The transfer in modular invariant theory 1

Transcription

1 Journal of Pure and Appled Algebra 4 ( The transfer n modular nvarant theory R. James Shank a;, Davd L. Wehlau a;b a Department of Mathematcs and Statstcs, Queen s Unversty, Kngston, Ont., Canada K7L N6 b Department of Mathematcs and Computer Scence, Royal Mltary College, Kngston, Ont., Canada K7K 7B4 Communcated by A.V. Geramta; receved 8 July 997; receved n revsed form 9 November 997 Abstract We study the transfer homomorphsm n modular nvarant theory payng partcular attenton to the mage of the transfer whch s a proper non-zero deal n the rng of nvarants. We prove that, for a p-group over F p whose rng of nvarants s a polynomal algebra, the mage of the transfer s a prncpal deal. We compute the mage of the transfer for SL n(f q and GL n(f q showng that both deals are prncpal. We prove that, for a permutaton group, the mage of the transfer s a radcal deal and for a cyclc permutaton group the mage of the transfer s a prme deal. c 999 Elsever Scence B.V. All rghts reserved. MSC: A50. Introducton We let V be a vector space of dmenson n over a eld k and we choose a bass, {x ;:::;x n }, for the dual, V, of V. Consder a nte subgroup G of GL(V. The acton of G on V nduces an acton on V whch extends to an acton by algebra automorphsms on the symmetrc algebra of V, k[v ]=k[x ;:::;x n ]. Speccally, for g G; f k[v ] and v V,(g f(v=f(g v. The rng of nvarants of G s the subrng of k[v ] gven by k[v ] G := {f k[v ] g f = f for all g G}: Correspondng author. E-mal: shank@mast.queensu.ca. Research partally supported by grants from ARP and NSERC /99/$ - see front matter c 999 Elsever Scence B.V. All rghts reserved. PII: S ( X

2 64 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( The transfer homomorphsm s dened by Tr G : k[v ] k[v ] G f g G g f and s a homomorphsm of k[v ] G -modules. If the order of G s nvertble n k, then the Reynolds operator, (= G Tr G, s a projecton onto k[v ] G. When the characterstc of k dvdes the order of G, the mage of the transfer s a proper, non-zero deal n k[v ] G see Theorem. and Remark 5.6. For each subgroup, H, ofg there are two factorzatons of Tr G. Dene the relatve transfer: Tr G H : k[v ] H k[v ] G f g G=H g f Clearly, Tr G = TrH G Tr H. For the second factorzaton choose a set of rght coset representatves for H, say R. Dene Tr G H : k[v ] k[v ]by Tr G H (f= g R g f. Although Tr G H does depend on the choce of R, t s easy to see that, regardless of the choce, Tr H Tr G H = Tr G. Throughout ths paper we assume that the characterstc of k s p and that p dvdes the order of G. In other words, ths s a paper on modular nvarant theory. We use I G to denote the mage of Tr G and F q to denote the eld wth q elements. In Secton we collect some consequences of the work of Feshbach [8]. In partcular we produce a formula for the radcal of the mage of the transfer. We use ths formula n Secton 6 to prove that the mage of the transfer for a cyclc permutaton group s a prme deal after showng that I G s radcal for any permutaton representaton. One way to construct generators for the mage of the transfer s to evaluate the transfer on a set of module generators for k[v ] as a k[v ] G -module. In Secton we gve sucent condtons for the exstence of a block bass for k[v ] over certan subalgebras. We use ths result to descrbe module generators for k[v ]asak[v ] P - module for any p-group P. Ths generatng set s used n Secton 4 to prove that, for a p-subgroup of GL n (F p whose rng of nvarants s a polynomal algebra, the mage of the transfer s a prncpal deal. We also show that I P s a prncpal deal for certan examples of p-subgroups of GL n (F q, wth q = p s and s, havng a polynomal rng of nvarants and we gve an example of a representaton of a p-group where the rng of nvarants s a hypersurface and the mage of the transfer s not prncpal. These results lend support to the followng conjecture. s a poly- Conjecture.. Suppose that P s a p-subgroup of GL(V. Then k[v ] P nomal algebra f and only f I P s a prncpal deal.

3 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( For the usual permutaton representaton of the symmetrc group p over F p,fp then the mage of the transfer s not a prncpal deal (see [5, Theorem 9.8] even though the rng of nvarants s a polynomal algebra. (If p = then the mage of the transfer s the prncpal deal generated by the dscrmnant (see [5, Theorem 9.7]. Therefore ths conjecture does not extend to arbtrary modular representatons. In Secton 5 we relate the mage of the transfer for G wth the mage of the transfer for ts p-sylow subgroup. Usng ths result we show that, for both SL n (F q and GL n (F q, the mage of the transfer s a prncpal deal. We thank Mara Neusel for suggestng the use of the addtvty of the pth power operaton to smplfy the proof of Theorem 6.. We also thank Eddy Campbell and Ian Hughes for ther assstance and encouragement.. The radcal of the mage of the transfer Suppose a s an element of some set on whch G acts. The sotropy subgroup of a s G a := {g G g a = a} and the orbt of a s the set Ga := {g a g G} = {g a g G= G a }. Usng the acton of G on V we dene the orbt space of V as V=G := {Gv v V }. For f k[v ], the norm of f; N(f, s the product of the elements n the orbt of f. Clearly N(f k[v ] G. When k s algebracally closed, the ntely generated Noetheran algebra k[v ] G s the rng of regular functons on the ane varety V=G (see [, Ch. III Secton ]. When k s nte, the elements of k[v ] G stll represent regular functons on V=G but dstnct polynomals may represent the same functon. For example, f k = F p then f and f p represent the same functon. Let k denote the algebrac closure of k and, for any vector space, U, over k, dene U := U k k. Extend the acton of G on V to a lnear acton of G on V by denng g(v c:=(g v c for g G; v V and c k. Smlarly extend the acton of G on k[v ] to an acton on k[v ]. It s not hard to prove that k[v ] G =(k[v ] G =(k[v ] G.In other words, takng the nvarants of a lnear representaton commutes wth extendng the eld. We wll dentfy k[v ] G wth the subrng k[v ] G k k k[v ] G. Usng ths dentcaton, elements of k[v ] G represent regular functons on the V=G and dstnct polynomals do represent dstnct functons (see [5, Ch. VII Secton ]. For any subset X of k[v ] G, let V(X :={Gv V=G f(v = 0 for all f X }: If I s an deal n k[v ] G then dentfyng I wth I k k k[v ] G allows us to dene V(I V=G. Of course V( I=V(I. By denton I s a vector space over k. Usng the fact that I s an deal n k[v ] G, t s easy to show that I s an deal n k[v ] G. Furthermore V(I=V(I. Thus V(I s a subvarety of V=G. The followng result s reasonably well known. We nclude a proof for completeness. Theorem.. V(I G ={Gv V=G p dvdes G v }.

4 66 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( Proof. Snce the transfer s a lnear map, the mage of Tr G appled to k[v ]si G. Therefore, snce V(I G =V(I G, we may assume that k = k and V = V. A pont G v V=G belongs to V(I G f and only f f(v = 0 for all f I G. Therefore we need to show that f(v = 0 for all f I G f and only f p dvdes G v. Observe that, for any h k[v ], (Tr Gv (h(v= g h (v= h(g v= h(v= G v h(v: g Gv g Gv g Gv Therefore (Tr G (h(v =(Tr Gv Tr G G v (h(v=tr Gv ( Tr G G v (h(v = G v g h (v= G v h(g v: g G=G v g G=G v Thus f p dvdes G v, then (Tr G (h(v = 0. However, snce the orbt Gv s nte, there exst h k[v ] such that h(v s non-zero and h(g v = 0 for all g= G v. One such functon s gven by h = x (w x (w (w; w Gv w v where (w s the least such that x (v x (w. Note that h s not homogeneous and (Tr G (h(v= G v h(g v= G v h(v: g G=G v Therefore, f p does not dvde G v, we conclude that (Tr G (h(v 0. We remnd the reader that the augmentaton deal of k[v ] G s the deal generated by the homogeneous elements of postve degree. Theorem.. If p dvdes the order of G; then I G s a non-zero deal whch s properly contaned n the augmentaton deal of k[v ] G. Proof. We rst show that I G s properly contaned n the augmentaton deal. Snce p dvdes the order of G, the restrcton of the transfer to elements of degree zero s the zero map. Therefore I G s a subset of the augmentaton deal. The varety assocated to the augmentaton deal s the orbt of the zero vector. On the other hand G has a non-trval p-sylow subgroup and every representaton of a p-group has a non-zero xed pont v V. By Theorem., the orbt Gv les n V(I G. Therefore the varety of the augmentaton deal s properly contaned n V(I G and, consequently, I G s properly contaned n the augmentaton deal. To see that I G s non-zero, start be extendng the acton of G to the eld of fractons k(v. The group acts on k(v by eld automorphsms. Snce G s a subgroup

5 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( of GL(V, the representaton of G s fathful and dstnct group elements gve dstnct eld automorphsms. Any set of eld automorphsms s lnearly ndependent. Therefore Tr G s non-zero on k(v. Thus there are polynomals f and h n k[v ] such that Tr G (f=h s non-zero. However N(hf=h k[v ] and N(h k[v ] G. Therefore Tr G (N(hf=h=N(hTr G (f=h s nonzero. We note that, for a non-fathful representaton, f the order of the kernel of the representaton s dvsble by p then the mage of the transfer s the zero deal. Followng Feshbach n [8], for every g G of order p, dene P g to be the deal n k[v ] generated by (g V. Observe that, snce t s generated by homogeneous elements of degree one, P g s a prme deal. Therefore P g k[v ] G s also a prme deal. If f and h are n k[v ], then (g (fh=h(g (f+(g f(g (h. Hence (g k[v ] P g. We also dene P G = P g where the ntersecton runs over all group elements, g, of order p n G. Lemma.. If r p; then P g = P g r. Proof. Factorng g r as(g (+g+ +g r shows that (g r V (g V. Therefore, snce P g r s generated by (g r V and P g s generated by (g V, we have P g r P g. However, g generates a subgroup of order p so, as long as g r s not the dentty, g r s another generator of the same subgroup. Therefore, f r p then, for some m, (g r m = g and P g = P (gr m P g r as requred. It follows from Lemma. that P g depends only on the subgroup generated by g and P G can be constructed by takng the ntersecton over subgroups of order p. Theorem.4. I G = P G k[v ] G. Proof. In [8, Theorem.4] Feshbach proves that I G P G k[v ] G I G. Snce P G k[v ] G s a nte ntersecton of prme deals, t s a radcal deal. Therefore I G = P G k[v ] G. Remark.5. Suppose G = Z=p and let g be a generator for G. Usng Lemma., we see that P g = P G. By Theorem.4, I G = P G k[v ] G. Therefore I G = P g k[v ] G s a prme deal. If the representaton of G on V s ndecomposable then the set of xed ponts n V; V g, s a one dmensonal subspace and P g s generated by the elements of V whch are zero on V g. If we choose our bass so that V g s the span of the dual of x n, then P g =(x ;:::;x n and I G =(x ;:::;x n k[v ] G.. Block bases A homogeneous system of parameters for a graded k-algebra A s a collecton of homogeneous elements, {a ;:::;a n }, such that {a ;:::;a n } s algebracally ndependent

6 68 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( and A s a ntely generated k[a ;:::;a n ]-module. When A = k[v ], elements of A represent regular functons on V = V k k and a homogeneous set {a ;:::;a n } s a homogeneous system of parameters f and only f the only common zero of the a s the orgn. Also, snce k[v ] s Cohen Macaulay, f {a ;:::;a n } s a homogeneous system of parameters then k[v ] s a free k[a ;:::;a n ]-module and a ;:::;a n s a regular sequence. For detals we refer the reader to [, Chs. 5&6]. Recall that k[v ] s a ntely generated k[v ] G -module and suppose that B s a set of module generators for k[v ]asak[v ] G -module. Snce Tr G s a map of k[v ] G - modules, we can construct a generatng set for I G by evaluatng Tr G on B. Suppose that {a ;:::;a n } k[v ] G s a homogeneous system of parameters for k[v ]. Snce k[v ]s Cohen-Macaulay, t s a free k[a ;:::;a n ]-module. Any bass for k[v ] over k[a ;:::;a n ] s a generatng set for k[v ]asak[v ] G -module and can be used to construct a generatng set for I G. The purpose of ths secton s to descrbe certan famles of bases, called block bases, whch wll be used n the later sectons to compute the mage of the transfer for varous examples. A block bass s a bass consstng of the monomal factors of a sngle monomal. The sngle monomal s called the generator of the block bass. We refer the reader to [5] for a more extensve dscusson of block bases. In the followng we make use of the theory of monomal orders. We refer the reader to [6, Ch. ] for the approprate dentons and a detaled dscusson of monomal orders. We use the conventon that a monomal s a product of varables and a term s a monomal wth a coecent. Theorem.. Suppose that dm k (V =n and that a ;:::;a n s a sequence of homogeneous elements n k[v ]. Further suppose that there exst ntegers d ;:::;d n such that; wth respect to some monomal order; the lead term of a s x d for all. Then a ;:::;a n s a regular sequence n k[v ] and n = xd generates a block bases for k[v ] over k[a ;:::;a n ]. Proof. We begn by provng that a ;:::;a n s a regular sequence. Snce k[v ] s Cohen Macaulay, t s sucent to prove that a ;:::;a n s a homogeneous system of parameters. We prove ths by showng that the only common zero of a ;:::;a n s the orgn. We remnd the reader that elements of k[v ] represent functons on V. Wthout loss of generalty we may assume that x x x n n the gven monomal order. Observe that the multplcatve property of the order mples that for each monomal appearng n a, other than x d, there exsts j such that x j dvdes. In partcular, a must equal x d. Thus the zero set of a s the hyperplane cut out by x = 0. The restrcton of a to ths hyperplane s just x d. Therefore the set of common zeros of a and a s the subspace dened by x = x = 0. Contnung n ths fashon we see that the only common zero of a ;:::;a n s the orgn. We remnd the reader that f M s a graded subspace of k[v ] and M d s the homogeneous component of degree d, then the Poncare seres of M s P(M; t= dm k (M d t d : =0

7 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( Snce a ;:::;a n s a regular sequence, k[a ;:::;a n ] s a polynomal algebra wth Poncare seres n = (. Therefore the Poncare seres of k[v ]=(a td ;:::;a n s n t d t : = In partcular, the rank of k[v ] as a free k[a ;:::;a n ]-module s n = d. Ths s equal to the number of monomal factors of n = xd. Thus t suces to prove that these monomal factors span k[v ]asak[a ;:::;a n ]-module. Let S denote the k[a ;:::;a n ]-module spanned by the factors of n = xd. Suppose, by way of contradcton, that S s a proper subset of k[v ]. Choose an element f k[v ] S wth smallest possble lead monomal. Let = x e xen n be the lead monomal of f and let c be the lead term of f. Note that c 0. For each, wrte e = q d + r where 0 r d. Dene ( n n f = f c : j= a qj = x r Clearly f k[v ] S and the lead monomal of f s less than, contradctng the choce of f. Remark.. Suppose that the sequence a ;:::;a n satses the hypotheses of Theorem.. Let I denote the deal n k[v ] generated by the sequence and let J be the deal generated by x d ;:::;xdn n. Clearly J s contaned n the lead term deal of I. Furthermore, the Poncare seres of k[v ]=I equals the Poncare seres of k[v ]=J. Therefore J s the lead term deal of I and a ;:::;a n sagrobner bass for I (for more on Grobner bases, see [6]. Also, the block bass generated by n = xd projects to a k-bass for k[v ]=I. It s not hard to show that k[v ]=I s a Hodge algebra governed by the deal of monomals generated by the projectons of x d ;:::;xdn n (for more on Hodge algebras, see [7]. Corollary.. Suppose that P s a p-subgroup of GL n (F q. Choose a bass n whch P s an upper-trangular group and let p m be the sze of the P-orbt of x. Then N (x ;:::;N(x n s a regular sequence n F q [V ] and n = xpm generates a block bass for F q [V ] over F q [N(x ;:::;N(x n ]. Proof. Usng the graded reverse lexcographc order wth x x x n, we see that the lead monomal of N(x sx pm. Now apply Theorem.. 4. Nakajma groups In [9] Nakajma characterzed the representatons of p-groups over F p wth polynomal nvarants. Nakajma s characterzaton prompts us to make the followng

8 70 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( dentons. Suppose P s a p-subgroup of GL n (F q and, for n, let P = {g P g x j = x j f j }: Clearly the subgroups P depend on the choce of bass. Denton 4.. P s a Nakajma group f, for some choce of bass, ( P s an upper trangular group and ( P = P n P n P. For any Nakajma group P = n = P. Furthermore, for any, the orbt of x under P s the same as the orbt of x under P. Note that P s an elementary abelan p-group and the orbt of x s {x + u u W } where W s closed under addton and s a subset of the span of {x ;:::;x }. Thus W s vector space over F p. However W not necessarly a vector space over F q. Let m = dm Fp (W and observe that the degree of N (x sp m. Snce {N(x ;:::;N(x n } s a homogeneous system of parameters for both F q [V ] and F q [V ] P and the product of the degrees s equal to the order of the group, F q [V ] P = F q [N(x ;:::;N(x n ]. Therefore, from Corollary., n = xpm generates a block bass for F q [V ] as a free F q [V ] P -module. Nakajma s characterzaton s expressed by the followng proposton. Proposton 4. (Nakajma [9, Proposton 4.]. Suppose P s a p-subgroup of GL n (F p. Then F p [V ] P s a polynomal algebra f and only f P s a Nakajma group. Suppose that r s a non-negatve nteger. We denote by p (r the sum of the dgts n the p-adc expanson of r. Also, f W s a nte dmensonal vector space over a nte eld then dene d(w := u W {0} u: We warn the reader that, n the followng proposton, W s a vector space over F p but not necessarly a vector space over F q. Proposton 4.. Suppose that V s a vector space over F q and that W s a subset of V whch s closed under addton so that W s a vector space over F p. Let m = dm Fp (W. Then u W u =0 unless p dvdes and p ( m(p. Furthermore; u pm = u = d(w : u W u W {0} Proof. The proof s a smple generalzaton of the proof of [5, Proposton 9.5]. Theorem 4.4. If P s a Nakajma group then I P s the prncpal deal generated by n d(w : = Proof. Usng Corollary., we see that the monomal factors of = n = xpm are a block bass for F q [V ] over F q [V ] P. Snce Tr P s an homomorphsm of F q [V ] P -modules,

9 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( applyng the transfer to the elements of a bass gves a generatng set for the mage. Consder ( n ( n Tr P x r = (x + u r ; = = u W where r p m for each. Expandng gves r (x + u r = u W u W j=0 ( r x r j j r u j = j=0 ( r j x r j u j : u W By Proposton 4., u W u j = 0 unless p dvdes j and p (j m (p. Snce j r p m, ths expresson s zero unless j =r =p m. Therefore Tr P ( n = xr =0 unless r = m for all. Agan by Proposton 4., ( n n Tr P x m = u p m n = d(w : = = u W = Nakajma s characterzaton does not extend to representatons of p-groups over F q for q = p s wth s. One of the reasons Nakajma s characterzaton does not extend s the fact that W s not necessarly a vector space over F q. Although all Nakajma groups have polynomal nvarants, there are p-groups wth polynomal nvarants whch are not Nakajma groups. One example of such a group s due to Stong (see []. We contnue our nvestgaton of the relatonshp between G and I G by computng I G for ths example. We wll use our bass to dentfy elements of G wth the correspondng matrces and we dentfy V wth the space of column vectors. Example 4.5. Let {;!;} be a bass for F p over F p, 0 0! T = 0 0 ; T = 0 0 and T = 0 0 : Let G be the group generated by T ;T and T. Then G s somorphc to (Z=p. Dene := ( p (x p x x p (! p!(x p x x p and := (x p x x p p (! p! p (x p x x p x p(p. Note that and are both elements of F p [V ] G. In the graded reverse lexcographc order, wth x x x, the lead monomal of s x p and the lead monomal of s xp. Applyng Theorem., we see that x ;; s a regular sequence n F p [V ] and the monomal factors of x p x p form a bass for F p [V ] over F p [x ;; ]. Thus {x ;; } s a homogeneous system of parameters for F p [V ] G and, snce the product of the degrees equals the order of the group, we conclude that F p [V ] G = F p [x ; ;] (or see []. We wll construct a generatng set for I G by evaluatng Tr G on the monomal factors of x p x p. Let H be the subgroup generated by T and T. Then Tr G = TrH G TrH.

10 7 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( Tr H (x x j = = a; b F p (x + ax (x + bx j j m=0 k=0 ( ( j x m+k x m x j k m k a; b F p a m b k : Recall that c F p c r s fp dvdes r and r 0; otherwse ths sum zero. Snce j p, Tr H (x x j = 0 unless j = p. Furthermore, when j = p only the terms wth k = p and m dvsble by p and m 0 contrbute. Wrtng m = m (p we have Thus Tr H (x x p = p+ m = Tr G (x x p =Tr G H Tr H (x x p = = = p+ m = p+ m = p+ m = ( m x (m +(p (p x m (p : ( m x (m +(p (p (x + c!x m c F p ( m ( x (m +(p m m t ( m As above, ths smples to p+ Tr G (xx p = = p m = t = x (m +(p c F p t=0 m ( m t t=0 ( ( m m t (p x (m +t +(p x (m +t (p! t (p p+ p ( m = t = m (p x (m +t +(p x (m+t! t : (p x m t c t! t x t x m t ( m (p t (p Lemma 4.6. Suppose p ; t 0 and m 0. Then ( ( m (p m (p t 0 (mod p: (p! t x t c t : c F p

11 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( Proof. We consder the three p-adc expansons: m (p =(m p +(p m, t (p =(t p +(p t and = p + 0. Now f ( m (p 0 (mod p then the ( p-adc expanson of m (p s ( (m p +( 0 (p m. Smlarly f m (p t (p 0 (mod p then the p-adc expanson of (m +t (p s ( (m (t p +( 0 (p m (p t. In partcular (m (t 0 and 0 (p m (p t 0. Therefore + 0 (p. However, snce p, + 0 (p. Therefore, f Tr G (x x j s non-zero, then = p, j = p and Tr G (x p x p = p+ m = t = p ( p ( p m m t x (m +t +(p x p (m+t! t : However, f m = p then, snce t 0, ( p m t = 0. Thus we may assume that m p. If ( ( p m t = p m (p t (p s non-zero then the p-adc expanson of p m (p t (p s (p m t +p +(m p + t. Therefore p m t + 0 and m p+t 0. Thus m +t = p+ and ( p m (p t (p =. Furthermore, ( p ( p ( ( p p = m m = (p m p m Therefore, =( m ( p m =( p =: Tr G (x p x p =( x (p+(p =( x (p+(p p t =! t (p! p (! p p! p : Observe that! s not n F p and therefore! p. Furthermore, snce F p has no non-trval pth roots of unty, (! p p and the coecent n the above expresson s non-zero. Note that ths also follows from the fact that Tr G s never the zero map. In concluson, I G s the prncpal deal generated by x p +p We nclude the followng example to llustrate the fact that, for a p-group whose rng of nvarants s a hypersurface, the mage of the transfer s not always a prncpal deal.

12 74 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( Example 4.7. For smplcty we restrct our attenton to p = but smlar examples exst for all prmes. Let = and = : Let G = ; and H =. Clearly G = Z= Z= and H = Z=. Furthermore G s a Nakajma group and F [V ] G = F [x ;N(x ;x ;N(x 4 ]: From Theorem 4.4, I G s the prncpal deal generated by x x. H s a subgroup of ndex n the Nakajma group G. Applyng [4, Theorem 4.4] we see that F [V ] H s a free F [V ] G -module wth bass {;a} where a = x x 4 + x x. In other words F [V ] H s a hypersurface. We wsh to descrbe I H. Note that Tr H s a homomorphsm of F [V ] G -modules and that F [V ]s a free F [V ] G -module wth a block bass generated by x x 4. Therefore I H s generated by Tr H (x =x, Tr H (x 4 =x and Tr H (x x 4 =x x 4 + x x = a. In partcular I H s not a prncpal deal. 5. The p-sylow transfer In ths secton we consder the relatonshp between the mage of the transfer for G and the mage of the transfer for a p-sylow subgroup of G. Our startng pont s the followng theorem. Theorem 5.. If p does not dvde [G : H]; then I G = I H k[v ] G. Proof. We rst show that I H k[v ] G I G. Suppose that f I H k[v ] G and Tr H (k=f. Then Tr G H (f=[g : H]f and therefore Tr G (k=[g : H] = f. In order to show that I G I H k[v ] G, we use the factorzaton Tr G = Tr H Tr G H. Suppose that f I G k[v ] G. Therefore f = Tr G (k for some k k[v ] and thus f = Tr H ( Tr G H (k I H. The heght of a prme deal P s the length of a maxmal chan, wth respect to ncluson, of prme deals contaned n P. The heght of an arbtrary deal I s the mnmal heght of a prme deal contanng I. We denote the heght of I by ht(i and we refer the reader to [, Appendx] for detals. Corollary 5.. If P s the the p-sylow subgroup of G; then ht(i G =ht(i P. Proof. Snce k[v ] P s ntegral over k[v ] G, we have both gong up and gong down (see, e.g., [, Theorem.4.4]. From Theorem 5., I P les over I G and thus they have the same heght.

13 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( We now use Theorem 5. to compute the mage of the transfer for SL n (F q. A p-sylow subgroup for both SL n (F q and GL n (F q s gven by U n (F q, the group of upper trangular matrces wth s along the dagonal. Clearly U n (F q s a Nakajma group (see Secton 4 and F q [V ] Un(Fq s the polynomal algebra F q [h ;:::;h n ] where h = N Un(F q(x has degree q. The rng of nvarants for GL n (F q, known as the Dckson algebra, s the polynomal algebra F q [d ;n ;d ;n ;:::;d n; n ] where d ; n has degree q n q n. We refer the reader to [, Secton 8.] for a detaled dscusson of the Dckson algebra. Fnally, choose a non-zero representatve from each lne n V and take the product to form L F q [V ]. It s well known that L q = d n; n and F q [V ] SLn(Fq = F q [d ;n ;:::;d n ;n ;L] (agan, see [, Secton 8.]. Theorem 5.. The mage of Tr SLn(Fq s the the prncpal deal generated by L (q (n. Proof. From Theorem 5., I SLn(Fq = I Un(Fq F q [V ] SLn(Fq. By [5, Corollary 9.7] or Theorem 4.4, I Un(Fq s the prncpal deal generated by (h n h n h n q. Suppose that f I SLn(Fq = I Un(Fq F q [V ] SLn(Fq. Therefore f = h (h n h n h n q for some h F q [V ] Un(Fq. Snce h = x, we see that x (q (n dvdes f. Suppose that s a non-zero lnear functonal. Clearly s n the SL n (F q -orbt of x. Because f s an SL n (F q -nvarant and x (q (n dvdes f, t follows that (q (n dvdes f. Snce x q = c F cx q {0} and d n; n s the product of all non-zero lnear functonals, we conclude that dn; n n dvdes f. InF q [V ] SLn(Fq, d n; n = L q. Thus f = L (q (n k for some k F q [V ]. Clearly k F q [V ] SLn(Fq. Fnally, snce d n; n =( n (h h n q and L (q (n = dn; n n we see that L (q (n I Un(Fq F q [V ] SLn(Fq = I SLn(Fq. Remark 5.4. Suppose G s a subgroup of GL n (F q and G contans SL n (F q. Then there exsts an m dvdng q such that G = {g GL n (F q (det g m =}. It s easy to see that L m s G-nvarant and that F p [V ] G s a polynomal algebra. Furthermore the proof of Theorem 5. shows that I G s the prncpal deal generated by (L m (n (q =m. In partcular ths gves us a smpler proof of [5, Corollary 9.4]. Theorem 5.5. The mage of Tr GLn(Fq s the the prncpal deal generated by d n n; n. Remark 5.6. For any modular representaton of a nte group G over k F p, there exsts a q such that G GL n (F q. Snce, Tr GLn(Fq = Tr GLn(Fq G Tr G, the kernel of Tr G s a subset of the kernel of Tr GLn(Fq. Usng the other factorzaton, Tr GLn(Fq = Tr G Tr GLn(Fq G, we see that I GLn(Fq I G. In partcular dn; n n I G and Tr G s non-zero. 6. Permutaton groups In ths secton we consder groups whch act by permutng a xed bass of V.We wll call such groups permutaton groups. For a monomal we denote the orbt sum of by #s(= g G=G g : It s well known that, for a permutaton group, the

14 76 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( orbt sums of monomals form a vector space bass for the rng of nvarants. Snce Tr G (= G #s(, the set {#s( p does not dvde G } forms a vector space bass for I G. In partcular #s( I G f and only f p does not dvde G. Therefore f f I G then f s a lnear combnaton of orbt sums each of whch s contaned n I G. Theorem 6.. Suppose that G s a permutaton group. Then I G s a radcal deal and I G = P G k[v ] G. Proof. Suppose f r I G. Wrte f = t = c #s( where c k and each s a monomal. Choose m so that p m r. Then f pm I G. However, ( t p m t f pm = c #s( = c pm #s( pm : = Therefore for each, #s( pm = I G. Thus p does not dvde the order of G p m. However. Hence, for each, #s( I G and therefore f I G. Thus I G s radcal and, G = G p m by Theorem.4, I G = P G k[v ] G. As a corollary we obtan the followng generalzaton of recent results of Neusel [0] and Smth [4]. Corollary 6.. If G s a cyclc permutaton group; then I G s a prme deal. Proof. If G s non-modular then the transfer s surjectve and I G s prme. If p dvdes the order of G then there s a unque subgroup of G wth order p. Let g be a generator for ths subgroup. Usng Lemma., we see that P g = P G. By Theorem.4, I G = P G k[v ] G. By Theorem 6., I G = I G. Therefore I G = P g k[v ] G s a prme deal. We return brey to Example 4.7. In ths example, after a change of bass, both G and H are permutaton groups. H s a cyclc permutaton group and I H s the prme deal generated by x, x and x x 4 + x x. G s somorphc to Z= Z= and I G s generated by x x. In partcular I G s radcal but not prme. References [] A. Adem, R.J. Mlgram, Cohomology of Fnte Groups, Sprnger, New York, 994. [] D.J. Benson, Polynomal Invarants of Fnte Groups, Lecture Note Seres, vol. 90, Cambrdge Unversty Press, London, 99. [] W. Bruns, J. Herzog, Cohen Macaulay Rngs, Cambrdge Studes n Adv. Math., vol. 9, Cambrdge Unversty Press, Cambrdge, 99. [4] H.E.A. Campbell, I.P. Hughes, Rngs of nvarants of certan p-groups over the eld F p, J. Alg., to appear. [5] H.E.A. Campbell, I.P. Hughes, R.J. Shank, D.L. Wehlau, Bases for rngs of convarants, Transformaton Groups (4 (

15 R. James Shank, D.L. Wehlau / Journal of Pure and Appled Algebra 4 ( [6] D. Cox, J. Lttle, D. O Shea, Ideals, Varetes, and Algorthms, Sprnger, New York, 99. [7] C. DeConcn, D. Esenbud, C. Proces, Hodge Algebras, Astersque 9, Socete Mathematque de France, 98. [8] M. Feshbach, p-subgroups of compact Le groups and torson of nnte heght n H (BG, II, Mch. Math. J. 9 ( [9] H. Nakajma, Regular rngs of nvarants of unpotent groups, J. Algebra 85 ( [0] M. Neusel, The transfer n the nvarant theory of modular permutaton representatons, preprnt, 997. [] J.P. Serre, Algebrac Groups and Class Felds, Sprnger, New York, 988. [] L. Smth, Polynomal Invarants of Fnte Groups, A.K. Peters, Wellesley, MA, 995. [] L. Smth, Modular representatons of q-groups wth regular rngs of nvarants, preprnt, 996. [4] L. Smth, Modular vector nvarants of cyclc permutaton representatons, preprnt, 997. [5] O. Zarsk, P. Samuel, Commutatve Algebra, vol. II, Sprnger, New York, 960.