Puzzles Littlewood-Richardson coefficients and horn inequalities - PDF document

Littlewood-Richardson(LR)coecientscarenon-negativeintegernumbersdependingonthreenon-negativeintegervectors,,ordereddecreasingly. Whocares? Howcanoneevaluatethem? Whatdotheycount? ArethereconditionstoseewhetherornotagivenLR-coecientisnon-zero? 3/92 Littlewood-Richardson(LR)coecientscarenon-negativeintegernumbersdependingonthreenon-negativeintegervectors,,ordereddecreasingly. Whocares? Howcanoneevaluatethem? Whatdotheycount? ArethereconditionstoseewhetherornotagivenLR-coecientisnon-zero? 5/92 Littlewood-Richardsoncoecients:c SchurfunctionsfsgformaZ-basisfortheringofsymmetricfunctionsss=Xcs: IForwhichdoessscontainsasa(positive)summand? IGiven,andwhendoesonehavec�0? ThetensorproductoftwoirreduciblepolynomialrepresentationsVandVofthegenerallineargroupGLd(C)decomposesintoirreduciblerepresentationsofGLd(C)V V=Xl()dcV: IGivenand,forwhichdoesVappear(withpositivemultiplicity)inV V?IGiven,andwhendoesonehavec�0? 7/92 Littlewood-Richardsoncoecients:c SchurfunctionsfsgformaZ-basisfortheringofsymmetricfunctionsss=Xcs: IForwhichdoessscontainsasa(positive)summand? IGiven,andwhendoesonehavec�0? ThetensorproductoftwoirreduciblepolynomialrepresentationsVandVofthegenerallineargroupGLd(C)decomposesintoirreduciblerepresentationsofGLd(C)V V=Xl()dcV: IGivenand,forwhichdoesVappear(withpositivemultiplicity)inV V?IGiven,andwhendoesonehavec�0? 9/92 Littlewood-Richardsoncoecients:c SchurfunctionsfsgformaZ-basisfortheringofsymmetricfunctionsss=Xcs: IForwhichdoessscontainsasa(positive)summand? IGiven,andwhendoesonehavec�0? ThetensorproductoftwoirreduciblepolynomialrepresentationsVandVofthegenerallineargroupGLd(C)decomposesintoirreduciblerepresentationsofGLd(C)V V=Xl()dcV: IGivenand,forwhichdoesVappear(withpositivemultiplicity)inV V?IGiven,andwhendoesonehavec�0? 11/92 Littlewood-Richardsoncoecients:c SchubertclassesformalinearbasisforH(G(d;n)),thecohomologyringoftheGrassmannianG(d;n)ofcomplexd-dimensionallinearsubspacesofCn,=Xd(n�d)c: ThereexistnnnonsingularmatricesA,BandC,overalocalprincipalidealdomain,withSmithinvariants=(1;:::;n),=(1;:::;n)and=(1;:::;n)respectively,suchthatAB=Cifandonlyifc�0. ThereexistnnHermitianmatricesA,BandC,withintegereigenvaluesarrangedinweaklydecreasingorder=(1;:::;n),=(1;:::;n)and=(1;:::;n)respectively,suchthatC=A+Bifandonlyifc;�0. 13/92 Littlewood-Richardsoncoecients:c SchubertclassesformalinearbasisforH(G(d;n)),thecohomologyringoftheGrassmannianG(d;n)ofcomplexd-dimensionallinearsubspacesofCn,=Xd(n�d)c: ThereexistnnnonsingularmatricesA,BandC,overalocalprincipalidealdomain,withSmithinvariants=(1;:::;n),=(1;:::;n)and=(1;:::;n)respectively,suchthatAB=Cifandonlyifc�0. ThereexistnnHermitianmatricesA,BandC,withintegereigenvaluesarrangedinweaklydecreasingorder=(1;:::;n),=(1;:::;n)and=(1;:::;n)respectively,suchthatC=A+Bifandonlyifc;�0. 15/92 1.SchurfunctionsPartitionsandYoungdiagrams Fixapositiveintegerr1. =(1;:::;r),with1


r�0positiveintegers,isapartitionoflengthl()=r. EachpartitionisidentiedwithaYoung(Ferrer)diagram consistingofjj=1+


+rboxesarrangedinrbottomleftadjustedrowsoflengths1


r�0. Example =(4;3;2),jj=9,l()=3= 17/92 1.SchurfunctionsPartitionsandYoungdiagrams Fixapositiveintegerr1. =(1;:::;r),with1


r�0positiveintegers,isapartitionoflengthl()=r. EachpartitionisidentiedwithaYoung(Ferrer)diagram consistingofjj=1+


+rboxesarrangedinrbottomleftadjustedrowsoflengths1


r�0. Example =(4;3;2),jj=9,l()=3= 19/92 YoungTableaux nr,=(1;:::;r),l()=r. AsemistandardtableauTofshapeisallingoftheboxesoftheFerrerdiagramwithelementsiinf1;:::;ngwhichisIweaklyincreasingacrossrowsfromlefttorightIstrictlyincreasingupcolumns Thastype=(1;:::;n)ifThasientriesequali. Example =(4;3;2),l()=3,n=6 T = 5 6 4 4 6 2 3 4 6 semistandardtableauTofshape=(4;3;2); =(0;1;1;3;1;3). 21/92 YoungTableaux nr,=(1;:::;r),l()=r. AsemistandardtableauTofshapeisallingoftheboxesoftheFerrerdiagramwithelementsiinf1;:::;ngwhichisIweaklyincreasingacrossrowsfromlefttorightIstrictlyincreasingupcolumns Thastype=(1;:::;n)ifThasientriesequali. Example =(4;3;2),l()=3,n=6 T = 5 6 4 4 6 2 3 4 6 semistandardtableauTofshape=(4;3;2); =(0;1;1;3;1;3). 23/92 YoungTableaux nr,=(1;:::;r),l()=r. AsemistandardtableauTofshapeisallingoftheboxesoftheFerrerdiagramwithelementsiinf1;:::;ngwhichisIweaklyincreasingacrossrowsfromlefttorightIstrictlyincreasingupcolumns Thastype=(1;:::;n)ifThasientriesequali. Example =(4;3;2),l()=3,n=6 T = 5 6 4 4 6 2 3 4 6 semistandardtableauTofshape=(4;3;2); =(0;1;1;3;1;3). 25/92 Schurfunctions Example n=7T= 5 6 4 4 6 2 3 4 6 x(T)=x01x2x3x34x5x36x07(T)=(0;1;1;3;1;3;0) 27/92 Schurfunctionscontinued Letx=(x1;:::;xn)beasequenceofvariables. Giventhepartition,theSchurfunction(polynomial)s(x)associatedwiththepartitionisthehomogeneouspolynomialofdegreejjonthevariablesx1:::;xns(x)=XTX(T)whereTrunsoverallsemistandardtableauxofshapeonthealphabetf1;:::;ng. Example =(2;1),jj=3 n=3 2 1 1 3 1 1 2 1 2 3 1 2 2 1 3 3 1 3 3 2 2 3 2 3 : s(x1;x2;x3)=x21x2+x21x3+x1x22+2x1x2x3+x1x23+x22x3+x2x23: d=2;s(x1;x2)=x21x2+x1x22: 29/92 Schurfunctionscontinued Letx=(x1;:::;xn)beasequenceofvariables. Giventhepartition,theSchurfunction(polynomial)s(x)associatedwiththepartitionisthehomogeneouspolynomialofdegreejjonthevariablesx1:::;xns(x)=XTX(T)whereTrunsoverallsemistandardtableauxofshapeonthealphabetf1;:::;ng. Example =(2;1),jj=3 n=3 2 1 1 3 1 1 2 1 2 3 1 2 2 1 3 3 1 3 3 2 2 3 2 3 : s(x1;x2;x3)=x21x2+x21x3+x1x22+2x1x2x3+x1x23+x22x3+x2x23: d=2;s(x1;x2)=x21x2+x1x22: 31/92 K=K,withanypermutationof. Corollary TheSchurfunctions(;x)=PweakcompositionofjjK;x;isahomogeneoussymmetricfunctioninx1;:::;xn. 33/92 ProductofSchurfunctions TheSchurfunctionssformanadditivebasisfortheringofthesymmetricfunctions. AproductofSchurfunctionssscanbeexpressedasanon-negativeintegerlinearsumofSchurfunctions:ss=Xcs:35/92 2.Littlewood-Richardsonrule=(3;1);=(2;2) 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 invalidtableaux 1 1 2 211 1 2 2 1 1221 37/92 2.Littlewood-Richardsonrule=(3;1);=(2;2) 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 invalidtableaux 1 1 2 211 1 2 2 1 1221 39/92 2.Littlewood-Richardsonrule=(3;1);=(2;2) 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 invalidtableaux 1 1 2 211 1 2 2 1 1221 41/92 2.Littlewood-Richardsonrule=(3;1);=(2;2) 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 invalidtableaux 1 1 2 211 1 2 2 1 1221 43/92 2.Littlewood-Richardsonrule=(3;1);=(2;2) 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 invalidtableaux 1 1 2 211 1 2 2 1 1221 45/92 Littlewood-Richardsonrule=(3;1);=(2;2) 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 ss=s53+s521+s431+s422+s4211+s332+s3221 47/92 Littlewood-Richardsonrule=(3;1);=(2;2) 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 ss=s53+s521+s431+s422+s4211+s332+s3221 49/92 Littlewood-Richardsonrule=(3;1);=(2;2) 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 ss=s53+s521+s431+s422+s4211+s332+s3221 51/92 Littlewood-Richardsonrule=(3;1);=(2;2) 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 ss=s53+s521+s431+s422+s4211+s332+s3221 53/92 =(3;1);=(2;1) 2 1 1 2 1 1 2 1 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 1 1 2 1 2 1 1 ss=s52+s511+s43+2s421+s4111+s331+s322+s3211 c421=2 55/92 =(3;1);=(2;1) 2 1 1 2 1 1 2 1 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 1 1 2 1 2 1 1 ss=s52+s511+s43+2s421+s4111+s331+s322+s3211 c421=2 57/92 =(3;1);=(2;1) 2 1 1 2 1 1 2 1 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 1 1 2 1 2 1 1 ss=s52+s511+s43+2s421+s4111+s331+s322+s3211 c421=2 59/92 =(3;1);=(2;1) 2 1 1 2 1 1 2 1 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 1 1 2 1 2 1 1 ss=s52+s511+s43+2s421+s4111+s331+s322+s3211 c421=2 61/92 =(3;1);=(2;1) 2 1 1 2 1 1 2 1 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 1 1 2 1 2 1 1 ss=s52+s511+s43+2s421+s4111+s331+s322+s3211 c421=2 63/92 =(3;1);=(2;1) 2 1 1 2 1 1 2 1 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 1 1 2 1 2 1 1 ss=s52+s511+s43+2s421+s4111+s331+s322+s3211 c421=2 65/92 =(3;1);=(2;1) 2 1 1 2 1 1 2 1 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 1 1 2 1 2 1 1 ss=s52+s511+s43+2s421+s4111+s331+s322+s3211 c421=2 67/92 Littlewood-Richardsonrule cisthenumberoftableauxwithshape=andcontentsatisfyingIIfonereadsthelabeledentriesinreversereadingorder,thatis,fromrighttoleftacrossrowstakeninturnfrombottomtotop,atanystage,thenumberofi'sencounteredisatleastaslargeasthenumberof(i+1)'sencountered,#10s#20s:::. 69/92 3.IntegerHives(99) Knutson-Tao(99) Ann-integerhiveisatriangulargraphmadeof(n+12)+(n2)=n2unitarytrianglesand(n+22)verticeswithnon-negativeedgelabelssatisfyingasetofconditionsgivenbylinearinequalitiescalledhiveconditionsn=5 70/92 (Edge)Hiveconditions Twodistincttypesofelementarytriangleswithnon-negativeintegeredgelabelling 71/92 (Edge)Hiveconditionscontinued Threedistincttypesofrhombiwithnon-negativeintegeredgelabelling 72/92 Knutson-TaoHives99 TheLittlewood-RichardsoncoecientscisthenumberofHiveswithboundary,and.75/92 4.Hornconjecture(62) ThereexistnnHermitianmatricesA,BandC,withintegereigenvaluesarrangedinweaklydecreasingorder=(1;:::;n),=(1;:::;n)and=(1;:::;n)respectively,suchthatC=A+Bifandonlyif,andsatisfyacertainhugesystemoflinearinequalities. 77/92 Horninequalities LetN=f1;2;:::;ng,thenforxedd,with1dn,letI=fi1;i2;:::;idgN. LetI;J;KNwith#I=#J=#K=dandordereddecreasingly.Onedenesthepartitions(I)=I�(d;:::;2;1);(J)=J�(d;:::;2;1); (K)=K�(d;:::;2;1): LetTndbethesetofalltriples(I;J;K)withI;J;KNand#I=#J=#K=dsuchthatc (K)(I);(J)�0: 79/92 Horninequalitiescontinued ;;aresaidtosatisfytheHorninequalitiesifnXk=1k=nXi=1i+nXj=1jXk2KkXi2Ii+Xj2Jjforalltriples(I;J;K)2Tndwithd=1;:::;n�1. NotallofHorn'sinequalitiesareessential.Theessentialinequalitiesarethoseforwhich(I;J;K)satisfyc (K)(I);(J)=1:81/92 WheredoHorninequalitiescomefrom? Imposeonan-hiveapuzzleofsizen.83/92 Partitionsand01-strings Fixpositiveintegers0dnandconsiderad(n�d)rectangle. d=4n=10 86/92 Puzzlerule (Knutson-Tao-Woodward)cisthenumberofpuzzleswith,andappearingas01-stringsalongtheboundary. 88/92 Example I=f1;3g;J=f1;4g;K=f2;4gisaHorntriplesinceI,J,andKspecifythepositionsofthe0'sontheboundaryofthepuzzle 90/92 Examplecontinued I=f1;3g;J=f1;4g;K=f2;4gisaHorntriple.Superimposethepuzzle,withpinkedgesspeciedbythosesetsI;J;K,onahiveofsize5andexplorethehiveconditions 91/92