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

Pub . Ma . UAB Vol . 30 N3 1 Maig 1986 ON THE POINCARE SERIES OF H*(GI,(2,2n),Z/2) G .R . Chapman INTRODUCTION . Le G be a ini e g oup, and A a Noe he ian G- ing . Then E ens [3], Venko [11] show ha H*(G,A), Che cohomology ing o G wi h coe icien s in A, is ini ely gene a ed . The p oo s a e essen ially non-cons uc i e, and gi e li le in o ma ion conce ning Che deg ees in which Che ing gene a o soccu ; a ques ion i s aised by Johnson [6] . Explici desc ip ions o p oduc s uc u es o cohomology ings a e no easy o ob ain, a majo di icul ybeing o de e mine when a se o gene a o s is comple e . In his pape , we exhibi a ci cums ance in which his di icul y may be o e come, and gi en an example . Le p be a p ime, and le Gp,(H*(G,A)]p deno e a Sylow p-subg oup o G, H*(G,A) espec i ely . Swan [10] shows ha when Gp is abelian, hen [H * (G,A)] p consis s o Che sub ing o H* (G P ,A) ixedunde he ac ion induced by inne au omo phisms o G .  Cbnside Che case when p=2, G2 is elemen a y abelian, and A is Z/2, he in ege s mod 2 wi h i ial G-ac ion . Then H*(G 2 ,Z/2) is isomo phic o R=(Z/2)[x1, . . .,x ], a polynomial ing in inde e mina es whe e is he ank o G 2 [7] . Consequen ly H * (G,Z/2) may be calcula edas a ing o in a ian s R H , whe e His a g oup whose o de is odd, and hence cop ime o he cha ac e is ic o he base ield o R . In exposi o y a icles, Sloane [8] and S anley [9] discuss .classical in a ian heo y, in which he base ield is he complex numbe s . A canonical o m is gi en o he ing o in a ian s, om which a comple e se o gene a o s and ela ions may be de i ed . In sec ion 2 we indica e how, wi h mino modi ica ion, hese esul s apply o he si ua ion desc ibed abo e . In sec ion 3 we apply hese esul s o G=GL(2,2 n ), and ob ain an exp ession o he Poinca é Se ies o H*(GL(2,2n),Z/2) . The addi i e * s uc u e o H (GL(2,2n),Z/2) has been desc ibed by Aguadé [1], bu he knowledge o he Poinca éSe ies leads, ía he canonical o m o he ing o ín a ian s, o a comple e se o gene a o s and ela ions . Fo n=2, he esul s a e well known [12] . Fo highe alues o n, he calcula ion becomes mo e complica ed, and he esul s o a machine compu a ionsa e p esen ed o n=3 . I would like o hank P .J . Webb o a se ies o enligh ening co espondences, and in pa icula o indica ed how he B aue li could be used o p o e he e sion o Molien's heo em gi en in heo em 1 . 2 . 18 MOLIEN'S THEOREM AND A BASIS OF INVARIANTS . Le F be a ield, H'a ini e g oup, V an F(H)-module wi h F-basis The polynomial ing R = F[x1, . . .,xn] is {x1, . . .,xn} and cha ac e X . g aded, wi h k- h componen R k ha ing F-basis he se o monomials o deg ee k in XI, . . .,x n  (k>0) .  Each hc H induces h :V + V  by h( )  = h . ( cV), and o each j>0 a map h j :R j } R j de ined by This makes R j an F(H)-module, whose cha ac e we deno e by X j . Deno e by R H he sub ing o R in a ian unde his ac ion, and le a j = dimF(RjH) . l n  l n hj(xl . . .x n ) = h(x j ) . . .h(x n )  ( l + . . .+ n = j) " In he classical heo y, whe e Fis aken o be he complex numbe s (C) Molien's Theo em yíelds an explici exp ession o £ aj j, he Poinca é Se ies o RH .  Mo eo e , RH is a Cohen-Macauley ing whichmeans he Poinca é Se ies may be w i en k i i II l (1- ) Consequen ly, he e exis ee in a ian s 1 , . . ., . (deg i = i ) and ansien in a ian s 9 11 . . .,gk (deg g i = u i )  such ha k RH = 1L g,C[ 1, . . ., ] i=1 and l, . . ., , k a e algeb aically independen o e C . The esul s ske chedhe e a e discussed mo e ully in [8], [9] . Now suppose ha cha (F)~ H . The ac ha R H is Cohen-Macauley ollows di ec ly om [5] P opn 13 p1033, so ha as in he complex case k R H ] =  LL gi F[ l , ". ., R . i=1 Molien's heo em may be modi ied by i s assuming ha F con ains IHI- h oo s o uni y . This may be achie ed by enso ing up o a sui able ield i necessa y, bu does no e ec wha ollows . Fo h ¿H, le B(h), B(h k ) deno e a B aue li o h, hk espec i ely, and B(X), B(X k ) he B aue cha ac e s o V, Rk espec i ely . We ha e he ollowing e sion o Molien's heo em . Theo em 1  I cha (F) i HI, hen whe e c(B(h)) is he cha ac e is ic polynomial o B(h) in he inde é mina e . PROOF  Le n l " "" .n n e  C deno e he eigen alues o B(h) . Then l n whe e X j =  E  n 1 . . .nn  . 1 + . . . + n = j 1 n Bu {n1 . . .n n ; 1 + ." .+ n = j} is a se o eigen alues o B(hk), so ha X j = B(X k ) (h) .  (2) Since . cha (F)j H, he o hogonali y ela ions o B aue cha ac e s [2] §18C a e simila o chose o o dina ycha ac e s . In pa icula , i ollows ha 1 de (B(h)) j=Oaj j . 7 T heH  c(B(h)) [de (I - B(h) )] -1 = i, l(1-ni )-1 aj = ~ .  E B(X j )(h) . heH W = j E 0 X j j  (1) Hence by (1), (2) and (3), and he heo em ollows . «0 0) ; BE GF(2 n )} cyclic g oup o o de 2 . I 1 +C + N iH i 1 . He e H ac s on G2 by E hEH 3 . THE POINCARE SERIES OF H*(GL(2,2n), Z/2) . Fo n>1, le GF(2 n ) deno e he ield o 2 n elemen s, and G be CL(2,2 n ), he g oup o 2x2 ma ices wi h en ies in GF(2 n ) . A Sylow 2-subg oup G 2 o G consís s o he ma ices de (I-B(h) ) and is isomo phic o he di ec p oduc o n copies o C2, he 0  ñ 1 hen H is cyclic o o de 2 n-1, and i N, C deno e he no malize o G 2 in G, and cen alize - o G 2 in G espec i ely, we ha e he ex ension (a  0  )  ( 1  0)  a  1  0)  -  ~1  a 2 0 ) 0  o¡- 1  0  1  0  a  0  1 and since Au (G2) = CL(n,2), we ha e a monomo phism O :H i GL( i,2) . This ís discussed mo e ully in [4] . n Le P n ( ) ° j E0c j+l 3 be a p imi i e, i educible, deg ee n polynomial o e GF(2), and,le p be a oo o P n ( ) . Then H is ó 0  2n-1 gene a ed by  ( 0  ó-1) whe e 6 = 11  .  Fu he , as a ec o space o e F2, G 2 has basis Since d . i ollows ha 0 (  o-  is he companion ma ix M o Pn ( ), and 0 d 1 ha 0(H) = M, he g oup gene a ed by M . Fo 1<i < n, le x i be he elemen o H2 (G2,Z/2) which co esponds unde his isomo phism o he homomo phism which maps o 1 i j = i-1 and 0 o he wíse . l is well known (see e.g . [7]) ha H* (G2,Z/2)is, he polynomial ing R = GF(2)[x1, . . .,xnl . l . ollows om he de ini ion o x, ha M induces a ans o ma ion so ha , iden i ying H wi h i s image unde 0, we H* (G,Z/2) as he ing o in a ian s RH . whe e B deno es he B aue li , and c he cha ac e is ic polynomial . To simpli y his exp ession, we i s no e ha i Q is a polynomial o e GF(2), hen Q(y2) _ [Q(y)J 2 . Hence i Q( ) is i educible o deg ee d, he oo s o Q( ) a e o he o m 2 2d-1 {y,y , . . .,y  } . Since he cha ac e is ic polynomial o M is 2 2 xi + xi+1 (l<i<n-1), n xn  i  j El  cjxj (0  ° -i )(0  1  (ó -1 °  °  (0 Tu ning o cohomology we no e ha H 1 (G 2 ,Z/2) Z Hom (G 2 ,Z/2) . By heo em 1, he Poinca é se ies is i+l u 1~ (Ocicn-1), (1 uj 0 1 may calcula e 1 2n-2 de B(Mi)  (4) 2n-1  1=0  c B(M i ) P n ( ),  í ollows ha  M  is simila o diag  (u,V2, . . .,u 2n-1 ),  and n-1 ha M i is simila o diag (pi,  ,u í2  ) . To simpli y (4), conside he ac ion o Z/n ( he in ege s mod . n) on Xn = {0,1, . . .,2n-2} gi en by z(i) = esidue o 2 zi mod . 2 n -1 (zEZ/n, ¡EX n ) . I O b(i) deno es he o bi o i and ¡O b(i)l = di , hen di is he exponen o 2 mod . e(i), whe e 2 n _1 (2 n -1 ,i) is he exponen o P i in GF(2 n ) . Mo eo e , pi is a oo o an i educible polynomial o deg ee d i o e GF(2) . Fo each din, le O d deno e a se o ep esen a i es o he o bi s o size d . I n is a p imi i e complex (2 n -1) h oo o uni y co esponding o V unde he B aue li , we ha e j de B(M i ) = ni l ni2 = 1 j =0 n-1 j d i -1 j n/d i cB(M i ) =  n (n i2 - ) =  n (n i2 - )  (o<i<2n-2) . j =o  J=O Thus om (4) we ob ain Theo em 2 . The Poinca é Se ies o H*(GL(2,2n),Z/2) is 1 E (  d  ) 2 n -1 din ¡EOd d-1  j 1[ (n2 i- ) n / d j=o whe e nand Od a e de ined abo e . We exhibi he Poinca é Se ies o some low alues o n . An explici exp ession seems ha d o ob ain o a bi a y n . (i) l n = 2, he o bi s o X 2 a e {0}, {1,2} and heo em 2 gi es [ 1 +2 1 (n3 =1) 3 (1- ) 2 (n- )(n2- ) as he Poinca é Se ies . This simpli ies o 1- + 2 24 (1- )(1- 3) as is well known (see e .g . (12}) . (íi) I n = 3, he o bi s o X3 a e {0}, {1,2,4}, {3,5,6} so we ob ain 1 1 +  3  +  3  } (n7=1) 7 (1- ) 3 (n- )(n2- )(n4- ) (n3- )(n5- )(n6- ) which can be w i en 1-2 + 2 + 3 + 4 -2 5+ 6 (1_0 2 (1- 7) (iii) Fo n = 4, he o bi s o X 4 a e {0} {5,10}, {1,2,4,8}, {3,6,9,12} and {7,11,13,14} . A leng hy calcula ion shows he Poinca é Se ies is 1-2 + 2 - 3 +3 4 + 5- 6- 7 + 8 - . 9 - 10 + ll +3 12- 13 + 14 -2 15+ 16 (1- )2(1- 3 )(1- 15) 4 .  The Cohomology Rings o n= 2,3 . As obse ed in sec ion 2, he Poinca é Se ies (as gi en by heo em 2) may be w i en in he o m R n (1- i) í=1 hough no necessa ilyuniquely . Howe e , i ee in a ian s can be ound in deg ees 1 ,  , L , and . ansien in a ian s in deg ees 1 , . . ., k hen we may conclude ha hese gene a e he en i e ing . (i) When n = 2, (6) may be w i en W i e x,y ins ead o xl,x2 . Since A=x 2 +xy+y 2 , B=xy(x+y) and  C=x 3 + x2 y + y 3 a e in a ian swi h C 2 = A3 + B 2 +C.B, i ollows ha hese h ee elemen sgene a e H*(GL(2,22),Z/2) as a commu a i e ing o exponen 2 subjec o he single ela iongi en . (ii) Fo n = 3, he Poinca é Se ies may be w i en 1 + 2 4 +3 5 +3 6 + 2 7 + il W i e x,y,z o x l ,x 2 ,x 3 . Sea ching in he ing o in a ian s, we ind ing gene a o s . 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+ 3 (1- 2 )(1- 3 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 ) .