THE PERMUTATION REPRESENTATION OF Sp(2m, F p ) ACTING ON THE VECTORS OF ITS STANDARD MODULE. Peter Sin University of Florida

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1 THE PERMUTATION REPRESENTATION OF Sp(2, F p ) ACTING ON THE VECTORS OF ITS STANDARD MODULE. Peter Sn Unversty of Florda Abstract. Ths paper studes the perutaton representaton of a fnte syplectc group over a pre feld of odd characterstc on the vectors of ts standard odule. The subodule lattce of ths perutaton odule s deterned. The results yeld addtve forulae for the p-ranks of varous ncdence atrces arsng fro the fnte syplectc spaces. Introducton In ths paper, we study the acton of the syplectc group Sp(2, p) on the set of vectors n ts standard odule. The coposton factors of ths perutaton odule have been known for soe te ([9], [12], [13]) and so the proble we address here s that of descrbng the subodule lattce. Ths turns out to be qute slar to the known structure of ths odule under the acton of the general lnear group (See [3] and references cted there). Ths structural nforaton yelds addtve forulae for the p-ranks of the ncdence atrces between ponts and sotropc subspaces of fxed denson n (2 1)-densonal projectve space over F p. Ths generalzes recent work [5] of de Caen and Moorhouse, who worked out the p-rank of the pont-lne ncdence when = 2. I wsh to thank Erc Moorhouse and Alex Zalessk for frutful dscussons and for supplyng e wth copes of ther work, out of whch ths paper grew. 1. Functons on a fnte vector space 1.1. Let p be an odd pre and let V be a 2-densonal F p -vector space wth a nonsngular alternatng blnear for,. We shall assue 2 to avod trval exceptons. We fx a syplectc bass e 1,... e, f,... f 1 and correspondng coordnates X 1,..., X, Y,... Y 1 so that e, f j = δ j. Supported by NSF grant DMS Typeset by AMS-TEX

2 Let k be an algebrac closure of F p and let A = k[x 1,..., X, Y,..., Y 1 ]/(X p X, Y p Y ) =1 (1) be the rng of functons on V. Ths s the prncpal object of our study Structure of A as a k GL(V )-odule. The the acton of GL(V ) on A s nduced fro ts acton on the polynoal rng k[x 1,..., X, Y,..., Y 1 ] through lnear substtutons of the varables. The k GL(V )-odule structure of A s well known; we wll gve a bref descrpton. When we factor out by the nhoogeneous deal (X p X, Y p Y ) =1 the gradng on k[x 1,...X, Y,..., Y 1 ] s destroyed, leavng only a fltraton {F e } 2(p 1) e=0, where F e = Iage n A of polynoals of degree e, (2) and a Z/(p 1)Z-gradng (fro the acton of the scalar atrces) A = p 2 d=0a[d], (3) where A[d] s the age of all hoogeneous polynoals of degree congruent to d odulo p 1. Denote by S(e) the coponent of degree e n the graded rng S = k[x 1,..., X, Y 1,..., Y ]/(X p, Y p ) =1. (4) Here e ranges fro 0 to 2(p 1). The denson of S(e) s e p ( )( ) e p s(e) = ( 1). (5) 2 1 =0 The odules S(e) are sple odules for GL(V ) and snce the graded odule of A assocated wth the fltraton {F e } s soorphc to S, ths fltraton s n fact a coposton seres of A. The fltraton {F e } also nduces coposton seres on each drect suand A[d]. We have F e A[d] = F e 1 A[d] unless e d od p 1, so we set A[d] t = F d+t(p 1) A[d] (and A[d] 1 = {0}). Then for [d] [0] s a coposton seres of A[d], wth {0} A[d] 0 A[d] 1 A[d] 2 1 (6) A[d] t /A[d] t 1 = S(d + t(p 1)); (7) the sae holds for A[0] except that the seres has one extra ter A[0] 2. The followng has been known for a long te n one for or other, e.g. [7], [10]. A generalzaton to all fnte felds and further references can be found n [3]. 2

3 Lea 1. (a) For d 0, A[d] s a unseral odule of denson p2n 1 p 1. (b) In A[0] the top and botto factors A[0] 0 = k and A[0]2 /A[0] 2 1 = k splt off and A[0] = k k M, (8) where M = A[0] 2 1 /A[0] 0 s unseral of denson p2n 1 p 1 1. Snce we wll be lookng closely at the odule M, we let M r be the age n M of A[0] r for 1 r 2 1. (Ths s the subspace of M generated by ages of polynoals of degree r(p 1).) Then the seres {0} M 1 M 2 1 = M (9) s the unque coposton seres of M as a k GL(V )-odule wth M t /M t 1 = S(t(p 1)), 1 t 2 1. (10) (By conventon we set M 0 = {0}.) The subodule lattce for SL(V ) s dentcal, snce ths subgroup has ndex pre to p and all coposton factors rean sple when restrcted to SL(V ) Structure of A as a k Sp(V )-odule. The k Sp(V )-coposton factors of A, or, what aounts to the sae thng, of the odules S(e), were gven ndependently n [9] and [12]. Lea 2. The odules S(e) all rean sple for Sp(V ) wth the sngle excepton of the ddle degree e = (p 1), where S((p 1)) s the drect su of two sple odules S + ((p 1)) and S ((p 1)) of densons 1 2 (s((p 1)+p )) and 1 2 (s((p 1) p )) respectvely. More nforaton can be found n [9] an [12]. Snce S(e) and S(2(p 1) e) are dual as k GL(V )-odules they are soorphc for Sp(V ). We are now ready to state our an result concernng the k Sp(V )-subodule lattce of A. Takng nto account the decopostons (3) and (8), t suffces to descrbe the subodule lattces of A[d], [d] [0] and of the nontrval suand M of A[0]. Snce the sple odules n the layers A[d] r /A[d] r 1 and M r /M r 1 have already been descrbed above, we wll not repeat that nforaton here. Theore 1. (a) For [d] [0] the odule A[d] s unseral. (b) The odules M t for the socle (and radcal) fltraton of M. The quotents M t /M t 1 = are sple except that M /M 1 s the drect su of two sple odules. 3

4 Rearks.(1) In other words, the theore says that the socle and radcal seres of A are the sae for GL(V ) and Sp(V ), wth all coposton factors reanng rreducble, except for one whch splts nto two. (2) Pctorally, M has the followng structure. S((2 1)(p 1)) S((2 2)(p 1)). S(( 1)(p 1)) S + ((p 1)) S ((p 1)) S(( 1)(p 1)). S(2(p 1)) S(p 1). (3) As has already been entoned, we have S((2 r)(p 1)) = S(r(p 1)), (1 r ) The perutaton odule on projectve space and p-rank probles. Now A[0] s the subspace of functons on V whch are unchanged by scalar ultplcaton of the coordnates, t ay be consdered as the space of functons on the dsjont unon of the projectve space P(V ) and the zero subspace of V. Thus, the perutaton odule k[p(v )] on P(V ) s soorphc to k M The structure of the perutaton odule k[p(v )] follows edately fro part (b) of the theore. In fnte projectve geoetry, the ncdence relatons between objects of two types s encoded n an ncdence atrx, wth rows labeled by the objects of one type and coluns by those of the second and entres 1 or 0 accordng to whether or not the correspondng row and colun labels are ncdent. It s natural to ask about the rank of ths atrx, over any feld. When the geoetry arses fro a feld of characterstc p, then one ay be nterested n the rank over a feld of characterstc p, or p-rank. An portant geoetry of ths knd s the geoetry of a syplectc vector space V n characterstc p and a typcal proble s to deterne the p-rank of the ncdence between ponts and sotropc r-densonal lnear subspaces (sotropc r-flat for short) of the projectve space P(V ). When 2 = 4 and r = 1, the geoetry of ponts and sotropc lnes s an exaple of a generalzed quadrangle In ths case, the p-rank has been found recently by de Caen and Moorhouse [5]. In general, the rank s equal to the denson of the k Sp(V )-subodule of k[p(v )] generated by the characterstc functon of a fxed r-flat. Ths subodule can be found wthout uch trouble thanks to part (b) of the theore, yeldng the followng nuercal result. 4

5 Theore 2. The p-rank of the ncdence atrx between ponts and sotropc r-flats (r = 0,..., 2 1) of P(V ) s equal to { r =1 s((p 1)) for r 1, 1 + =1 s((p 1)) (s((p 1)) + p ) for r = 1. (The nubers s(e) were defned n (5) above.) The forula for r 1 agrees wth Haada s forula [7] for the p-rank of the ncdence between ponts and all r-flats Ths shows that the code generated by sotropc r-flats s equal to the code generated by all r-flats, except when r = Techncal prelnares 2.1. Characters of the dagonal subgroup. The dagonal subgroup H of the syplectc group Sp(V ) conssts of all atrces of the for dag(µ 1,..., µ, µ 1..., µ 1 1 ) where the µ are nonzero eleents of F p. Let δ, 1, be the ap whch sends such a atrx to ts dagonal entry µ 1. The aps δ generate the group of characters of H and each character s unquely expressble as a product =1 δc, wth c Z/(p 1)Z. Each onoal =1 Xa Y b (or ts age n A) s a sultaneous egenvector for H, affordng the character =1 δ[a b ], where [a] denotes the congruence class odulo p 1 of the nteger a Induced odules. Let G = Sp(V ). For a subgroup X of G and a kx-odule L, Let nd G X L denote the nduced kg-odule kg kx L. If L = k then the nduced odule s just the perutaton odule of G actng on the left cosets of X. In ths subsecton and the next, we collect together soe facts about odules nduced fro one-densonal odules of stablzers of flags of sotropc subspaces. These subgroups are parabolc subgroups and so our stateents are really specal cases of the general theory of such odules developed n [4]. Let G r G denote the stablzer of the r-densonal sotropc subspace W r = e 1,..., e r of V. Lea 3. There exsts a unque nontrval sple kg-odule L(r, k) whch contans a trval kg r -subodule. Moreover dfferent choces of r gve nonsoorphc odules. Proof. Ths s a standard fact n the theory of representatons of Chevalley groups [11], so we wll gve a bref suary of the relevant facts n leu of a proof. Let ω r be the r-th fundaental weght n the usual order where ω 1 s the hghest weght of V. We vew ω r as a character of the dagonal subgroup of the algebrac group Sp(V Fp k). The sple odules for Sp(V Fp k) are paraetrzed by ther hghest weghts. Each nonnegatve ntegral cobnaton n j ω j s the hghest weght of a unque sple odule. The odules 5

6 whose hghest weghts are one of the p cobnatons wth 0 n j p 1, rean sple on restrcton to Sp(V ) and for a full set of sple k Sp(V )-odules. The sple odules on whch G r leaves a lne nvarant are precsely those wth hghest weght a ultple of ω r and the stable lne s the hghest weght space. Thus, the ω r weght space n the odule wth hghest weght ω r s a one-densonal kg r -odule. Ths representaton generates the group (under tensor product) of one-densonal representatons of G r. Thus, the hgh weght spaces n the odules of hghest weght nω r, 0 n p 1 gve all p 1 one densonal representatons of G r, wth the trval representaton occurng for the two values n = 0 and n = p 1. So the odule L(r, k) s the sple odule wth hghest weght (p 1)ω r. Lea 4. nd G G r k = k M(r, k) (11) where M(r, k) has a unque sple subodule and a unque sple quotent. Both subodule and quotent are soorphc to L(r, k). Proof. These stateents follow fro Lea 3 by applyng Frobenus recprocty and selfdualty of nd G G r k, usng the fact that all sple kg-odules are soorphc to ther duals. To relate ths to our earler notaton, we observe that nd G G r k s the perutaton odule on the set of r-densonal subspaces of V, so n partcular M(1, k) = M, the nontrval suand of k[p(v )]. The sple odules L(r, k) can also be recognzed; coparson of hghest weghts shows that L(r, k) = S(r(p 1)), for r and L(, k) = S + ((p 1)) Incdence aps. For any subset X of V we denote by χ X A ts characterstc functon. If X happens to be a hoogeneous subset, such as a lnear subspace, we wll use the sae notaton for the correspondng characterstc functon on P(V ). Thus, we ay regard characterstc funtons of subspaces as eleents of the perutaton odule nd G G 1 (k) = k[p(v )]. For each r = 1,...,, we have ncdence aps fro the perutaton odule on sotropc r-subspaces to the perutaton odule on projectve space gven by α r : nd G G r (k) k[p(v )], W χ W (12) and β r : nd G G r (k) k[p(v )], W χ W. (13) Of course, α = β. These are aps of kg-odules. All the odules nd G G r (k) have a trval suand (the constant functons) and a ore nterestng suand M(r, k). Moreover t s easly checked that the ncdence aps ap the 6

7 constant functons onto the constant functons. Therefore, t s techncally ore convenent to work odulo the trval suands and consder the aps M(r, k) nd G G r (k) α r,β r nd G G1 (k) M(1, k) = M, (14) nduced by restrcton and projecton. We shall call these aps α r and β r and refer to the also as ncdence aps. Let C r 1 the age of α r and let C 2 r 1 be the age of β r. (The subscrpts are the projectve densons of the supports of the characterstc functons.) The ages of α r and β r wll be denoted by C r 1 and C 2 r 1 respectvely. Thus C t = k Ct (15) for all t = 0,..., 2 2. An portant property of the odule C r 1 s that, beng a hooorphc age of M(r, k), t has a unque axal subodule. Lea 5. For 0 t < t 2 1, we have C t C t and C t C t. Proof. We ay assue t = t + 1. If 0 r 1, C r s spanned by (characterstc functons of) sotropc r-flats, whle f r 2 2, t s generated by the orthogonal copleents of sotropc 2 2 r-flats. These wll be the only two types of flats we wll consder n ths proof. Let W be a fxed t + 1-flat and P a pont of W. Then t s a sple count to check that the nuber of t-flats n W whch contan P s congruent to 1 od p. Thus the su of the characterstc functons of those t-flats s equal to the characterstc functon of W. Ths proves that C t+1 C t and strct contanent follows fro the fact that the unque sple quotents of C t and C t+1 are not soorphc, by the last asserton of Lea 3. The lea s proved. In the proof of Theore 1(a) wll also want to consder aps fro nduced odules nto the odules A[d]. The relevant parabolc subgroups are the stablzers G r 1,r = G r G r 1 of the flags W r 1 W r. Let λ be a one-densonal representaton of G r 1,r. By Frobenus recprocty, Ho G (nd G G r 1,r λ, A[d]) = Ho Gr,r 1 (λ, A[d]). (16) Thus, such hooorphss exst precsely when A[d] has a one-densonal kg r 1,r -subodule soorphc to λ and n ths case, the age of the kg-hooorphs s generated by ths one-densonal space. Now G r 1,r acts on W r leavng the hyperplane W r 1 nvarant. Hence n ts acton on the dual of W r, G r 1,r stablzes the one-densonal subspace spanned by the age of X r. We denote ths one-densonal representaton by λ r 7

8 Consder the functon χ Wr X d r for 0 d p 2. Its age n A les n A[d] (snce χ Wr A[0]). Moreover, the span of ths functon s G r 1,r -stable and affords the representaton λ d r. In ths way, we have for each r = 1,...,, and 0 d p 2 we have hooorphss α r [d] : nd G G r 1,r λ d r A[d], (17) wth age generated by χ Wr Xr. d Now the one-densonal kg r 1,r -odule Wr 1 /W r s spanned by the age of Y r, and soorphc to the dual λ r of λ r. It follows that for 0 d p 2 the functon χ W r 1 Yr d spans a kg r 1,r -subodule of A[d] soorphc to λ d r and generates the age of a kg-odule hooorphs β r [d] : nd G G r 1,r λ d r A[d], (18) Note that for d = 0, the age s C r. In fact, one can show that α r [0] (resp. β r [0]) s the coposte of α r (resp. β r ) wth the natural projecton nd G G r 1,r k nd G G r k sendng a flag to ts r-densonal (resp. (n r)-densonal) eber. In ths sense, we have generalzed the ncdence aps above. The an property we shall need s the followng [4, Th 6.13]. Lea 6. For d 0 the odules nd G G r 1,r λ d r and ndg G r 1,r λ d r subodules. have unque axal Thus, the ages of the aps α r [d] and β r [d] provde 2 subodules of A[d], each wth a unque axal subodule. These subodules wll be portant n our proof of Theore 1(a) because, as we already know, A[d] has exactly 2 coposton factors. 3. Proofs of Theores The followng leas wll be cobned to prove Theore 1(b) and Theore 2. The characterstc functon of the lnear subspace W defned by the vanshng of coordnates X, I and Y j, j J s gven by χ W = j J (1 Y p 1 j ) (1 ) (19) I Therefore, the characterstc functon of an (r 1)-flat of P(V ) s the age of a polynoal of degree 2 r, akng t clear that C t M 2 1 t for 1 t 2 1. Lea 7. For t 1 C 2 1 t = M t Proof. It s edate fro Lea 5 and (10) that the subodule C 2 1 t of M t has at least as any coposton factors as M t tself, so the two ust be equal. 8

9 Lea 8. M 1 and M/M are unseral odules. Proof. Snce the odules C r are ages of the odules M(r, k), they have unque axal subodules. Therefore, by Lea 7, each of the odules n the seres M 1 M 2 M 1 has a unque axal subodule. The unseralty of M/M follows by dualty. In vew of Lea 8, we can focus attenton on the subquotent M +1 /M 1. In partcular, we wsh to deterne the subodules C 1 /M 1 M /M 1 and C 2 /M 1 M +1 /M 1 Lea 9. C 1 /M 1 = S + ((p 1)). Proof. By defnton, C 1 /M 1 s the subodule of M /M 1 = S + ((p 1)) S ((p 1)) generated by the age of χ 1,...,. But odulo M 1 we have χ 1,..., = ( 1) 1 X p 1, by (19). Ths onoal s n the S+ ((p 1)) suand of M /M 1 (where t s n fact the hghest weght vector). We now coe to the key lea. Lea 10. C 2 = M +1 Proof. We begn by exanng the case = 2. In ths case C 2 = k P so the result s trval. For > 2 we shall argue by nducton. Let v + denote the onoal 1 X p 1 Y 1. and let v be the bnoal 1 2 Xp 2 1 Xp 1 Y 1 2 Xp 1 1 Xp 2 The results of [9] and [12] show that the ages n M /M 1 = S + (( 1)(p 1)) S (( 1)(p 1)) of v + and v le n the S + (( 1)(p 1)) and S + (( 1)(p 1)) coponents respectvely. C 2 s the k Sp(V )-subodule of M generated by the age of the characterstc functon of the -plane where X 1 = = X = Y = 0. Ths functon s gven by χ 1,...,, = (1 Y p 1 The age of ths functon n M/M 1 s χ 1,...,, = ( 1) +1 ( 1...X p 1 Y p 1 ) (1 ) (20) =1 Xp X p 1 I = 1 I where X I denotes the product of the varables X wth I. Let = {1,..., } 9 Y p 1 ), (21)

10 Applyng the syplectc transvecton X 1 X 1 + ηy 1 to ( 1) +1 χ 1,...,, yelds ( ) ) p 1 η r r 1 Y1 r r r=0 ( p 1 Xp 1 \{1} Y p 1 2 X p 1 1/ J J = 2 The δ [ 2] 1, coponent s ( η(p 1)X p 2 1 Y 1 + η p+1 2 ( p 1 p+1 2 ) X p Y p ) J Y p 1 Xp 1 \{1} Y p 1 Xp 1 2 X p 1 1/ J J = 2 Xp 1 \{1} Y p 1 J Y p 1. (22) (23) By subtractng ths expresson for η = 1 fro the correspondng expresson when η s a prtve p-th root of unty (so that η p 1 2 = 1) after frst dvdng the latter by η, we see that the k Sp(V )-odule generated by χ 1,...,, contans the eleent X p 2 1 Y 1 Xp 1 \{1} Y p 1 2 1/ J J = 2 J Y p 1 On applyng the syplectc transvecton Y 1 Y 1 +X 1 and takng fxed ponts of H, we see that the odule n queston contans 1 Xp 1 \{1} Y p 1 Xp 1 2 X p 1 1/ J J = 2 J Y p 1 (24). (25) Wrte V = V 1 V 1 where V 1 s the 2-densonal hyperbolc space where X = Y = 0 for > 1 and V 1 s ts orthogonal copleent. Insde Sp(V ) s the subgroup Sp(V 1 ) 10

11 Sp(V 1 ), preservng ths decoposton. Wth respect to ths group, we have a tensor product decoposton A = A 1 A 1, (26) wth A 1 = k[x 2,...X, Y,..., Y 2 ]/(X p X, Y p Y ) =2. Then (25) s equal to 1 χ 2,...,,, where χ 2,...,, A 1 s the characterstc functon of an 2 densonal subspace of V 1. By nducton, we know that the ages n A 1 of the onoal v 1 + = Xp 1 \{1} and the bnoal v 1 = Xp 1 \{1, 1,} Xp 2 1 Xp 1 Y \{1, 1,} Xp 1 1 Xp 2 Y 1 (27) le n the k Sp(V 1 )-odule generated by χ 2,...,,, odulo ters of degree less than ( 2)(p 1). Therefore, n A, the ages of v + = 1 v + 1 and v = 1 v 1, le n the k Sp(V 1)- subodule generated by 1 χ 2,...,,, odulo ters of degree less than ( 1)(p 1). Therefore, we have proved that the ages of both v + and v le n the k Sp(V (p))- subodule of M +1 generated by χ 1,...,,, whch shows that both coposton factors of S((p 1)) are coposton factors of C 2 and hence proves that C 2 M. Snce the characterstc functon of an sotropc ( 2)-flat les n C 2 and has nonzero age n the sple odule M +1 /M, t follows that C 2 = M +1. Proof of Theore 1(b). It follows fro Lea 10 that M s the unque axal subodule of M +1 and by dualty that M 1 /M 2 s the unque sple subodule of M/M 2. The subodule lattce of M s deterned. Proof of Theore 2. The p-rank for ponts versus r-flats s the denson of C r. Fro (15) and Leas 8, 9 and 10 we obtan the coposton factors and fro (5) we have the denson of each coposton factor. Proof of Theore 1(a). Fnally, we turn to the proof of Theore 1 (a). Frst, we note that the age of α r [d] les n A[d] 2 r and has nonzero age od A[d] 2 r 1. Thus, the unque sple quotent of nd G G r 1,r λ d r s soorphc to S(d + (2 r)(p 1)). Slarly, β r [d] aps nd G G r 1,r λ d R nto A[d] r 1 and the age aps onto the sple odule A[d] r 1 /A[d] r 2 = S(d+(r 1)(p 1)). In order to prove the unseralty of A[d], t suffces to prove that of A[d] r /A[d] r 2 for 1 r 2 1. We start by assung that 1 r 1. Then we know that the age of β r+1 [d] has a unque axal subodule, so t t s enough to show that A[d] r /A[d] r 2 s generated as a k Sp(V )-subodule by the age of Yr+1 d χ W, whch n turn wll follow f we can show that ths subodule contans a r 11

12 nonzero age of a polynoal of degree d + (r 1)(p 1). We observe that Y d polynoal n the Y s only. We consder the decoposton r+1χ W r s a V = V e V f = e 1,..., e f 1,..., f, (28) whch gves the factorzaton A = A(e) A(f), where A(e) and A(f) are the ages of A of all polynoals n the X s and Y s respectvely. The stablzer of ths decoposton nduces the full general lnear group on V e, hence any lnear substtuton aong the Y s. Snce A(f) s soorphc to the rng of functons on V f we can ake use of the known structure of ths odule for GL(V e ). Naely, the coponent A(f)[d], whch s the age of polynoals wth degrees congruent to d od p 1 s unseral. Ths eans the k GL(V f )-subodule generated by any polynoal n the Y s of degree d+r(p 1) whch s nonzero od A[d] r 1 contans the ages of all onoals n the Y s of degree d + (r 1)(p 1) Thus, the sae s true for the k Sp(V ) subodule of A[d], generated by such an eleent. Therefore, we have establshed that A[d] r /A[d] r 2 s unseral for 1 r 1. Snce A[d] s dual to A[p 1 d] for all d t follows that A[d] r /A[d] r 2 s unseral for + 1 r 2 1. It reans to show that A[d] /A[d] 2 s unseral. Snce the subodule genearted by the age of Xχ d W has a unque axal subodule, t s enough to show that ths subodule contans the nonzero age of a polynoal of degree d+( 1)(p 1). Wthout loss, we ay replace Xχ d W by X1 dχ W. Modulo A[d] 2, we have ( ( 1) X1 d χ W = X1 d Y p 1 Y p 1 ) Y p 1 1 Y p 1 J (29) =1 =2 J where J runs over all subsets of sze 2 of {2,...}. Snce ters of degree d+( 2)(p 1) are zero, the rght hand sde of (29) can be wrtten as X1 d p 1 (Y1 1)q(Y 2,..., Y ) (30) for soe polynoal q(y 2,..., Y ), of degree ( 1)(p 1). We shall consder agan the decopostons V = V 1 V 1, A = A 1 A 1 and Sp(V ) = Sp(V 1 ) Sp(V 1 ) fro (26) above. The subrng A 1 s soorphc to the rng of functons on a 2-densonal F p -vector space and ts structure under the acton of Sp(V 1 ) = SL(V 1 ) s gven by 1.2. In partcular, the coponent A 1 [d] (consstng of ages of polynoals n X 1 and Y 1 of degree congruent to d od p 1) s a nonsplt extenson of two sple odules. The sple subodule has a bass of ages of onoals of degree d and the quotent has a bass of ages of onoals of degree d + (p 1). Moreover, A 1 [d] s generated by X1 d p 1 (Y1 1). (Note that (Y p 1 1 1) s the characterstc functon of the subspace spaaned by e 1 n V 1.) Returnng to (30), we now see that snce the k Sp(V 1 )- subodule of A 1 [d] generated by X1 d p 1 (Y1 1) contans X1 d, the k Sp(V 1)-subodule of A[d] /A[d] 2 generated by (30) contans X1 dq(x Y,..., Y ), whch s a nonzero eleent of A[d] 1 /A[d] 2. Our proof s coplete. 12

13 Concludng rearks. The cases of the sae proble for p = 2 or wth p replaced by a pre power are, to y knowledge, unsolved. For p = 2, the odules S(e) are the exteror powers of the standard odule, so an answer would nclude the subodule structure of these odules. The coposton factors can be found usng [8] and are gven explctly n [1]. The paper [2] also gves the subodule structure of the Weyl odules wth fundaental hghest weghts. The exteror powers are known to be fltered by these odules [6, Appendx A]. References 1. A. M. Adaovtch, Analogues of spaces of prtve fors over a feld of postve characterstc, Moscow Unversty Matheatcs Bulletn 39, No. 1 (1984), A. M. Adaovtch, The subodule lattces of Weyl odules for syplectc groups wth fundaenstal hghest weghts, Moscow Unversty Matheatcs Bulletn 41, No. 2 (1986), M. Bardoe, P. Sn, The perutaton odules for GL(n + 1, F p ) actng on P n (F p ) and F p n+1, J. Lond. Math. Soc. 61 (2000), C. W. Curts, Modular representatons of fnte groups wth splt BN-pars, Senar on Algebrac Groups and Related Fnte Groups, Lecture Notes n Matheatcs 131, Sprnger, Berln, 1969, pp D. de Caen, E. Moorhouse, The p-rank of the Sp(4, p) generalzed quadrangle, Preprnt (1998). 6. S. Donkn, On tltng odules and nvarants for algebrac groups, Representatons of Algebras and Related Topcs, V. Dlab and L.L. Scott, (Ed.), Kluwer, Dordrecht/Boston/London, 1994, pp N. Haada, The rank of the ncdence atrx of ponts and d-flats n fnte geoetres, J. Sc. Hrosha Unv. Ser. A-I 32 (1968), J. C. Jantzen, Darstellungen halbenfacher algebrascher Gruppen un zugeordnete kontravarante Foren, Bonner Math. Schr. 67 (1973). 9. J. Lahtonen, On the subodules and coposton factors of certan nduced odules for groups of type C n, J. Algebra 140 (1991), F. J. MacWllas, N. J. A. Sloane, Theory of Error Correctng Codes, vol. 2, North Holland, New York, R. Stenberg, Representatons of algebrac groups, Nagoya Math. J. 22 (1963), I. D. Supunenko, A. E. Zalessk, Reduced syetrc powers of natural realzatons of the groups SL (P) and Sp (P) and ther restrctons to subgroups, Sberan Matheatcal Journal (4) 31 (1990), I. D. Supunenko, A. E. Zalessk, Perutaton Representatons and a fragent of the decoposton atrx of syplectc and specal lnear groups over a fnte feld., Sberan Matheatcal Journal (4) 31 (1990), Departent of Matheatcs, Unversty of Florda, Ganesvlle, FL 32611, USA e-al: sn@ath.ufl.edu 13