THE PERMUTATION REPRESENTATION OF Sp(2m, 𝔽_{p})

ACTING ON THE VECTORS OF ITS STANDARD MODULE.

Peter Sin

University of Florida

Abstract. This paper studies the permutation representation of a finite symplectic group over a prime field of odd characteristic on the vectors of its standard module. The submodule lattice of this permutation module is determined. The results yield additive formulae for the p-ranks of various incidence matrices arising from the finite symplectic spaces.

Introduction

In this paper, we study the action of the symplectic group Sp(2m, p) on the set of vectors in its standard module. The composition factors of this permutation module have been known for some time ([9], [12], [13]) and so the problem we address here is that of describing the submodule lattice. This turns out to be quite similar to the known structure of this module under the action of the general linear group (See [3] and references cited there). This structural information yields additive formulae for the p-ranks of the incidence matrices between points and isotropic subspaces of fixed dimension in (2m−1)-dimensional projective space over 𝔽_{p}. This generalizes recent work [5] of de Caen and Moorhouse, who worked out the p-rank of the point-line incidence when m = 2.

I wish to thank Eric Moorhouse and Alex Zalesskii for fruitful discussions and for supplying me with copies of their work, out of which this paper grew.

§1. Functions on a finite vector space

1.1. Let p be an odd prime and let V be a 2m-dimensional 𝔽_{p}-vector space with a nonsingular alternating bilinear form ⟨−, −⟩. We shall assume m ≥ 2 avoid trivial exceptions. We fix a symplectic basis e_{1}, … e_{m}, f_{m}, … f_{1} and corresponding coordinates coordinates X_{1}, …, X_{m}, Y_{m}, … Y_{1} so that ⟨e_{i}, f_{j}⟩ = δ_{ij}.

Typeset by AMS-TEX

Supported by NSF grant DMS9701065

Let k be an algebraic closure of 𝔽_{p} and let

A = k[X_{1}, …, X_{m}, Y_{m}, …, Y_{1}]/(X_{i}^{p} − X_{i}, Y_{i}^{p} − Y_{i})_{i=1}^{m}  (1)

be the ring of functions on V. This is the principal object of our study.

1.2. Structure of A as a k GL(V)-module. The the action of GL(V) on A is induced from its action on the polynomial ring k[X_{1}, …, X_{m}, Y_{m}, …, Y_{1}] through linear substitutions of the variables. The k GL(V)-module structure of A is well known; we will give a brief description. When we factor out by the inhomogeneous ideal (X_{i}^{p} − X_{i}, Y_{i}^{p} − Y_{i})_{i=1}^{m} the grading on k[X_{1}, … X_{m}, Y_{m}, …, Y_{1}] is destroyed, leaving only a filtration {F_{e}}_{e=0}^{2m(p−1)}, where

F_{e} = Image in A of polynomials of degree ≤ e,  (2)

and a ℤ/(p − 1)ℤ-grading (from the action of the scalar matrices)

A = ⊕_{d=0}^{p−2}A[d],  (3)

where A[d] is the image of all homogeneous polynomials of degree congruent to d modulo p − 1.

Denote by S(e) the component of degree e in the graded ring

S = k[X_{1}, …, X_{m}, Y_{1}, …, Y_{m}]/(X_{i}^{p}, Y_{i}^{p})_{i=1}^{m}.  (4)

Here e ranges from 0 to 2m(p − 1). The dimension of S(e) is

s(e) = ∑_{i=0}^{⌊ \frac{e}{p} ⌋}(−1)^{i}\binom{2m}{i}\binom{2m − 1 + e − ip}{2m}.  (5)

The modules S(e) are simple modules for GL(V) and since the graded module of A associated with the filtration {F_{e}} is isomorphic to S, this filtration is in fact a composition series of A. The filtration {F_{e}} also induces composition series on each direct summand A[d]. We have F_{e} ∩ A[d] = F_{e−1} ∩ A[d] unless e ≡ d mod p − 1, so we set A[d]_{t} = F_{d+t(p−1)} ∩ A[d] (and A[d]_{−1} = {0}). Then for [d] ≠ [0]

{0} ⊂ A[d]_{0} ⊂ A[d]_{1} ⊂ ⋯ ⊂ A[d]_{2m−1}  (6)

is a composition series of A[d], with

A[d]_{t}/A[d]_{t−1} ≅ S(d + t(p − 1));  (7)

the same holds for A[0] except that the series has one extra term A[0]_{2m}.

The following has been known for a long time in one form or other, e.g. [7], [10]. A generalization to all finite fields and further references can be found in [3].

Lemma 1.

(a) For d ≠ 0, A[d] is a uniserial module of dimension \frac{p^{2n}−1}{p−1}.

(b) In A[0] the top and bottom factors A[0]_{0} ≅ k and A[0]_{2m}/A[0]_{2m−1} ≅ k split off and

A[0] ≅ k ⊕ k ⊕ M,  (8)

where M ≅ A[0]_{2m−1}/A[0]_{0} is uniserial of dimension \frac{p^{2n}−1}{p−1} − 1.

Since we will be looking closely at the module M, we let M_{r} be the image in M of A[0]_{r} for 1 ≤ r ≤ 2m − 1. (This is the subspace of M generated by images of polynomials of degree ≤ r(p − 1).) Then the series

{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{2m−1} = M  (9)

is the unique composition series of M as a k GL(V)-module with

M_{t}/M_{t−1} ≅ S(t(p − 1)),  1 ≤ t ≤ 2m − 1.  (10)

(By convention we set M_{0} = {0}.)

The submodule lattice for SL(V) is identical, since this subgroup has index prime to p and all composition factors remain simple when restricted to SL(V).

1.3. Structure of A as a k Sp(V)-module. The k Sp(V)-composition factors of A, or, what amounts to the same thing, of the modules S(e), were given independently in [9] and [12].

Lemma 2. The modules S(e) all remain simple for Sp(V) with the single exception of the middle degree e = m(p − 1), where S(m(p − 1)) is the direct sum of two simple modules S^{+}(m(p − 1)) and S^{−}(m(p − 1)) of dimensions \frac{1}{2}(s(m(p − 1) + p^{n})) and \frac{1}{2}(s(m(p − 1) − p^{n})) respectively.

More information can be found in [9] an [12]. Since S(e) and S(2m(p − 1) − e) are dual as k GL(V)-modules they are isomorphic for Sp(V).

We are now ready to state our main result concerning the k Sp(V)-submodule lattice of A. Taking into account the decompositions (3) and (8), it suffices to describe the submodule lattices of A[d], [d] ≠ [0] and of the nontrivial summand M of A[0]. Since the simple modules in the layers A[d]_{r}/A[d]_{r−1} and M_{r}/M_{r−1} have already been described above, we will not repeat that information here.

Theorem 1.

(a) For [d] ≠ [0] the module A[d] is uniserial.

(b) The modules M_{t} form the socle (and radical) filtration of M. The quotients M_{t}/M_{t−1} ≅ are simple except that M_{m}/M_{m−1} is the direct sum of two simple modules.

Remarks.(1) In other words, the theorem says that the socle and radical series of A are the same for GL(V) and Sp(V), with all composition factors remaining irreducible, except for one which splits into two.

(2) Pictorially, M has the following structure.

S((2m − 1)(p − 1))
S((2m − 2)(p − 1))
⋮
S((m − 1)(p − 1))
S^{+}(m(p − 1)) ⊕ S^{−}(m(p − 1))
S((m − 1)(p − 1))
⋮
S(2(p − 1))
S(p − 1).

(3) As has already been mentioned, we have S((2m − r)(p − 1)) ≅ S(r(p − 1), (1 ≤ r ≤ m).

1.4. The permutation module on projective space and p-rank problems.

Now A[0] is the subspace of functions on V which are unchanged by scalar multiplication of the coordinates, it may be considered as the space of functions on the disjoint union of the projective space ℙ(V) and the zero subspace of V. Thus, the permutation module k[ℙ(V)] on ℙ(V) is isomorphic to k ⊕ M

The structure of the permutation module k[ℙ(V)] follows immediately from part (b) of the theorem.

In finite projective geometry, the incidence relations between objects of two types is encoded in an incidence matrix, with rows labeled by the objects of one type and columns by those of the second and entries 1 or 0 according to whether or not the corresponding row and column labels are incident. It is natural to ask about the rank of this matrix, over any field. When the geometry arises from a field of characteristic p, then one may be interested in the rank over a field of characteristic p, or p-rank. An important geometry of this kind is the geometry of a symplectic vector space V in characteristic p and a typical problem is to determine the p-rank of the incidence between points and isotropic r-dimensional linear subspaces (isotropic r-flat for short) of the projective space ℙ(V). When 2m = 4 and r = 1, the geometry of points and isotropic lines is an example of a generalized quadrangle In this case, the p-rank has been found recently by de Caen and Moorhouse [5].

In general, the rank is equal to the dimension of the k Sp(V)-submodule of k[ℙ(V)] generated by the characteristic function of a fixed r-flat. This submodule can be found without much trouble thanks to part (b) of the theorem, yielding the following numerical result.

Theorem 2. The p-rank of the incidence matrix between points and isotropic r-flats (r = 0, …, m − 1) of ℙ(V) is equal to

\{ \begin{matrix}1 + ∑_{i=1}^{2m−1−r} s(i(p − 1))  for r ≠ m − 1, \\ 1 + ∑_{i=1}^{m−1} s(i(p − 1)) + \frac{1}{2}(s(m(p − 1)) + p^{n})  for r = m − 1. \end{matrix}

(The numbers s(e) were defined in (5) above.)

The formula for r ≠ m−1 agrees with Hamada’s formula [7] for the p-rank of the incidence between points and all r-flats This shows that the code generated by isotropic r-flats is equal to the code generated by all r-flats, except when r = m − 1.

§2. Technical preliminaries

2.1. Characters of the diagonal subgroup. The diagonal subgroup H of the symplectic group Sp(V) consists of all matrices of the form diag(μ_{1}, …, μ_{m}, μ_{m}^{−1} …, μ_{1}^{−1}) where the μ_{i} are nonzero elements of 𝔽_{p}. Let δ_{i}, 1 ≤ i ≤ m, be the map which sends such a matrix to its diagonal entry μ_{i}^{−1}. The maps δ_{i} generate the group of characters of H and each character is uniquely expressible as a product ∏_{i=1}^{m} δ_{i}^{c_{i}}, with c_{i} ∈ ℤ/(p − 1)ℤ.

Each monomial ∏_{i=1}^{m} X_{i}^{a_{i}}Y_{i}^{b_{i}} (or its image in A) is a simultaneous eigenvector for H, affording the character ∏_{i=1}^{m} δ_{i}^{[a_{i}−b_{i}]}, where [a] denotes the congruence class modulo p − 1 of the integer a.

2.2. Induced modules. Let G = Sp(V). For a subgroup X of G and a kX-module L, Let ind_{X}^{G}L denote the induced kG-module kG ⊗_{kX} L. If L = k then the induced module is just the permutation module of G acting on the left cosets of X. In this subsection and the next, we collect together some facts about modules induced from one-dimensional modules of stablizers of flags of isotropic subspaces. These subgroups are parabolic subgroups and so our statements are really special cases of the general theory of such modules developed in [4].

Let G_{r} ≤ G denote the stabilizer of the r-dimensional isotropic subspace W_{r} = ⟨e_{1}, …, e_{r}⟩ of V.

Lemma 3. There exists a unique nontrivial simple kG-module L(r, k) which contains a trivial kG_{r}-submodule. Moreover different choices of r give nonisomorphic modules.

Proof. This is a standard fact in the theory of representations of Chevalley groups [11], so we will give a brief summary of the relevant facts in lieu of a proof. Let ω_{r} be the r-th fundamental weight in the usual order where ω_{1} is the highest weight of V. We view ω_{r} as a character of the diagonal subgroup of the algebraic group Sp(V ⊗_{𝔽_{p}} k). The simple modules for Sp(V ⊗_{𝔽_{p}} k) are parametrized by their highest weights. Each nonnegative integral combination ∑ n_{j}ω_{j} is the highest weight of a unique simple module. The modules whose highest weights are one of the p^{m} combinations with 0 ≤ n_{j} ≤ p − 1, remain simple on restriction to Sp(V) and form a full set of simple k Sp(V)-modules. The simple modules on which G_{r} leaves a line invariant are precisely those with highest weight a multiple of ω_{r} and the stable line is the highest weight space. Thus, the ω_{r} weight space in the module with highest weight ω_{r} is a one-dimensional kG_{r}-module. This representation generates the group (under tensor product) of one-dimensional representations of G_{r}. Thus, the high weight spaces in the modules of highest weight nω_{r}, 0 ≤ n ≤ p − 1 give all p − 1 one dimensional representations of G_{r}, with the trivial representation occuring for the two values n = 0 and n = p − 1. So the module L(r, k) is the simple module with highest weight (p − 1)ω_{r}.

Lemma 4.

ind_{G_{r}}^{G} k ≅ k ⊕ M(r, k)  (11)

where M(r, k) has a unique simple submodule and a unique simple quotient. Both submodule and quotient are isomorphic to L(r, k).

Proof. These statements follow from Lemma 3 by applying Frobenius reciprocity and selfduality of ind_{G_{r}}^{G} k, using the fact that all simple kG-modules are isomorphic to their duals.

To relate this to our earlier notation, we observe that ind_{G_{r}}^{G} k is the permutation module on the set of r-dimensional subspaces of V, so in particular M(1, k) = M, the nontrivial summand of k[ℙ(V)]. The simple modules L(r, k) can also be recognized; comparison of highest weights shows that L(r, k) ≅ S(r(p − 1)), for r ≠ m and L(m, k) ≅ S^{+}(m(p − 1)).

2.3. Incidence maps. For any subset X of V we denote by χ_{X} ∈ A its characteristic function. If X happens to be a homogeneous subset, such as a linear subspace, we will use the same notation for the corresponding characteristic function on ℙ(V). Thus, we may regard characteristic funtions of subspaces as elements of the permutation module ind_{G_{1}}^{G}(k) = k[ℙ(V)].

For each r = 1, …, m, we have incidence maps from the permutation module on isotropic r-subspaces to the permutation module on projective space given by

α_{r} : ind_{G_{r}}^{G}(k) → k[ℙ(V)],  W ↦ χ_{W}  (12)

and

β_{r} : ind_{G_{r}}^{G}(k) → k[ℙ(V)],  W ↦ χ_{W^{⊥}}.  (13)

Of course, α_{m} = β_{m}. These are maps of kG-modules.

All the modules ind_{G_{r}}^{G}(k) have a trivial summand (the constant functions) and a more interesting summand M(r, k). Moreover it is easily checked that the incidence maps map the constant functions onto the constant functions. Therefore, it is technically more convenient to work modulo the trivial summands and consider the maps

M(r, k) → ind_{G_{r}}^{G}(k) ⟶^{α_{r},β_{r}} ind_{G_{1}}^{G}(k) → M(1, k) = M,  (14)

induced by restriction and projection. We shall call these maps \overline{α}_{r} and \overline{β}_{r} and refer to them also as incidence maps.

Let C_{r−1} the image of α_{r} and let C_{2m−r−1} be the image of β_{r}. (The subscripts are the projective dimensions of the supports of the characteristic functions.) The images of \overline{α}_{r} and \overline{β}_{r} will be denoted by \overline{C}_{r−1} and \overline{C}_{2m−r−1} respectively. Thus

C_{t} ≅ k ⊕ \overline{C}_{t}  (15)

for all t = 0, …, 2m − 2. An important property of the module \overline{C}_{r−1} is that, being a homomorphic image of M(r, k), it has a unique maximal submodule.

Lemma 5. For 0 ≤ t < t′ ≤ 2m − 1, we have C′_{t} ⊊ C_{t} and \overline{C}′_{t} ⊊ \overline{C}_{t}.

Proof. We may assume t′ = t + 1. If 0 ≤ r ≤ m − 1, C_{r} is spanned by (characteristic functions of) isotropic r-flats, while if m ≤ r ≤ 2m − 2, it is generated by the orthogonal complements of isotropic 2m − 2 − r-flats. These will be the only two types of flats we will consider in this proof. Let W be a fixed t + 1-flat and P a point of W. Then it is a simple count to check that the number of t-flats in W which contain P is congruent to 1 mod p. Thus the sum of the characteristic functions of those t-flats is equal to the characteristic function of W. This proves that C_{t+1} ⊆ C_{t} and strict containment follows from the fact that the unique simple quotients of C_{t} and C_{t+1} are not isomorphic, by the last assertion of Lemma 3. The lemma is proved.

In the proof of Theorem 1(a) will also want to consider maps from induced modules into the modules A[d]. The relevant parabolic subgroups are the stabilizers G_{r−1,r} = G_{r} ∩ G_{r−1} of the flags W_{r−1} ⊂ W_{r}.

Let λ be a one-dimensional representation of G_{r−1,r}. By Frobenius reciprocity,

Hom_{G}(ind_{G_{r−1,r}}^{G} λ, A[d]) ≅ Hom_{G_{r,r−1}}(λ, A[d]).  (16)

Thus, such homomorphisms exist precisely when A[d] has a one-dimensional kG_{r−1,r}-submodule isomorphic to λ and in this case, the image of the kG-homomorphism is generated by this one-dimensional space.

Now G_{r−1,r} acts on W_{r} leaving the hyperplane W_{r−1} invariant. Hence in its action on the dual of W_{r}, G_{r−1,r} stablizes the one-dimensional subspace spanned by the image of X_{r}. We denote this one-dimensional representation by λ_{r}

Consider the function χ_{W_{r}}X_{r}^{d} for 0 ≤ d ≤ p − 2. Its image in A lies in A[d] (since χ_{W_{r}} ∈ A[0]). Moreover, the span of this function is G_{r−1,r}-stable and affords the representation λ_{r}^{d}.

In this way, we have for each r = 1, …,m, and 0 ≤ d ≤ p − 2 we have homomorphisms

α_{r}[d] : ind_{G_{r−1,r}}^{G} λ_{r}^{d} → A[d],  (17)

with image generated by χ_{W_{r}}X_{r}^{d}.

Now the one-dimensional kG_{r−1,r}-module W_{r−1}^{⊥}/W_{r}^{⊥} is spanned by the image of Y_{r}, and isomorphic to the dual λ_{r}^{∗} of λ_{r}. It follows that for 0 ≤ d ≤ p−2 the function χ_{W_{r−1}^{⊥}}Y_{r}^{d} spans a kG_{r−1,r}-submodule of A[d] isomorphic to λ_{r}^{∗d} and generates the image of a kG-module homomorphism

β_{r}[d] : ind_{G_{r−1,r}}^{G} λ_{r}^{∗d} → A[d],  (18)

Note that for d = 0, the image is C_{r}. In fact, one can show that α_{r}[0] (resp. β_{r}[0]) is the composite of α_{r} (resp. β_{r}) with the natural projection ind_{G_{r−1,r}}^{G} k → ind_{G_{r}}^{G} k sending a flag to its r-dimensional (resp. (n − r)-dimensional) member. In this sense, we have generalized the incidence maps above.

The main property we shall need is the following [4, Thm 6.13].

Lemma 6. For d ≠ 0 the modules ind_{G_{r−1,r}}^{G} λ_{r}^{d} and ind_{G_{r−1,r}}^{G} λ_{r}^{∗d} have unique maximal submodules.

Thus, the images of the maps α_{r}[d] and β_{r}[d] provide 2m submodules of A[d], each with a unique maximal submodule. These submodules will be important in our proof of Theorem 1(a) because, as we already know, A[d] has exactly 2m composition factors.

§3. Proofs of Theorems

The following lemmas will be combined to prove Theorem 1(b) and Theorem 2.

The characteristic function of the linear subspace W defined by the vanishing of coordinates X_{i}, i ∈ I and Y_{j}, j ∈ J is given by

χ_{W} = ∏_{j∈J}(1 − Y_{j}^{p−1}) ∏_{i∈I}^{m}(1 − X_{i}^{p−1})  (19)

Therefore, the characteristic function of an (r − 1)-flat of ℙ(V) is the image of a polynomial of degree 2m − r, making it clear that \overline{C}_{t} ⊆ M_{2m−1−t} for 1 ≤ t ≤ 2m − 1.

Lemma 7. For t ≤ m − 1 \overline{C}_{2m−1−t} = M_{t}

Proof. It is immediate from Lemma 5 and (10) that the submodule \overline{C}_{2m−1−t} of M_{t} has at least as many composition factors as M_{t} itself, so the two must be equal.

Lemma 8. M_{m−1} and M/M_{m} are uniserial modules.

Proof. Since the modules \overline{C}_{r} are images of the modules M(r, k), they have unique maximal submodules. Therefore, by Lemma 7, each of the modules in the series M_{1} ⊂ M_{2} ⊂ ⋯ M_{m−1} has a unique maximal submodule. The uniseriality of M/M_{m} follows by duality.

In view of Lemma 8, we can focus attention on the subquotient M_{m+1}/M_{m−1}. In particular, we wish to determine the submodules \overline{C}_{m−1}/M_{m−1} ≤ M_{m}/M_{m−1} and \overline{C}_{m−2}/M_{m−1} ≤ M_{m+1}/M_{m−1}

Lemma 9. \overline{C}_{m−1}/M_{m−1} ≅ S^{+}(m(p − 1)).

Proof. By definition, \overline{C}_{m−1}/M_{m−1} is the submodule of M_{m}/M_{m−1} ≅ S^{+}(m(p − 1)) ⊕ S^{−}(m(p − 1)) generated by the image of χ_{1,... ,m}. But modulo M_{m−1} we have χ_{1,... ,m} = (−1)^{m}X_{1}^{p−1} ⋯ X_{m}^{p−1}, by (19). This monomial is in the S^{+}(m(p−1)) summand of M_{m}/M_{m−1} (where it is in fact the highest weight vector).

We now come to the key lemma.

Lemma 10. \overline{C}_{m−2} = M_{m+1}

Proof. We begin by examining the case m = 2. In this case C_{m−2} = k^{P} so the result is trivial. For m > 2 we shall argue by induction. Let v^{+} denote the monomial X_{1}^{p−1} ⋯ X_{m}^{p−1} and let v^{−} be the binomial X_{1}^{p−1} ⋯ X_{m−2}^{p−1}X_{m−1}^{p−2}X_{m}^{p−1}Y_{m} −X_{1}^{p−1} ⋯ X_{m−2}^{p−1}X_{m−1}^{p−1}X_{m}^{p−2}Y_{m−1}.

The results of [9] and [12] show that the images in M_{m}/M_{m−1} ≅ S^{+}((m − 1)(p − 1)) ⊕ S^{−}((m − 1)(p − 1)) of v^{+} and v^{−} lie in the S^{+}((m − 1)(p − 1)) and S^{+}((m − 1)(p − 1)) components respectively.

\overline{C}_{m−2} is the k Sp(V)-submodule of M generated by the image of the characteristic function of the m-plane where X_{1} = ⋯ = X_{m} = Y_{m} = 0. This function is given by

χ_{1,... ,m,m} = (1 − Y_{m}^{p−1}) ∏_{i=1}^{m}(1 − X_{i}^{p−1})  (20)

The image of this function in M/M_{m−1} is

\overline{χ}_{1,... ,m,m} = (−1)^{m+1}(X_{1}^{p−1} … X_{m}^{p−1}Y_{m}^{p−1} − X_{1}^{p−1} … X_{m}^{p−1} − ∑_{|I|=m−1} X_{I}^{p−1}Y_{m}^{p−1}), (21)

where X_{I} denotes the product of the variables X_{i} with i ∈ I. Let m = {1, …, m} Applying the symplectic transvection X_{1} ↦ X_{1} + ηY_{1} to (−1)^{m+1}\overline{χ}_{1,... ,m,m} yields

(∑_{r=0}^{p−1} η^{r}\binom{p − 1}{r}X_{1}^{p−1−r}Y_{1}^{r}) (X_{m\\{1\}}^{p−1}Y_{m}^{p−1} − ∑_{\begin{matrix}1∉J⊆m \\ |J|=m−2 \end{matrix}} X_{J}^{p−1}Y_{m}^{p−1}) − X_{m\\{1\}}^{p−1}Y_{m}^{p−1}. (22)

The δ_{1}^{[−2]}, component is

(η(p − 1)X_{1}^{p−2}Y_{1} + η^{\frac{p+1}{2}} \binom{p − 1}{\frac{p+1}{2}}X1^{\frac{p−3}{2}} Y1^{\frac{p+1}{2}} ) (X_{m\\{1\}}^{p−1}Y_{m}^{p−1} − ∑_{\begin{matrix}1∉J⊆m \\ |J|=m−2 \end{matrix}} X_{J}^{p−1}Y_{m}^{p−1})

(23)

By subtracting this expression for η = 1 from the corresponding expression when η is a primitive p-th root of unity (so that η^{\frac{p−1}{2}} = −1) after first dividing the latter by −η, we see that the k Sp(V)-module generated by \overline{χ}_{1,... ,m,m} contains the element

X_{1}^{p−2}Y_{1} (X_{m\\{1\}}^{p−1}Y_{m}^{p−1} − ∑_{\begin{matrix}1∉J⊆m \\ |J|=m−2 \end{matrix}} X_{J}^{p−1}Y_{m}^{p−1})  (24)

On applying the symplectic transvection Y_{1} ↦ Y_{1} + X_{1} and taking fixed points of H, we see that the module in question contains

X_{1}^{p−1} (X_{m\\{1\}}^{p−1}Y_{m}^{p−1} − ∑_{\begin{matrix}1∉J⊆m \\ |J|=m−2 \end{matrix}} X_{J}^{p−1}Y_{m}^{p−1}) .  (25)

Write V = V_{1} ⊕ V_{m−1} where V_{1} is the 2-dimensional hyperbolic space where X_{i} = Y_{i} = 0 for i > 1 and V_{m−1} is its orthogonal complement. Inside Sp(V) is the subgroup Sp(V_{1}) × Sp(V_{m−1}), preserving this decomposition. With respect to this group, we have a tensor product decomposition

A = A_{1} ⊗ A_{m−1},  (26)

with A_{m−1} = k[X_{2}, … X_{m}, Y_{m}, …, Y_{2}]/(X_{i}^{p} − X_{i}, Y_{i}^{p} − Y_{i})_{i=2}^{m}. Then (25) is equal to X_{1}^{p−1}\overline{χ}_{2,... ,m,m}, where χ_{2,... ,m,m} ∈ A_{m−1} is the characteristic function of an m − 2 dimensional subspace of V_{m−1}. By induction, we know that the images in A_{m−1} of the monomial v_{m−1}^{+} = X_{m\{1}}^{p−1} and the binomial

v_{m−1}^{−} = X_{m\{1,m−1,m}}^{p−1}X_{m−1}^{p−2}X_{m}^{p−1}Y_{m} − X_{m\{1,m−1,m}}^{p−1}X_{m−1}^{p−1}X_{m}^{p−2}Y_{m−1}  (27)

lie in the k Sp(V_{m−1})-module generated by \overline{χ}_{2,... ,m,m}, modulo terms of degree less than (m − 2)(p − 1).

Therefore, in A, the images of v^{+} = X_{1}^{p−1}v_{m−1}^{+} and v^{−} = X_{1}^{p−1}v_{m−1}^{−}, lie in the k Sp(V_{m−1})- submodule generated by X_{1}^{p−1}\overline{χ}_{2,... ,m,m}, modulo terms of degree less than (m − 1)(p − 1). Therefore, we have proved that the images of both v^{+} and v^{−} lie in the k Sp(V (p))- submodule of M_{m+1} generated by \overline{χ}_{1,... ,m,m}, which shows that both composition factors of S(m(p − 1)) are composition factors of \overline{C}_{m−2} and hence proves that \overline{C}_{m−2} ⊇ M_{m}. Since the characteristic function of an isotropic m − 2-flat lies in \overline{C}_{m−2} and has nonzero image in the simple module M_{m+1}/M_{m}, it follows that \overline{C}_{m−2} = M_{m+1}.

Proof of Theorem 1(b).

It follows from Lemma 10 that M_{m} is the unique maximal submodule of M_{m+1} and by duality that M_{m−1}/M_{m−2} is the unique simple submodule of M/M_{m−2}. The submodule lattice of M is determined.

Proof of Theorem 2. The p-rank for points versus r-flats is the dimension of C_{r}. From (15) and Lemmas 8, 9 and 10 we obtain the composition factors and from (5) we have the dimension of each composition factor.

Proof of Theorem 1(a). Finally, we turn to the proof of Theorem 1 (a).

First, we note that the image of α_{r}[d] lies in A[d]_{2m−r} and has nonzero image mod A[d]_{2m−r−1}. Thus, the unique simple quotient of ind_{G_{r−1,r}}^{G} λ_{r}^{d} is isomorphic to S(d + (2m − r)(p − 1). Similarly, β_{r}[d] maps ind_{G_{r−1,r}}^{G} λ_{R}^{∗d} into A[d]_{r−1} and the image maps onto the simple module A[d]_{r−1}/A[d]_{r−2} ≅ S(d + (r − 1)(p − 1)) . In order to prove the uniseriality of A[d], it suffices to prove that of A[d]_{r}/A[d]_{r−2} for 1 ≤ r ≤ 2m−1. We start by assuming that 1 ≤ r ≤ m − 1. Then we know that the image of β_{r+1}[d] has a unique maximal submodule, so it it is enough to show that A[d]_{r}/A[d]_{r−2} is generated as a k Sp(V)-submodule by the image of Y_{r+1}^{d}χ_{W_{r}^{⊥}}, which in turn will follow if we can show that this submodule contains a nonzero image of a polynomial of degree d + (r − 1)(p − 1). We observe that Y_{r+1}^{d}χ_{W_{r}^{⊥}} is a polynomial in the Y_{i}’s only. We consider the decomposition

V = V_{e} ⊕ V_{f} = ⟨e_{1}, …, e_{m}⟩ ⊕ ⟨f_{1}, …, f_{m}⟩,  (28)

which gives the factorization A = A(e) ⊗ A(f), where A(e) and A(f) are the images of A of all polynomials in the X_{i}’s and Y_{i}’s respectively. The stabilizer of this decomposition induces the full general linear group on V_{e}, hence any linear substitution among the Y_{i}’s. Since A(f) is isomorphic to the ring of functions on V_{f} we can make use of the known structure of this module for GL(V_{e}). Namely, the component A(f)[d], which is the image of polynomials with degrees congruent to d mod p − 1 is uniserial. This means the k GL(V_{f})-submodule generated by any polynomial in the Y_{i}’s of degree d + r(p − 1) which is nonzero mod A[d]_{r−1} contains the images of all monomials in the Y_{i}s of degree d + (r − 1)(p − 1) Thus, the same is true for the k Sp(V) submodule of A[d], generated by such an element. Therefore, we have established that A[d]_{r}/A[d]_{r−2} is uniserial for 1 ≤ r ≤ m − 1. Since A[d] is dual to A[p − 1 − d] for all d it follows that A[d]_{r}/A[d]_{r−2} is uniserial for m + 1 ≤ r ≤ 2m − 1.

It remains to show that A[d]_{m}/A[d]_{m−2} is uniserial. Since the submodule genearted by the image of X_{m}^{d}χ_{W_{m}} has a unique maximal submodule, it is enough to show that this submodule contains the nonzero image of a polynomial of degree d+(m−1)(p−1). Without loss, we may replace X_{m}^{d}χ_{W_{m}} by X_{1}^{d}χ_{W_{m}}. Modulo A[d]_{m−2}, we have

(−1)^{m}X_{1}^{d}χ_{W_{m}} = X_{1}^{d} (∏_{i=1}^{m} Y_{i}^{p−1} − ∏_{i=2}^{m} Y_{i}^{p−1} − ∑_{J} Y_{1}^{p−1}Y_{J}^{p−1})  (29)

where J runs over all subsets of size m−2 of {2, … m}. Since terms of degree d+(m−2)(p−1) are zero, the right hand side of (29) can be written as

X_{1}^{d}(Y_{1}^{p−1} − 1)q(Y_{2}, …, Y_{m})  (30)

for some polynomial q(Y_{2}, …, Y_{m}) , of degree (m − 1)(p − 1).

We shall consider again the decompositions V = V_{1}⊕V_{m−1}, A = A_{1}⊗A_{m−1} and Sp(V) = Sp(V_{1}) × Sp(V_{m−1}) from (26) above. The subring A_{1} is isomorphic to the ring of functions on a 2-dimensional 𝔽_{p}-vector space and its structure under the action of Sp(V_{1}) = SL(V_{1}) is given by 1.2. In particular, the component A_{1}[d] (consisting of images of polynomials in X_{1} and Y_{1} of degree congruent to d mod p − 1) is a nonsplit extension of two simple modules. The simple submodule has a basis of images of monomials of degree d and the quotient has as basis of images of monomials of degree d + (p − 1). Moreover, A_{1}[d] is generated by X_{1}^{d}(Y^{p−1} − 1). (Note that −(Y^{p−1} − 1) is the characteristic function of the subspace spaaned by e_{1} in V_{1}.) Returning to (30), we now see that since the k Sp(V_{1})- submodule of A_{1}[d] generated by X_{1}^{d}(Y^{p−1} − 1) contains X_{1}^{d}, the k Sp(V_{1})-submodule of A[d]_{m}/A[d]_{m−2} generated by (30) contains X_{1}^{d}q(X_{2}, …, X_{m}), which is a nonzero element of A[d]_{m−1}/A[d]_{m−2}. Our proof is complete.

Concluding remarks. The cases of the same problem for p = 2 or with p replaced by a prime power are, to my knowledge, unsolved. For p = 2, the modules S(e) are the exterior powers of the standard module, so an answer would include the submodule structure of these modules. The composition factors can be found using [8] and are given explcitly in [1]. The paper [2] also gives the submodule structure of the Weyl modules with fundamental highest weights. The exterior powers are known to be filtered by these modules [6, Appendix A].

References

1. A. M. Adamovitch, Analogues of spaces of primitive forms over a field of positive characteristic, Moscow University Mathematics Bulletin 39, No. 1 (1984), 53-56.

2. A. M. Adamovitch, The submodule lattices of Weyl modules for symplectic groups with fundamenstal highest weights, Moscow University Mathematics Bulletin 41, No. 2 (1986), 6-9.

3. M. Bardoe, P. Sin, The permutation modules for GL(n + 1, 𝔽_{p}) acting on ℙ^{n}(𝔽_{p}) and 𝔽_{p}^{n+1}, J. Lond. Math. Soc. (To Appear) (1999).

4. C. W. Curtis, Modular representations of finite groups with split BN-pairs, Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Mathematics 131, Springer, Berlin, 1969, pp. 57-95.

5. D. de Caen, E. Moorhouse, The p-rank of the Sp(4, p) generalized quadrangle, Preprint (1998).

6. S. Donkin, On tilting modules and invariants for algebraic groups, Representations of Algebras and Related Topics , V. Dlab and L.L. Scott, (Ed.), Kluwer, Dordrecht/Boston/London, 1994, pp. 59-77.

7. N. Hamada, The rank of the incidence matrix of points and d-flats in finite geometries, J. Sci. Hiroshima Univ. Ser. A-I 32 (1968), 381-396.

8. J. C. Jantzen, Darstellungen halbeinfacher algebraischer Gruppen un zugeordnete kontravariante Formen, Bonner Math. Schr. 67 (1973).

9. J. Lahtonen, On the submodules and composition factors of certain induced modules for groups of type C_{n}, J. Algebra 140 (1991), 415-425.

10. F. J. MacWilliams, N. J. A. Sloane, Theory of Error Correcting Codes, vol. 2, North Holland, New York, 1977.

11. R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33-56.

12. I. D. Suprunenko, A. E. Zalesskii, Reduced symmetric powers of natural realizations of the groups SL_{m}(P) and Sp_{m}(P) and their restrictions to subgroups, Siberian Mathematical Journal (4) 31 (1990), 33-46.

13. I. D. Supunenko, A. E. Zalesskii, Permutation Representations and a fragment of the decomposition matrix of symplectic and special linear groups over a finite field., Siberian Mathematical Journal (4) 31 (1990), 46-60.

Department of Mathematics, University of Florida, Gainesville, FL 32611, USA e-mail: sin@math.ufl.edu