1997]A.BjornerandM.L.Wachs,\Shellablenonpurecomplexesandposets,II",Trans.Amer.Math.Soc.:10(1997),3945{3975.[Bjorneretal.1993]A.Bjorner,M.LasVergnas,B.Sturmfels,N.White,andG.M.Ziegler,Orientedmatroids,EncyclopediaofMathematicsanditsApplicationsCambridgeUniversityPress,NewYork,1993.[Bohne1992]J.Bohne,EinekombinatorischeAnalysezonotopalerRaumaufteilungenDissertation,UniversitatBielefeld,1992.AlsoavailableasPreprint92{041,Sonder-forschungsbereich343\DiskreteStruktureninderMathematik",1992.[ConnellyandHenderson1980]R.ConnellyandD.W.Henderson,\Aconvex3-complexnotsimpliciallyisomorphictoastrictlyconvexcomplex",Math.Proc.CambridgePhilos.Soc.:2(1980),299{306.[Destainvilleetal.1997]N.Destainville,R.Mosseri,andF.Bailly,\Congurationalentropyofcodimension-onetilingsanddirectedmembranes",J.Statist.Phys.:3-4(1997),697{754.[Edelman1984]P.H.Edelman,\Apartialorderontheregionsofdissectedbyhyperplanes",Trans.Amer.Math.Soc.:2(1984),617{631.[Edelman1998]P.H.Edelman,\Orderingpointsbylinearfunctionals",preprint,1998.[EdelmanandReiner1996]P.H.EdelmanandV.Reiner,\ThehigherStashe{Tamariposets",Mathematika:1(1996),127{154.[EdelmanandReiner1998]P.H.EdelmanandV.Reiner,\VisibilitycomplexesandtheBauesproblemfortriangulationsintheplane",DiscreteComput.Geom.(1998),35{59.[EdelmanandWalker1985]P.H.EdelmanandJ.W.Walker,\Thehomotopytypeofhyperplaneposets",Proc.Amer.Math.Soc.:2(1985),221{225.[Edelmanetal.1997]P.H.Edelman,J.Rambau,andV.Reiner,\Onsubdivisionposetsofcyclicpolytopes",preprint1997-030,MSRI,1997.Seehttp://www.msri.org/publications/preprints/online/1997-030.html.[EdelsbrunnerandShah1992]H.EdelsbrunnerandN.Shah,\Incremental ippingworksforregulartriangulations",pp.43{52inProceedingsoftheEighthAACMSymposiumonComputationalGeometry(Berlin,1992),ACMPress,NewYork,1992.[Elnitsky1997]S.Elnitsky,\RhombictilingsofpolygonsandclassesofreducedwordsinCoxetergroups",J.Combin.TheorySer.A:2(1997),193{221.[FirlaandZiegler1997]R.T.FirlaandG.M.Ziegler,\Hilbertbases,unimodulartriangulations,andbinarycoversofrationalpolyhedralcones",preprint,1997.ToappearinDiscreteComput.Geom. THEGENERALIZEDBAUESPROBLEM333byP.GritzmannandB.Sturmfels,DIMACSSeriesinDiscreteMathematicsandTheoreticalComputerScience,Amer.Math.Soc.,Providence,RI,1991.[deLoera1995a]J.A.deLoera,\PUNTOS",1995.Seeftp://geom.umn.edu/priv/deloera/PUNTOS.tar.AMapleprogramforcomputingtriangulationsofpointcongurations.[deLoera1995b]J.A.deLoera,TriangulationsofpolytopesandcomputationalalgebraPh.D.thesis,CornellUniversity,Ithaca,NY,1995.[deLoera1996]J.A.deLoera,\Nonregulartriangulationsofproductsofsimplices",DiscreteComput.Geom.:3(1996),253{264.[deLoeraetal.1995]J.A.deLoera,B.Sturmfels,andR.R.Thomas,\Grobnerbasesandtriangulationsofthesecondhypersimplex",Combinatorica:3(1995),[deLoeraetal.1996]J.A.deLoera,S.Hosten,F.Santos,andB.Sturmfels,\Thepolytopeofalltriangulationsofapointconguration",Doc.Math.(1996),103{[deLoeraetal.1999]J.A.deLoera,F.Santos,andJ.Urrutia,\Thenumberofgeometricbistellarneighborsofatriangulation",DiscreteComput.Geom.(1999),131{142.[MacMahon1915{16]P.A.MacMahon,Combinatoryanalysis(2v.),CambridgeUniversityPress,1915{16.ReprintedbyChelsea,NewYork,1960.[MacPherson1993]R.D.MacPherson,\Combinatorialdierentialmanifolds:asymposiuminhonorofJohnMilnor'ssixtiethbirthday",pp.203{221inTopologicalmethodsinmodernmathematics(StonyBrook,NY,1991),editedbyL.R.GoldbergandA.Phillips,PublishorPerish,Houston,[ManinandSchechtman1989]Y.I.ManinandV.V.Schechtman,\Arrangementsofhyperplanes,higherbraidgroupsandhigherBruhatorders",pp.289{308inAlge-braicnumbertheory:inhonorofK.Iwasawa,Advancedstudiesinpuremathematics,AcademicPress,andTokyo,Kinokuniya,Boston,1989.[Milgram1966]R.J.Milgram,\Iteratedloopspaces",Ann.ofMath.(2)(1966),[MnevandRichter-Gebert1993]N.E.MnevandJ.Richter-Gebert,\Twoconstruc-tionsoforientedmatroidswithdisconnectedextensionspace",DiscreteComput.Geom.:3(1993),271{285.[MnevandZiegler1993]N.E.MnevandG.M.Ziegler,\Combinatorialmodelsforthenite-dimensionalGrassmannians",DiscreteComput.Geom.:3(1993),241{250.[MosseriandBailly1993]R.MosseriandF.Bailly,\Congurationalentropyinoctagonaltilingmodels",Internat.J.ModernPhys.B:6-7(1993),1427{1436.[MRIn.d.]Hexa-grid,atoypuzzleproducedbytheMathematicalResearchInstituteinTheNetherlands(MRI).Information:Prof.D.Siersma,Budapestlaan6,3584CDUtrecht,TheNetherlands,mri@math.ruu.nl.[Nabutovsky1996]A.Nabutovsky,\Geometryofthespaceoftriangulationsofacompactmanifold",Comm.Math.Phys.:2(1996),303{330. THEGENERALIZEDBAUESPROBLEM335[Schonhardt1928]E.Schonhardt,\UberdieZeulegungvonDreieckspolyederninTetraeder",Math.Annalen(1928),309{312.[Sleatoretal.1988]D.D.Sleator,R.E.Tarjan,andW.P.Thurston,\Rotationdistance,triangulations,andhyperbolicgeometry",J.Amer.Math.Soc.:3(1988),[Stanley1977]R.P.Stanley,\Eulerianpartitionsofaunithypercube",pp.49inHigherCombinatorics,editedbyM.Aigner,NATOAdvancedStudyInstituteSeries.Ser.C,D.Reidel,Dordrecht,Holland,1977.AppendixtoarticlebyD.Foata,\Distributionseuleriennes".[Stanley1997]R.P.Stanley,Enumerativecombinatorics,vol.1,CambridgeStudiesinAdvancedMathematics,CambridgeUniversityPress,Cambridge,1997.Cor-rectedreprintofthe1986original.[Stanley1999]R.P.Stanley,Enumerativecombinatorics,vol.2,CambridgeStudiesinAdvancedMathematics,CambridgeUniversityPress,Cambridge,1999.[StantonandWhite1986]D.StantonandD.White,ConstructivecombinatoricsUndergraduateTextsinMathematics,Springer,NewYork,1986.[Stashe1963]J.D.Stashe,\Homotopyassociativityof-spaces,I",Trans.Amer.Math.Soc.(1963),275{292.[Stembridge1994]J.R.Stembridge,\Somehiddenrelationsinvolvingthetensymme-tryclassesofplanepartitions",J.Combin.TheorySer.A:2(1994),372{409.[Sturmfels1991]B.Sturmfels,\Fiberpolytopes:abriefoverview",pp.117{124inSpecialdierentialequations,editedbyM.Yoshida,KyushuUniversity,Fukuoka,[Sturmfels1996]B.Sturmfels,Grobnerbasesandconvexpolytopes,UniversityLectureSeries,Amer.Math.Soc.,Providence,RI,1996.[SturmfelsandZiegler1993]B.SturmfelsandG.Ziegler,\Extensionspacesoforientedmatroids",DiscreteComput.Geometry(1993),23{45.[Tamari1951]D.Tamari,Monodespreordonnesetcha^nesdeMal`cev,Ph.D.thesis,Paris,1951.[Tamari1962]D.Tamari,\Thealgebraofbracketingsandtheirenumeration",Arch.Wisk.(3)(1962),131{146.[Todd1976]M.J.Todd,Thecomputationofxedpointsandapplications,LectureNotesinEconomicsandMathematicalSystems,Springer,Berlin,1976.[Tonks1997]A.Tonks,\Relatingtheassociahedronandthepermutohedron",pp.33{36inOperads:ProceedingsofRenaissanceConferences(Hartford,CTandLuminy,France,1995)),editedbyJ.-L.Lodayetal.,Contemporarymathematics,Amer.Math.Soc.,Providence,RI,1997.[Walker1988]J.W.Walker,\Canonicalhomeomorphismsofposets",EuropeanJ.Combin.:2(1988),97{107.[Ziegler1993]G.M.Ziegler,\HigherBruhatordersandcyclichyperplanearrange-ments",Topology:2(1993),259{279.[Ziegler1995]G.M.Ziegler,Lecturesonpolytopes,GraduateTextsinMathematics,Springer,NewYork,1995. NewPerspectivesinGeometricCombinatoricsMSRIPublicationsVolume,1999Littlewood{RichardsonSemigroupsANDREIZELEVINSKYAbstract.WediscusstheproblemofndinganexplicitdescriptionofthesemigroupLRoftriplesofpartitionsoflengthatmostsuchthatthecor-respondingLittlewood{Richardsoncoecientisnon-zero.Afterdiscussingthehistoryoftheproblemandpreviouslyknownresults,wesuggestanewapproachbasedonthe\polyhedral"combinatorialexpressionsfortheLittlewood{Richardsoncoecients.ThisarticleisbasedonmytalkattheworkshoponRepresentationTheoryandSymmetricFunctions,MSRI,April14,1997.Ithanktheorganizers(SergeyFomin,CurtisGreene,PhilHanlonandSheilaSundaram)forbringingtogetheragroupofoutstandingcombinatorialistsandforgivingmeachancetobringtotheirattentionsomeoftheproblemsthatIndveryexcitingandbeautiful.Inpreparingthenoteforthisvolume(October1998),Imadeafewsmallchangesintheoriginalversion[Zelevinsky1997],andaddedintheendabrief(andundoubtedlyincomplete)accountofsomeexcitingprogressachievedsinceApril1997.Iamgratefultotherefereeforhelpfulsuggestions.For1,let;:::;


bethesemigroupofpartitionsoflengthatmost.Ourmainobjectofstudywillbetheset;;;;whereistheLittlewood{Richardsoncoecient.RecallthatisthesetofhighestweightsofpolynomialirreduciblerepresentationsofGL);iftheirreduciblerepresentationofGL)withhighestweightthenisthemultiplicityof.Equivalently,thearethestructureconstantsofthealgebraofsymmetricpolynomialsinvariableswithrespecttothebasisofSchurpolynomials.WecallLRLittlewood{Richardsonsemigroupoforderthisnameisjustiedbythefollowingresult: ThisworkissupportedinpartbyNSFgrantDMS-9625511;researchatMSRIissupportedinpartbyNSFgrantDMS-9022140. LITTLEWOOD{RICHARDSONSEMIGROUPS339Theorem2.isapolyhedralconvexconeinThistheoremwasannouncedbyseveralauthors(seebelow)butapparentlytherstcompleteproofwasgivenbyA.Klyachko[1996].ProblemB.DescribeHEexplicitly.ProblemsAandBarecloselyrelatedtoeachother.Theyhavealonghistory.ProblemBwasprobablyrstposedbyI.M.Gelfandinthelate40's(eigenvaluesofthesumoftwoHermitianmatriceswerestudiedalreadybyH.Weylin1912,butIbelievethatI.M.Gelfandwastherstwhosuggestedstudyingtheconeasawholeratherthanconcentrateonindividualeigenvalues).AsolutionwasannouncedbyV.B.Lidskii[1950],butthedetailsoftheproofwereneverpublished.F.A.BerezinandI.M.Gelfand[1956]discussedtherelationshipsbetweenProblemsAandB;inparticular,theysuggestedtheremarkableequality=LR(1)wherestandsforthesetofnonnegativeintegers.A.Horn[1962]solvedProblemBfor4andconjecturedageneralanswer.Toformulatehiscon-jectureweneedsometerminology.Let[1]denotetheset;:::;r.Forasubsett;r],wedenotebythepartition)=(s;:::;iAtriple(I;J;K)ofsubsetsof[1]willbecalledHE-consistentI;J;Khavethesamecardinalityand(.Forandd;r],wewillwrite;inparticular,,;r]=jj=1+


+r.Horn'sConjecture.Let;,andbevectorsinwithweaklydecreasingcomponents.Then(;;ifandonlyifandforallHE-consistenttriples(I;J;K)ofsubsetsof[1TheproofsofHorn'sConjectureandequality(1)wereannouncedbyB.V.Lid-skii[1982];unfortunately,asinthecaseofthepaperbyV.B.Lidskii[1950]mentionedearlier,thedetailedproofsneverappeared.Wenowdiscusstheresultsin[Klyachko1996].FirsttheauthorprovesThe-orem2;moreover,hegivesthefollowingdescriptionofasetofdeninglinearinequalitiesforHE,whichisveryclose(butnottotallyequivalent)toHorn'sConjecture.ModifyingthedenitionofHE-consistenttriples,wewillcallatripleI;J;K)ofsubsetsof[1LR-consistentI;J;Khavethesamecardinalityand(Theorem3[Klyachko1996]Horn'sconjecturebecomestrueifHE-consistencyintheformulationisreplacedbyLR-consistency.Thefactthatany(;;satisestheinequalitiesforallLR-consistenttriples(I;J;K)wasprovedindependendlyin[Helmkeand LITTLEWOOD{RICHARDSONSEMIGROUPS341elegant,thisprocedureisnotveryexplicitfromcombinatorialpointofview.Thus,wewouldliketoformulatethefollowingproblem:ProblemC.Findanon-recursivedescriptionofLREquivalently,ProblemCasksforanon-recursivedescriptionofLR-consistenttriples.Wewouldliketosuggestanelementarycombinatorialapproachtothisproblembasedonthe\polyhedral"expressionsforthecoecientsgivenin[BerensteinandZelevinsky1992].Topresentsuchanexpression,itwillbeconvenienttomodifyLittlewood{Richardsoncoecientsasfollows.Wewillconsidertriples()ofdominantintegralweightsforthegroup.Letbetheirreducible-modulewithhighestweight,andletdenotethedimensionofthespaceof-invariantsinthetripletensorproduct.TherelationshipbetweentheandtheLittlewood{Richardsoncoecientsisasfollows.Wewillwriteeachoftheweightsandasanonnegativeintegerlinearcombinationoffundamentalweights;:::;!(inthestandardnumeration):(2)Thedenitionsreadilyimplythatif;;aresuchthatthen,wherethecoordinatesandin(2)aregivenby(3)Thus,theknowledgeofLRisequivalenttotheknowledgeofthesemigroup PassingfromLR hastwoimportantadvantages.First,thecoecientsaremoresymmetricthantheoriginalLittlewood{Richardsoncoecients:theyareinvariantunderthe12-elementgroupgeneratedbyallpermutationsofthreeweightsand,togetherwiththetransformationreplacingeachoftheseweightswithitsdual(i.e.,sending()to()).Second,thedimensionoftheambientspacereducesby2,from31to3(1).Ontheotherhand, hasatleastonepotentialdisadvantage:theconditionisreplacedbyamorecomplicatedconditionthat)isdivisible(inmoreinvariantterms,thismeansthatmustbearadicalweight,i.e.,belongstotherootlattice).Toillustratebothphenomena,onecancomparethedescriptionofLRgivenabovewiththefollowingdescriptionof whichisequivalenttotheclassicalClebsch{Gordanrule: consistsoftriplesofnonnegativeintegers()satisfyingthetriangleinequalityandsuchthatiseven. LITTLEWOOD{RICHARDSONSEMIGROUPS343thelinearinequalities0foralllinearformsasinCorollary6.ThissuggeststhefollowingstrategyfordeterminingthesetofLR-consistenttriples.Takeatripleofsubsets(I;J;K)in[1]ofthesamecardinality,considerthecorrespondinglinearform,writethisformasandcomputetheform.Astraightforwardcalculationgivesijk(7)where#()standsforthenumberofelementsofwhichare.TakingintoaccountTheorem3,weobtainthefollowingnewcriterionforLR-consistency.Theorem7.AtripleofsubsetsI;J;Kofthesamecardinalityy;r]isLR-consistentifandonlyifandtheformin(7)istail-positive.Inparticular,sinceeverytail-positivelinearformisobviouslyanonnegativelinearcombinationoftheijk,weobtainthefollowingnecessaryconditionforLR-consistency.Corollary8.IfatripleofsubsetsI;J;KK;r]isLR-consistentthen)+#()(8)foralli;j;kItwouldbeinterestingtodeducethiscorollarydirectlyfromtheLittlewood{Richardsonrule.Onecanshowthat(8)isnotsucientforLR-consistency.Infact,Theorem7canbeusedtoproduceothernecessaryconditionsforLR-consistency.OnecanhopetosolveProblemCbygeneratingasystemofneces-saryandsucientconditionsforLR-consistencyusingthismethod.AddedinOctober1998:SinceApril1997,importantprogresshasbeenachievedintheproblemsdiscussedabove.Hereisaverybriefandincompletediscussionofsomeofthesedevelopments.First,aniceself-containedexpositionofKlyachko'sresultswasgivenintheseminartalk[Fulton1999].Onecanalsondthereanaccountofsomenewdevelopmentsinrelatedareas,andanexpandedlistofreferences.AbeautifularmativesolutiontotheSaturationProblemhasbeenan-nouncedin[KnutsonandTao1998].Theproofisentirelycombinatorial,anditbasicallyfollowsthe\polyhedral"approachdiscussedabove.Themainnewin-gredientisageometricreformulationoftheLittlewood{Richardsonruleintermsofcertainplanarcongurationsoflinesegments(thehoneycombmodel).SeveralinterestinganaloguesandgeneralizationsofthepolyhedralconesHEandLRwereintroducedandstudiedin[Brion1998;BerensteinandSjamaar1998;AgnihotriandWoodward1997].Itwouldbeinterestingtouseageometricapproachdevelopedin[Brion1998]forasolutionofProblemCabove. LITTLEWOOD{RICHARDSONSEMIGROUPS3455Elashvili1992]A.G.Elashvili,\Invariantalgebras",pp.57{64inLiegroups,theirdiscretesubgroups,andinvarianttheory,editedbyE.B.Vinberg,AdvancesinSovietMath.,Amer.Math.Soc.,Providence,RI,1992.[Fulton1997]W.Fulton,Youngtableaux,LondonMath.Soc.StudentTextsCambridgeUniversityPress,Cambridge,1997.[Fulton1999]W.Fulton,\EigenvaluesofsumsofHermitianmatrices(afterA.Klyachko)",pp.255{269ineminaireBourbaki1997/98(exposes835{849),Astrisque,Soc.math.France,Paris,1999.[HelmkeandRosenthal1995]U.HelmkeandJ.Rosenthal,\EigenvalueinequalitiesandSchubertcalculus",Math.Nachr.(1995),207{225.[Horn1962]A.Horn,\EigenvaluesofsumsofHermitianmatrices",PacicJ.Math.(1962),225{241.[Klyachko1996]A.A.Klyachko,\Stablebundles,representationtheoryandHermitianoperators",Reportno.1,1996/97,InstitutMittag-Leer,1996.Earlierversion,UniversityofMarne-La-Vallee,1994.[KnutsonandTao1998]A.KnutsonandT.Tao,\ThehoneycombmodeloftheBerenstein-ZelevinskypolytopeI.Klyachko'ssaturationconjecture",preprint,1998.Availableathttp://xxx.lanl.gov/abs/math.RT/9807160.[Lidskii1950]V.B.Lidskii,\Onthecharacteristicnumbersofthesumandproductofsymmetricmatrices",DokladyAkad.NaukSSSR(1950),769{772.InRussian.[Lidskii1982]B.V.Lidskii,\SpectralpolyhedronofasumoftwoHermitianmatrices",Funktsional.Anal.iPrilozhen.:2(1982),76{77.InRussian;translationinFunct.Anal.Appl.(1982),139{140.[Macdonald1995]I.G.Macdonald,SymmetricfunctionsandHallpolynomials,2nded.,OxfordUniv.Press,1995.[Thompson1989]R.C.Thompson,\Divisibilityrelationssatisedbytheinvariantfactorsofamatrixproduct",pp.471{491inTheGohberganniversarycollection(Calgary,1988),vol.1,editedbyH.Dymetal.,OperatorTheory,AdvancesandApplications,Birkhauser,Basel,1989.[Zelevinsky1997]A.Zelevinsky,\Littlewood{Richardsonsemigroups",preprint1997-044,MathematicalSciencesResearchInstitute,Berkeley,1997.Availableathttp://www.msri.org/publications/preprints/online/1997-044.html.AndreiZelevinskyDepartmentofMathematicsNortheasternUniversityBoston,MA02115UnitedStatesandrei@neu.edu