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gener­ating functions a background reader. a a a ad hoc picture a a a AMS (MOS) classifications © a R a G R = C[x..., x a R n G. A G R F = R a a R a G R a a priori a R a Ra RF RF Q(R a N P C [n] n) n E T c S T a T =0 T = S, V © W V W, V V • • ,yj)y ,y V x a V. M: V-*V. x x m X m R x..., x R = x..., x R V. M G V) VR, x ƒ R, (Mf)(x) = x = x\ Mf(x = x V2 x X\ i jLX\Xy* G R Mf = ƒ M G c absolute invariant, G. G R, R algebra R G R:Mf f G X = G. G R R T a R x £ R* isotypical isotypic) G R. R% R = R R e T f E R,�T-fT ƒ R c R a C f B R = x(M)f A G R x relative invariant, semi-invariant, X"^ tf^y G R a ƒ G X(G). G R R A a degree G c m = V order of G to \G\ If G has degree then there exist but not m +algebraically {over Equivalently, R Krull di­ • Let G have and C by not more than ( • A R R (\/g)lL ƒ( x a R a a R R R a Let x £ Then R aIn fact, RD degrees a N-graded a B a B © * * B k C B B nth B, ƒ E B homogeneous degree n n. R = C[x..., x a R = R R R • • • , R n, ƒ Mf G R M G G c R R tff © • • •R£ R R R R B Z-graded a A U Bi&j C A A = U (R R a R A a B B A a Hubert junction A H(A, ri) = HubertPoincaré A H(A,n)\ A a = R Molien F = F(RMolien series G F = F(R G F G R 2 R H(R n\ H(R ri) (R A F Let G be a of and let G X(G). the Molien ! where x & ^e conjugate W Ha h. W% x H W. h M(EH M E G c p p M my p G R nth R. y ? ' . ,y%» n M R p%". M • +a ± S E PV---P p M= piX) g = a R£ ^0. to M m X = FR° G = M = G FcW-i 9R£, 0 Rf, E Rf E R p(0 + q(9 q C[x R © c[9 = e © C[9 0 x\ + x\ t\ R G V). 0 .. d.., d G R i\ e . . , e R S = 0 © i\ © • ®r\ F F Oj. G G F R C[0 some F R 0j G R R 0/s m 0 . . , 0 R a free C[0 B = B B • • B Krull dimension B B k. B a F(Bm = B 0 0 m a homogeneous of parameters B a k[0 0 0 0 0 0 B/(0.., 0 a k. A Noether normalization lemma, B Let B be as let..., 9 an h.s.o.p. for B. The following two B is a free module k[9 9other words, there exist t\ B (which may be chosen to be homo- For every h.s.o.p. \}/., B, B is a free k[^..., \p (and holds, then the..., v\ B satisfy if and only if their images in B/(9 — , 9 form a vector space Q A B a Cohen-Macaulay R G c V)! For G c R a R * R U, U ƒ e R $(f) = Reynolds &#xj000; = E R a R integral R R a R R R ƒ e R, P = ïï M(f)). P M(f)%R P P = 0 t - ƒ a Pj(t). ƒ R 0 R R C[9..., 9 R R R . 9 9..., 9 R. x R R a C[x., R aC[9.., R = R U R/(9 ...,0 = R 9 0 U/(9 + • • • +9 a r}.., % R/(9.., 9 % a R., 9 fj., % aU/(9 • • • +9 rjj a R 1 i / a U t + 1 i s. R = . . . , so R • 9 R a a \p \p R R a R a Let B an N-graded let 0j B positive C = k[0 0\. B/(0 for 0 i j — Moreover, a free C-module, then = m. j = 0 = 0 W a B 0B. a B B = W + 0W + 0 + • • • , B a W). / - B a k[0 B a k[0 B/(0., a j = m y B (0.., 0 B = u 0., Y Y D Y = D fu B� 0 u 0., 0 m B. • f V ¥" f V G. V a f distinct f H G a = fj R (and is a free a R/fyi* • • • Y, Zj G Y G p = f R ha,vef \f/ p. 7 G f c F G n = • D \p \p R \p g R F P - P a R G 0 0 G a + • • • + x\. x • • • + x a polynomial G a G a G = M = e x R R X x + R C[x + ®xl). F = * + = 1 -• F F - G 4 V = (x x x x x x x x x x x x x f x x f x x x X T ^ \ * + + + +(x xf4 x d a R 0 e 0 = e e • • • e a e Let o be the G o(M) = as rational functions we have = 1 ^ 1 8 g ~= D Let R • • = 0 = e • • • e Let be the least the least ƒ G R = M)~ for all Then m 2 - -i F = - \ = d • • • + rf = m + • notation, we have e if and G G M = M e • R R R a Cohen-Macaulay Let 0 an h.s.o.p.for R x £ X(G). Let . . , be R the R • • • + 0 a C-basisfor S Then obtained R = R U. R = UR a rj..., fjR/(0.., 0 SR 0 • R j 0 a R 0 R m G m X m a R C[a a a elementary symmetric function, = 'SXJXJ • • • x1 • • • R 0 a G a V) M M a pseudo- M Le/ G be a finite m independent that RC[0 6 if a be called G R G. a G G reflections a a a Let G be V) of and let r be the number of G. Then the LaurentA begins F = *r \r = 7 2 a m 1 \)~ M = I - a m — M a - - X) - p = M. P Gp = P. - P a P ~ p For G c let an h.s.o.p.for d Let R homogeneous Let g = \G\ and let r be theG. Then tg = d • drt + 2(e • • • +e = + • • • +d + If R . . with d then g = d r = / = 1 e Q Let \f/ xp homogeneous algebraically independent polynomials in R = C[x . x Let 0 in R which are polynomials in the xfr/s (so 0j E . Then there is a of m such that ^ all i E [m], equality all i if and only if = ^ fym \p • a T 9.., 9 ^ = k k am i E • R d H G G. R C[\p ip R R a \pe ^ d r G H. e d = d \G\ = \H| G = H. • Let G c GL{V) be f.g.g.r., let b the M with exactly i to one. Suppose R ...,0 with d 0 Then m 2 bji' = + « - D 1 g = 7Hd r = a R G a A % c V a reflecting I E G a% G % a C P s x( P%f. Sy{x) x % P L % % = {a E V: Lyfjx) = f T[ V. ƒ a R^% R£ a Let G c be an let x be a linear R R ƒ , i.e., M)~f Let G G be an let o be the linear by o(M) = My C[0 9 let J(0..., = (dOt/dxj), Jacobian matrix of 9 x some 0 ^ a E C we have af J(0 0 • G m X mR C[a.., a ax x transpositions a i = • m = mlr = 0 + 1 + • • • + = a M = (m^) vr m] ir(i) =j nty = M a = 0 + W + • + - m) m — a s{m group G a = = A alternating f G R, x = x IT. R% = . . , x A = discriminant. a + • • • R det(xj- = A(x a Let, 6 an h.s.o.p.for R is of Set d 0 t = d -d Then the action of G on the quotient ring S = R/{0 0 is t times the regular G. (I.e. multiplicity of x in S is equal to S xG S. H*{ri) x G n S S = H S S * g X = x S = D Let G be an fg-g.r. with R Then the action of G on R/(0., 0 is the G. D H Let G be the all m X mand let x be the G the partition of m. Then H is equal to the m) of such that n is equal to the sum of of Y for a the left of i + • G T = R/R+, R° = Rf © Rf © T R R a T x a linear G a T 1 R a Let x be a linear character of G, and define f {Though is anf.g.g.r., the sense for any G.) Then R a free R {of rankand f R£, in R% =f Q a f g a g — = 1 ^ S{g) right-hand a 2 T\ ' g w — all F group G g f a S{g) = G a F x = x y = G x b E N 0 c g R R a a xy. n\ 1 + - • + a = a co =£ =S = 23g 5 •S = (g S 2k gx g 2 = 1 2 X n C a X error-correcting C self-dual,C = {v e X: v • w = 0 w Ev • n = C r W(x,y) = C W((x + y)/V2 , - y)/V2 ) - a w E C w w =W(x,y) GG a a G R C[0 0 6 6 0 + = x _ a a = G P a a n without = + + ++ + + + + P a a X \j*i. 2 2 Pn(«)W= V = V • • • © V V G c Mx G V M G G x G V V e x G V x..., x a V R = C[x 1 i 1 j V, R = R R Ilij x$ a 1 i G R R R R R R F = 2 R 2 M V M V s n, R = 1 i 1 j group &„ [n] V a T ® s c C c A 2 = • X[y - 1 = 0 0 = P a T a = a • • • + a a a « = a a n a R£\ /� aN R B aj(x x G G [n], 0 je*.R % ^ e We have (\ - \fit\~ V QnW*" ' where Q is a • Q A distinct + R^ x T Abelian V/ E [m]. M G G x V a V I a R a x F = a F R 7 2 ô M = s RX\\ • • • 9 d R x£»] x£ 0 fa i E d g. aR a ô a = 0 ô \ = R G G 1 + ! - X - - - + ! 1 - - J + R © © x^xj). xf R a a^ViPi(^\y • • • Fr}/$ a R syzygies. y R R a y d F = \ syzygies of the first A F \ S • 9 0 a = f. F = X F = ~ XS; = S S S 0 syzygiesthe second aF F = - 2X + -Hilbert syzygy theorem s F -\ s F R F a R a Aa G R R G R x + x A S = S F = - - - F a kth a B = B B • • • a y B, d� y ,y A A = k[y ,y A A • • • = dj. A B y J = yJƒ e B. B 1 y B a B a A/1 of A, ƒ a A I I Throughout paper, A, and B = A/1 a A Ph PO 0^M� M h s M a free A a finite free B A M A M d M d. Af a d d a F{B F(B, X) = j=\ F(B, ft.j module1 i h. M M M a M a minimal a minimal free B A Bunique, M Nj ^ h = j A M Ph , , PO 0 M M 0 N N� N N 5 M d a rank) M Betti B P?(B). • • • y minimal B a fi d B, fi pf(B) y.., y problem fi BA p = A = 0 B a B B • • • h h /3^(B) homological dimension B hd B y y A m = B B = A/1 A = k[y s B s. B = s - m B Let G be a finite C[y let R A/1 as hd = s- m. D R 0 G pj (j� r X s r = Mj s = Mj_ M A. pj 0 pj a k. R a y y F a d^ dj/s F a Suppose the is generated by elements.., y of the same degree e that F(B = + pX - X by Lemma we have + p)g = e B = R Then in the of B (with y have that h = p and that the i E [p] a basis i(^+l) + • Suppose that the N-graded k-algebra B is a Cohen-Macaulay of the same degree e, and that F(B, = + pX X - X by Theorem we have + p)g = e = R the minimal free of B (with respect to y have that h =p and that the i e [p — has + l W + 1 / elements of + while M one element of + 2). U G A = C[y = y Q^A� � A-R [y\ — y\y by?\. M A 0 B) = A 8 y\ - y\y 4y A). - - R/i = h G 23 R Ji x x ^^ a + = + - R 3 M 6 M 8 M 3 += ��0-A*$A**A*2A-*R = = y y\y-. y y 0 0 0 y* y y* 0 -y% ,-y\ �-yl -ytfs -y - y^y* 0 0 y 0 0 -y -y 0 y 9 0 0 0 0 y 0 •y y 0 0 0 -y y 0 y* y 0 -y\ 0 0 0 y\ y -y* -y 0 -y\ 9 y ( -yt 0 Pi G - - F \ + + - R x i j R fl . . , fa) = fa = group G3F = + - . . . , fi - R R A R R A R R fa(R R hd s - A a ring A =... = A), A A-*A. a A u( +&#xj000; u =&#xj000; (uf)) u E e A. a a uf i»f: ty.) AA A . . , v / X q A. A f u Evf,..., * transpose a B y y A =.. ,y y y B A) = pf a BÜ Ü = A ti Ati canonical B. B = A,Q A, & a a ti A B = A/I I ti ti a B a y y A. E Ü ti A a fi B, A, type B, B. a B. 6 BT = k[0 0 B = ViTîoY E B. t T) rj* fy.&#xt000; (gf) ƒ E&#xt000; B E g EHom Q(B) a T) a B a ti(B) s T) Hom (T/T+) ® Hom T T 0 S = a t\ socle S S S ƒ B positive B = S. ti(R Let G be finite and let be the given o(M) = Then ti(R s R B a a F(ti(B) q E ti(R s R G e R C[x =x*y © xy Ü(R Q(R = E R:y) ti(R a xy a + b = 1 Q(R R xy. Ü(R = xR yR C[x\y xY). Q(R RS = R a x*y S x^ S = R = y xy A = R 0-^A A yi -y A, y\ ] y\yA-yl yxyi-yiy* 2 = R �0-A —� A\ [y\y4-y3y2y3-y4yiy2-y3y4] r yi y* [~y* y A ] y\ ~y* J ti(R s A J A (y x y, 2(R s T = a = xy x*y. T) = = 7)*x = y*x = = \*^ y*^ _ = = 0 = = = (x^)i*y nn*y = - (x = 0 = fy*y T) = Si(R ti(R Ü(R » R fc-algebra B Gorenstein =B Q(B) » B a B. MP*-i h. B M B tf = Q s-m. = p£ ptf (B) = pf i = 0 B F(B, B B a F(B, B m, F(B, l/X) = (-l) / E a B = i\ik[0.., 0 d 9 e 0 = ee • • • + = 1 / t. m l P(X) = \ B sufficient, B a B = R G a R aR F a / = m + r, m = V r G. R R R f R f R, o = Let G be a finite Let o be the = M)~ Let % % a reflecting V, and define the L R by % = {a E V: Lc^a) = Let c the order of the (cyclic) subgroup of G which fixes % pointwise, and define f J[c over all in V. (This the preceding when Then the R X 1 - XM) - XM) ' where is the G. f R which case we have R • If G c (i.e., M = for all M E G), then R G C r = 0 • If r = 0 and RG r = X = 0 g = G M a = a G 3 6 - - e R x = x = x R = x y y xyz, xy\ xY. R 7 = a y\ - - y\ - - ~ - - ~ = = P ft + fifio = = fi a G G x..., xjfr.G c x^" R R xffc) ax^~ x%-\ R R a R R a a y B, B = A/I A = k[y y ofy A m� s - pf (B). /3f (B) A s s - m B a complete a A y a B A B z a = (1?(B). a Pfi Pi P\ Po �0-M� M M a {S: S = M A). M M ft 1 ft({öi, • • a,}) = 2 iy*.., â, • • • a ^ a ftp,., = Koszul complex B z$ I. 0 ± a a B a B is a intersection and thecomplex is the free • B a (B) = = s — B, B B = B R B. S M S = 2 e e eej = Zj. d y X) V S SCZ[J3] 1 1 B a F F(B, a G 6 f e R r) = x\, y x%, y x^x^ y x il = 5 = 2 R a 0-+A R z z = y y z a = - - - - - - - a G 4 - - - y x\ x\ x A = = =� *i = = X\X y RG = ƒ z 7 - 4 = R a ~ = -Z 0 0 ^ 0 A* FG(V = - + - - + + + - - - - a s — m = B a ring) y yminimal B fi = 1 B = a unique B. B a hypersurface. A y y = m + = P(y y t� 2 P a y y y B = • • • • © © 7s + a ring G R R a H' c G c H, H H ' H. G c V) R a a I Let m = V = 2 G is a finite Then R a • a G c R a H = G n a G ail m X m R ƒ (x x E R. a a discriminant A = - Xj). R a . . . , aP(a a P. R a Suppose H = G n G c is an the index [G : H] is a prime power, then R a complete [G : H] is a prime p, then R a C[0..., 0 = f the = (as defined preceding then R © • • • 0 • V = V • • • © M (V) = MEG. MEG MEG a m = m M(V = V^y V 1 / = m). x a V M x V a monomial a A G a monomial M-*m a G % [m]. G c V) ax.. x If M E G, C a TTC = c = i / - tf = ot i / M(x = y = a \c\ - ƒ = a ƒ a = ) R R = x a. a M a a a MR R R R? = R R U R a a R„R? R R a L (X... ) » 2 a = (o a R„ a V. Let G c be a finitea V) of Then = 7 2 + + + Ö C ranges all M E G C m C C = aflW(tf/) = / / - *n = = i / = a = a R a M R a u E R M(u) = au a u = ( n E Mu = Û (t a \C t \C o Xf fi + + y + • a G = L I = M M 0 : + • • • ) , - \\ + \l - + • • • - • • • f 3 0 : - - + • • • - + - = + 2 + 2 + 2 W i = l i = l Ki xf + the Xj xfxf, x[x{ + x{x i x[x{ / permutation groups, all m X m M[m] M x^ C[x xM(x e % ƒ ~ ƒ MgMEG. patterns. ƒ g f(m) type, type a a a a, 8 GRf 8 R„ E a, Let be a Let the patterns f : N of Define Then a Fc(» l 2 S C where • a a a I Decompositions exterior and of in and relations to invariants, The theory Introduction Some results theWeyl Gleason's theoremself Groupes Lie, 5 Pólya's Theory Invariants finite generated by Finite reflection 9 Advanced Representation theory finite and associative Invariants finite groups Two Permutation statistics Problem solution, Der Modul eines Ueber die Theorie der Algebraischen Cohen-Macaulay theperfection Polynomial invariants finite groups two Fine Dualitàt und New for , Calculation the Molien function A the a Generalization On the a Theory finite Uber die Invarianten der Kombinatorische und Polynomial point The theory of An Combinatorial Groupes , Algèbre Finite 6 Error-correcting and invariant theory: New a Dimension Invariants finite , Partition A Regular elements of finite , Invariant Relative finite , Hilbert Invariants finite On , A the , Complete of the Equations Certain , Certain The Partitions of 7 Commutative Current