On the poincare series of H*(GL,(2,2[superscript]n),Z/2)

Pub . Mat . UAB Vol . 30 N3 1 Maig 1986 ON THE POINCARE SERIES OF H*(GI,(2,2n),Z/2) G .R . Chapman INTRODUCTION . Let G be a finite group, and A a Noetherian G-ring . Then Evens [3], Venkov [11] show that H*(G,A), Che cohomology ring of G with coefficients in A, is finitely generated . The proofs are essentially non-constructive, and give little information concerning Che degrees in which Che ring generatorsoccur ; a question first raised by Johnson [6] . Explicit descriptions of product structures of cohomologyrings are not easy to obtain, a major difficultybeing to determine when a set of generators is complete . In this paper, we exhibit a circumstance in which this difficulty may be overcome, and given an example . Let p be a prime, and let Gp,(H*(G,A)]p denote a Sylow p-subgroup of G, H*(G,A) respectively . Swan [10] shows that when Gp is abelian, then [H * (G,A)] p consists of Che subring of H* (G P ,A) fixedunder the action induced by inner automorphisms of G .  Cbnsider Che case when p=2, G2 is elementary abelian, and A is Z/2, the integers mod 2 with trivial G-action . Then H*(G 2 ,Z/2) is isomorphic to R=(Z/2)[x1, . . .,xr], a polynomialring in r indeterminates where r is the rank of G 2 [7] . Consequently H * (G,Z/2) may be calculatedas a ring of invariants R H , where His a group whose order is odd, and hence coprime to the characteristic of the base field of R . In expository articles, Sloane [8] and Stanley [9] discuss .classical invariant theory, in which the base field is the complex numbers . A canonical form is given for the ring of invariants, from which a complete set of generators and relations may be derived . In section 2 we indicate how, with minor modification, these results apply to the situation described above . In section 3 we apply these results to G=GL(2,2 n ), and obtain an expression for the Poincaré Series of H*(GL(2,2n),Z/2) . The additive * structure of H (GL(2,2n),Z/2) has been described by Aguadé [1], but the knowledge of the PoincaréSeries leads, vía the canonical form for the ring of ínvariants, to a complete set of generators and relations . For n=2, the results are well known [12] . For higher values of n, the calculation becomes more complicated, and the results of a machine computationsare presented for n=3 . I would like to thank P .J . Webb for a series of enlightening correspondences, and in particular for indicated how the Brauer lift could be used to prove the version of Molien's theorem given in theorem 1 . 2 . 18 MOLIEN'S THEOREM AND A BASIS OF INVARIANTS . Let F be a field, H'a finite group, V an F(H)-module with F-basis The polynomial ring R = F[x1, . . .,xn] is {x1, . . .,xn} and character X . graded, with k-th component R k having F-basis the set of monomials of degree k in XI, . . .,x n  (k>0) .  Each hc H induces h :V + V  by h(v)  = h . v (vcV), and for each j>0 a map h j :R j } R j defined by This makes R j an F(H)-module, whose character we denote by X j . Denote by R H the subring of R invariant under this action, and let a j = dimF(RjH) . r lr n r l rn hj(xl . . .x n ) = h(x j ) . . .h(x n )  (r l + . . .+r n = j) " In the classical theory, where Fis taken to be the complex numbers (C) Molien's Theorem yíelds an explicit expression for £ ajtj, the Poincaré Series of RH .  Moreover, RH is a Cohen-Macauley ring whichmeans the Poincaré Series may be written k v i i II l (1-t ) Consequently, there exist free invariants f 1 , . . .,f . (deg f i = v i ) and transient invariants 9 11 . . .,gk (deg g i = u i )  such that k RH = 1L g,C[f1, . . .,ft] i=1 and fl, . . .,f, k are algebraically independent over C . The results sketchedhere are discussed more fully in [8], [9] . Now suppose that char(F)~ H . The fact that R H is Cohen-Macauley follows directly from [5] Propn 13 p1033, so that as in the complex case k R H ] =  LL gi F[f l , ". .,f R . i=1 Molien's theorem may be modified by first assuming that F contains IHI-th roots of unity . This may be achieved by tensoring up to a suitable field if necessary, but does not effect what follows . For h ¿H, let B(h), B(h k ) denote a Brauer lift of h, hk respectively, and B(X), B(X k ) the Brauer characters of V, Rk respectively . We have the following version of Molien's theorem . Theorem 1  If char (F)fi HI, then where c(B(h)) is the characteristic polynomial of B(h) in the indetérminate t . PROOF  Let n l " "" .n n e  C denote the eigenvalues of B(h) . Then r l r n where X j =  E  n 1 . . .nn  . r 1 + . . . +r n = j r 1 r n But {n1 . . .n n ; r 1 + ." .+ rn = j} is a set of eigenvalues for B(hk), so that X j = B(X k ) (h) .  (2) Since . char(F)j H, the orthogonality relations for Brauer characters [2] §18C are similar to chose for ordinarycharacters . In particular, it follows that 1 det(B(h)) j=Oajtj . 7 T heH  c(B(h)) [det(I - B(h)t)] -1 = i,rl(1-nit)-1 aj = ~ .  E B(X j )(h) . heH W = j E 0 X j tj  (1) Hence by (1), (2) and (3), and the theoremfollows . «0 0) ; BE GF(2 n )} cyclic group of order 2 . If 1 +C + N iH i 1 . Here H acts on G2 by E hEH 3 . THE POINCARE SERIES OF H*(GL(2,2n), Z/2) . For n>1, let GF(2 n ) denote the field of 2 n elements, and G be CL(2,2 n ), the group of 2x2 matrices with entries in GF(2 n ) . A Sylow 2-subgroup G 2 of G consísts of the matrices det(I-B(h)t) and is isomorphic to the direct product of n copies of C2, the 0  ñ 1 then H is cyclic of order 2 n-1, and if N, C denote the normalizer of G 2 in G, and centralizer - of G 2 in G respectively, we have the extension (a  0  )  ( 1  0)  ra  1  0)  -  ~1  a 2 0 ) 0  o¡- 1  0  1  0  a  0  1 and since Aut(G2) = CL(n,2), we have a monomorphism O :H i GL(ri,2) . This ís discussed more fully in [4] . n Let P n (t) ° j E0c j+l t3 be a primitive, irreducible, degree n polynomial over GF(2), and,let p be a root of P n (t) . Then H is ó 0  2n-1 generated by  ( 0  ó-1) where 6 = 11  .  Further, as a vector space over F2, G 2 has basis Since d . it follows that 0 (  o-  is the companion matrix M of Pn (t), and 0 d 1 that 0(H) = M, the group generated by M . For 1<i < n, let x i be the element of H2 (G2,Z/2) which corresponds under this isomorphism to the homomorphism which maps to 1 if j = i-1 and 0 otherwíse . l t is well known (see e.g . [7]) that H* (G2,Z/2)is,the polynomial ring R = GF(2)[x1, . . .,xnl . lt . follows from the definition of x, that M induces a transformation so that, identifying H with its image under 0, we H* (G,Z/2) as the ring of invariants RH . where B denotes the Brauer lift, and c the characteristic polynomial . To simplify this expression, we first note that if Q is a polynomial over GF(2), then Q(y2) _ [Q(y)J 2 . Hence if Q(t) is irreducible of degree d, the roots of Q(t) are of the form 2 2d-1 {y,y , . . .,y  } . Since the characteristic polynomial of M is 2 2 xi + xi+1 (l<i<n-1), n xn  i  j El  cjxj (0  ° -i )(0  1  (ó -1 °  °  (0 Turning to cohomology we note that H 1 (G 2 ,Z/2) Z Hom (G 2 ,Z/2) . By theorem 1, the Poincaré series is i+l u 1~ (Ocicn-1), (1 uj 0 1 may calculate 1 2n-2 det B(Mi)  (4) 2n-1  1=0  c B(M i ) P n (t),  ít follows that  M  is similar to diag  (u,V2, . . .,u 2n-1 ),  and n-1 that M i is similar to diag (pi,  ,u í2  ) . To simplify (4), consider the action of Z/n (the integers mod . n) on Xn = {0,1, . . .,2n-2} given by z(i) = residue of 2 zi mod . 2 n -1 (zEZ/n, ¡EX n ) . If Orb(i) denotes the orbit of i and ¡Orb(i)l = di , then di is the exponent of 2 mod . e(i), where 2 n _1 (2 n -1 ,i) is the exponent of P i in GF(2 n ) . Moreover, pi is a root of an irreducible polynomial of degree d i over GF(2) . For each din, let O d denote a set of representatives for the orbits of size d . If n is a primitive complex (2 n -1)th root of unity corresponding to V under the Brauer lift, we have j det B(M i ) = nirl ni2 = 1 j =0 n-1 j d i -1 j n/d i cB(M i ) =  n (n i2 -t) =  n (n i2 -t)  (o<i<2n-2) . j =o  J=O Thus from (4) we obtain Theorem 2 . The Poincaré Series of H*(GL(2,2n),Z/2) is 1 E (  d  ) 2 n -1 din ¡EOd d-1  j 1[ (n2 i-t) n / d j=o where nand Od are defined above . We exhibit the Poincaré Series for some low values of n . An explicit expression seems hard to obtain for arbitrary n . (i) lf n = 2, the orbits of X 2 are {0}, {1,2} and theorem 2 gives [ 1 +2 1 (n3 =1) 3 (1-t) 2 (n-t)(n2-t) as the Poincaré Series . This simplifies to 1-t + t2 24 (1-t)(1-t3) as is well known (see e .g . (12}) . (íi) If n = 3, the orbits of X3 are {0}, {1,2,4}, {3,5,6} so we obtain 1 1 +  3  +  3  } (n7=1) 7 (1-t) 3 (n-t)(n2-t)(n4-t) (n3-t)(n5-t)(n6-t) which can be written 1-2t+t 2 +t 3 +t 4 -2t 5+t6 (1_0 2 (1-t 7) (iii) For n = 4, the orbits of X 4 are {0} {5,10}, {1,2,4,8}, {3,6,9,12} and {7,11,13,14} . A lengthy calculation shows the Poincaré Series is 1-2t+t 2 -t 3 +3t 4 +t 5-t6-t 7 +t 8 - .t 9 -t 10 +t ll +3t 12-t 13 +t 14 -2t 15+t 16 (1-t)2(1-t3 )(1-t15) 4 .  The Cohomology Rings for n= 2,3 . As observed in section 2, the Poincaré Series (as given by theorem 2) may be written in the form R v n (1-t i) í=1 though not necessarilyuniquely . However, if free invariants can be found in degrees v 1 ,  ,v L , and .transient invariants in degrees v 1 , . . .,v k then we may conclude that these generate the entire ring . (i) When n = 2, (6) may be written Write x,y instead of xl,x2 . Since A=x 2 +xy+y 2 , B=xy(x+y) and  C=x 3 + x2 y + y 3 are invariantswith C 2 = A3 + B 2 +C.B, it follows that these three elementsgenerate H*(GL(2,22),Z/2) as a commutative ring of exponent 2 subject to the single relationgiven . (ii) For n = 3, the Poincaré Series may be written 1 + 2t4 +3t 5 +3t6 + 2t 7 +til Write x,y,z for x l ,x 2 ,x 3 . Searching in the ring of invariants, we find ring generators . A = x 3 + y 3 + z 3 + xz 2 + y 2 z + xy 2+xyz B 1 = x 4 + y4 + z 4 + x 2 y 2 + y 2z 2 + z 2 x2 + xyz(x+y±z), B2 = x 3 (y+z) + xyz(Y+z) + z 3 (x+y) + x2 y 2 + y a z , B3 = x 3y + x2 z 2 + xy3 + xz 3 + ya z + y 2 z2 , C 1 = x5 + y5 + z5 + xyz(xy+yz+zx) +xy4 + xz4 + y4 z , C2 = xy4,+ yz4 + zx 4 + x2Y 3 + y2z3 + z 2 x3+x2 Yz(Y+z)  , C 3 = x 4 y + y4 z + z 4 x + x3y 2 + y 3 z 2 + z 3 x2 + . xy3(x+Y) 4 2  4 2  4 2  2 4  2 4  24  3 3 3  2 2 2 D l =x y +y z +z x +xy +y z +z x +xyz(x+y+z ) +xy z , D 2 = x5y + y5 z +z 5 x + x2y 4 + y 2 z4 + z 2 x4 + xy 2(x3+Y 3 ),  ' D 3 = x 5 y+ysz + z s x + x2y 2 z2 + x s z + x4 y2 + x4yz + yzs, 1+t 3 (1-t 2 )(1-t3 xyz(x 3 y + yaz +z3 x + xy 3 + yz3 + zx 3 ) , x 6 y + y6z + z 6 x + x 5 y2 + y5z2 + z S x2 + x5y(x+y) x 6 y + y6z + z6x + x3 y4 + y3z 4 + z 3 x4 + xy 3(x3+y 3 ) .