On the Doubly Transitive Permutation Representations of Sp(2n, 𝔽_{2})

N.S. Narasimha Sastry^{1} Math.-Stat. Division, Indian Statistical Institute 8th Mile, Mysore Road, R.V. College Post Bangalore - 560 059, India

Peter Sin^{2} Department of Mathematics, University of Florida, 358, Little Hall, PO Box 118105 Gainesville, FL 32611–8105, USA

1This work was done during a visit of the second named author to the Indian Statistical Institute, Bangalore in 2000 and a visit of the first named author to the University of Florida in 2001. We wish to thank both institutions for their hospitality. We also thank the International Mathematical Union Committee for Development and Exchange for a travel grant to the first named author.

2Supported by NSF grant DMS0071060

Abstract

Each symplectic group over the field of two elements has two exceptional doubly transitive actions on sets of quadratic forms on the defining symplectic vector space. This paper studies the associated 2-modular permutation modules. Filtrations of these modules are constructed which have subquotients which are modules for the symplectic group over an algebraically closed field of characteristic 2 and which, as such, have filtrations by Weyl modules and dual Weyl modules having fundamental highest weights. These Weyl modules have known submodule structures. It is further shown that the submodule structures of the Weyl modules are unchanged when restricted to the finite subgroups Sp(2n, 2) and O^{±}(2n, 2).

1 Introduction

The problem of determining the submodule structure of the doubly transitive permutation modules was first made explicit in [18]. In that paper, a list of the known doubly transitive groups was given and the cases where the permutation module has three or fewer composition factors were determined. Some of the other cases had already been considered in a different context (see for example [16]) or could be treated by existing methods, while others ([11], [4]) have been solved since, so that the submodule structures of many of the doubly transitive permutation modules in all characteristics are now known. Important classes where the submodule structure has not yet been investigated in detail include the two famous series of doubly transitive actions of the symplectic groups over the field of two elements. One class of these permutation representations is studied in Jordan’s famous treatise [13] and the other first appeared in work of Steiner. We shall therefore follow Mortimer (see [18, 3(D), p.8]) in calling them the Jordan-Steiner actions. These are the actions on the quadratic forms of non-maximal and maximal index respectively which polarize to the given symplectic form. Papers in which these actions have been studied include [14] and [19].

We know from [18] that a Jordan-Steiner permutation module will have a complicated structure only in characteristic 2. As we shall see, the structure is then extremely complicated and there does not seem to us to be any reasonable way to describe the complete submodule lattice. This is not very surprising, since for modular representations in general, it is the exception rather than the rule when such a complete answer is possible. In view of this, we may look instead for suitable filtrations of the modules keeping in mind the following two conflicting aims. The subquotients of the filtration must be simple enough that we can give a detailed description of their submodule structure. At the same time the filtration must be as coarse as possible in order for much of the structure of the whole module to be captured in the subquotients. For example we could consider the radical and socle filtrations. Another idea, from the representation theory of reductive algebraic groups is the notion of a filtration by Weyl modules ([12, p.251]) or one by their duals (called a good filtration; see [12, p.238]).

In this paper, we will construct and study filtrations of the Jordan-Steiner permutation modules which are closely related to Weyl filtrations and good filtrations and have a mixture of Weyl modules and dual Weyl modules as subquotients (Theorem 7.2 and Corollary 6.3). The group G = Sp(2n, 2) is embedded in the obvious way in the algebraic group G(k) = Sp(2n, k), where k is an algebraically closed extension of 𝔽_{2}. Therefore each Weyl module for G(k) is a kG-module by restriction. Now the permutation modules (with coefficients in k) are certainly not modules for G(k) but we shall construct kG-filtrations on them such that the subquotients are restrictions of G(k)- modules which have either Weyl or good filtrations. The Weyl modules involved turn out to have fundamental highest weights and consequently, we can show that their submodule structures with respect to G and G(k) are identical. Moreover, the submodule structure of these Weyl modules has been determined in [2]. In this way, we obtain an incomplete but still useful picture of the permutation modules, which includes the characters and multiplicities of the composition factors and some information about how they fit together.

It has long been known (See (5) below) that restriction of a Jordan-Steiner action to the orthogonal group O(f) ≅ O^{±}(2n, 2) fixing one of the forms f being permuted is isomorphic to the natural action of the orthogonal group on the set of zeroes of f. Our filtrations of the Jordan-Steiner permutation modules therefore yield similar kO(f)-filtrations on the module of functions on a quadric. In §8 we show that the submodule structures of the Weyl modules remain unchanged upon this further restriction from G to O(f).

2 Notation and background

2.1

Let V be a vector space of dimension 2n over a perfect field K of characteristic 2. To avoid trivial exceptions, we shall assume n ≥ 2, except in §5. The spaces S^{2}(V^{∗}) and ∧^{2}(V)^{∗} ≅ ∧^{2}(V^{∗}) are, respectively, the vector space of all quadratic forms and the space of all symplectic bilinear forms on V. By definition, a quadratic form has an associated bilinear form

θ(q)(v, u) = q(v + u) − q(v) − q(u)  (1)

and this formula defines the polarization homomorphism from S^{2}(V^{∗}) to ∧^{2}(V^{∗}). We recall that the “Frobenius twist” V^{(2)} of V is the vector space with the same additive group as V but scalar multiplication of v ∈ V^{(2)} by λ ∈ K gives the same element as scalar multiplication by \sqrt{λ} when v is considered as an element of V. The quadratic forms with zero polarization can be identified with V^{(2)∗} and we have a short exact sequence of GL(V)- modules

\xymatrix{0 \ar[r] & V^{(2)∗} \ar[r] & S^{2}(V^{∗}) \ar[r]^{θ} & ∧^{2}(V^{∗}) \ar[r] & 0.}  (2)

2.2

Fix a nondegenerate symplectic bilinear form b on V and let Sp(V) be its symplectic group. Every element of V^{(2)∗} = ker θ can be written as b(−, x)^{2} for some x ∈ V. So if q is any quadratic form polarizing to b, then

θ^{−1}(b) = {q + b(−, x)^{2} | x ∈ V }.  (3)

Now the group GL(V) acts on S^{2}(V^{∗}) by the formula (gq)(v) = q(g^{−1}v) for all g ∈ G, q ∈ S^{2}(V^{∗}) and v ∈ V. Two quadratic forms in the same orbit are called equivalent. The equivalence class of a quadratic form f is determined by its Arf invariant Δ(f) ∈ K/J [15, Theorem 27, p.33], where J = {λ^{2} + λ | λ ∈ K}. If two forms q and q′ ∈ θ^{−1}(b) are equivalent, the conjugating element of GL(V) must belong to Sp(V), so the Sp(V)-orbits in θ^{−1}(b) are also determined by the Arf invariant. By (3) we know that there is a unique x ∈ V such that q′ = q + b(−, x)^{2}, and it follows from the formula for the Arf invariant [15, p.30] that Δ(q) − Δ(q′) = q(x) in K/J. (See also [6, p. 65], [9, Lemma 1].)

The map λ ↦ λ^{2} + λ is an additive homomorphism of K with kernel {0, 1}. So if K is finite there are two Sp(V)-orbits.

Let e_{1}, …, e_{n}, f_{1}, …, f_{n} be a symplectic basis for V with respect to b and let x_{1}, …, x_{n}, y_{1}, …, y_{n} be the dual basis for V^{∗}. Thus,

b(e_{i}, f_{j}) = x_{i}(e_{j}) = y_{i}(f_{j}) = δ_{i,j}, b(e_{i}, e_{j}) = b(f_{i}, f_{j}) = 0.  (4)

Then when K is finite the quadratic forms q^{+} = ∑_{i=1}^{n} x_{i}y_{i} and q^{−} = x_{1}^{2}+αy_{1}^{2}+∑_{i=1}^{n} x_{i}y_{i}, where α ∉ J, are representatives of the two Sp(V)-orbits. These orbits consist of the forms of Witt index n and n − 1 respectively.

Our main interest is in the case K = 𝔽_{2}. In this case we denote Sp(V) by G and we choose q^{−} with α = 1. We denote the G-orbits containing q^{+} and q^{−} by Q^{+} and Q^{−} respectively, and we denote by Q_{f} the G-orbit of f ∈ θ^{−1}(b). The foregoing discussion shows that

Q_{f} = {f + b(−, v)^{2} | v ∈ V, f(v) = 0}.  (5)

The stabilizer in G of f is the orthogonal group O(V, f). By Witt’s Lemma this group has two orbits on V \ {0} consisting of isotropic and anisotropic vectors. Further, the map sending v to f + b(−, v)^{2} is an isomorphism of O(V, f)-sets from V to θ^{−1}(b) taking the set of isotropic vectors to Q_{f}. It follows that G acts doubly transitively on Q_{f}, which we shall call a Jordan-Steiner set.

The first proof of the double transitivity of the G-action on the forms of maximal index appears in [13, p. 236]. The argument we have given is standard.

Our approach to the study of the permutation modules for the Jordan-Steiner sets will be based on the characterization (5), which will allow us to view these sets as affine subsets of a suitable module for G.

Remark 2.1 We note in passing that when K = 𝔽_{2} Witt’s Lemma and the O(V, f)-isomorphism above also imply that each of the groups O(V, q^{+}) and O(V, q^{−}) acts transitively on the Jordan-Steiner set of the opposite type, namely Q^{−} and Q^{+} respectively.

Our study of the Jordan Steiner permutation modules will be motivated in part by the following heuristical remarks. The quotient of the module θ^{−1}(Kb) by the submodule V^{(2)∗} is a one-dimensional trivial module. Then if K is finite, the KSp(V)-permutation modules given by the action of the group on each of the cosets of V^{(2)∗} in θ^{−1}(Kb), will have the same composition factors. This is because the composition factors are determined by the elements of odd order and the actions of an element of odd order on the |K| cosets are all isomorphic. When K = 𝔽_{2}, we have V^{(2)∗} ≅ V^{∗} and the nontrivial coset is θ^{−1}(b) = Q^{+} ∪ Q^{−}. Therefore, in the Grothendieck group of KSp(V)-modules, the sum of the two Jordan-Steiner permutation modules is equal to the permutation module on the set V^{∗}. The latter is well known to have a filtration (by polynomial degree) such that the graded module is isomorphic to the exterior algebra ∧(V^{∗}) of V^{∗}. (This is in fact the classical construction of the Reed-Muller codes [17].) The exterior powers ∧^{r}(V^{∗}) are fundamental and have been studied in detail by several authors (e.g. [2], [3],

[7] and [10]).

The remainder of this paper is an attempt to make the relation between the Jordan-Steiner permutation modules and ∧(V^{∗}) precise.

3 Orthogonal spaces of dimension 2n + 1

In this section, we study the well known correspondence which exists over perfect fields of characteristic 2 between the geometry of an odd-dimensional orthogonal space and that of a symplectic space of one dimesion lower. Our purpose in doing this is to produce a kG-module where the action of G on the Jordan-Steiner set can be described in coordinates and directly compared with the action of G on ∧(V^{∗}). In the process, we also give proofs of some classical facts since this can be done with no extra effort.

3.1

Let V̂ denote a vector space of dimension 2n + 1 over K. Fix a nonzero vector d̂∈ V̂ and take V above to be the space V̂/Kd̂, with π : V̂ → V the natural map. Let b̂ be the symplectic bilinear form on V̂ defined by

b̂(u, v) = b(π(u), π(v)) for u, v ∈ V̂.  (6)

Let Q̂ denote the set of all nondegenerate quadratic forms q̂ on V̂ such that q̂(d̂) = 1 and b̂(u, v) = q̂(u + v) − q̂(u) − q̂(v) for u, v ∈ V̂. Let q̂ ∈ Q̂ and v ∈ V. Then the map

v̂ ↦ ṽ = v̂ + \sqrt{q̂(v̂)}d̂  (7)

is constant on π^{−1}(v). The element ṽ satisfies π(ṽ) = v and q̂(ṽ) = 0 and is the unique element of V̂ with these properties.

Let Γ(V̂) denote the group of linear automorphisms of V̂ which preserve d̂ and b̂. Let H be the set of all hyperplanes of V̂ which do not contain d̂.

Lemma 3.1 (i) For any q̂ ∈ Q̂, we have Q̂ = {q̂+λ^{2} | λ ∈ V̂^{∗}, λ(d̂) = 0}.

(ii) Γ(V̂) acts transitively on Q̂.

(iii) Γ(V̂) acts transitively on H.

Proof: Part (i) follows from the fact that the difference of two quadratic forms with the same polarization is the square of a linear functional. For λ ∈ V̂^{∗} with λ(d̂) = 0, the transformation t_{λ} : v ↦ v + λ(v)d̂ of V̂ belongs to Γ(V̂) and its induced action on quadratic forms sends q̂+ λ^{2} to q̂. This proves (ii). To prove (iii), we identify H with {ν ∈ V̂^{∗} | ν(d̂) = 1}. For μ, ν in this set, let λ = μ − ν. Then in its action on V̂^{∗} the element t_{λ} sends μ to ν.

Lemma 3.2 Let q̂ ∈ Q̂ and H ∈ H.

(i) The set of O(q̂)-orbits on H is in bijection with the set of Sp(H, b̂|_{H})- orbits on Q̂.

(ii) The permutation action of Sp(H, b̂|_{H}) on Q̂ is isomorphic to the action of Sp(V) on θ^{−1}(b).

(iii) The set of O(q̂)-orbits on H is in bijection with the set of Sp(V)-orbits on θ^{−1}(b).

Proof: The stabilizer in Γ(V̂) of q̂ is the orthogonal group O(q̂) and the stabilizer of H is the symplectic group Sp(H, b̂|_{H}). Therefore, (i) follows from (ii) and (iii) of Lemma 3.1. Each quadratic form on H polarizing to b̂|_{H} has a unique extension to an element of Q̂ and since π induces an isometry of H with V, we have (ii). Then (iii) follows from (i) and (ii).

Example 3.3 Let d̂, ê_{i}, f̂_{i}, i = 1, …, n, be a basis of V̂ such that π(ê_{i}) = e_{i} and π(f̂_{i}) = f_{i} and let ẑ, x̂_{i}, ŷ_{i}, i = 1, …, n, be the dual basis for V̂^{∗}. If K is finite and H is the hyperplane ẑ = 0 then the restrictions to H of q̂^{+} = ∑_{i=1}^{n} x̂_{i}ŷ_{i} and q̂^{+} + x̂_{1}^{2} + αŷ_{1}^{2} (α ∉ J) correspond under π to the representatives q^{+} and q^{−} of the Sp(V)-orbits in θ^{−1}(b).

3.2

Fix q̂ ∈ Q̂. Then π defines a group homomorphism σq̂ from O(q̂) to Sp(V). (The well known fact that this is an isomorphism will follow from our discussion.) Let U = θ^{−1}(Kb). Via σq̂, we may think of the Sp(V)-module U as a module for O(q̂). Also we have natural injections i : V^{(2)∗} → U and π^{(2)∗} : V^{(2)∗} → V̂^{(2)∗}.

Theorem 3.4 There exists a KO(q̂)-module isomorphism ζq̂ from V̂^{(2)∗} to U such that the following diagram commutes.

[figure: V̂^{(2)∗}; ζq̂; U; V^{(2)∗}; π^{(2)∗}; V^{(2)∗}; i]  (8)

For λ ∈ V̂^{(2)∗} the quadratic form ζq̂(λ) is given by the formula

ζq̂(λ)(v) = λ(v̂) + q̂(v̂)λ(d̂),  (9)

where v̂ ∈ V̂ is any preimage of v ∈ V under π.

Proof: For v ∈ V let ṽ be its preimage in V̂ given in (7). For λ ∈ V̂^{(2)∗}, we define Tq̂(λ) : V → K given by Tq̂(λ)(v) = λ(ṽ). Then Tq̂(λ) is a quadratic form which polarizes to λ(d̂)^{2}b, so Tq̂(λ) ∈ U. Then the map ζq̂ : V̂^{(2)∗} → U given by ζq̂(λ) = Tq̂(λ) is an injective homomorphism of KO(q̂)-modules, hence an isomorphism since both V^{(2)∗} and U have dimension 2n + 1. The commutativity of the diagram is easily checked.

Remark 3.5 Theorem 3.4 gives us the following commutative diagram

\xymatrix{O(q̂) \ar[d]^{σq̂} \ar[r]^{ρ} & GL(V̂^{(2)∗}) \\ Sp(V) \ar[r] & GL(U) \ar[u]^{≈}}  (10)

in which ρ is the representation of O(q̂) on V^{(2)∗} and the right vertical isomorphism is induced by ζq̂. The commutativity of the square shows that ρ may be factored through σq̂. Thus V^{(2)∗} has the structure of a Sp(V)- module. As a homomorphism of abstract groups, the injectivity of ρ implies that σq̂ is injective. Since Sp(V) acts faithfully on U, it follows that σq̂ is also surjective. Thus, we have proved the well known isomorphism (of abstract groups) of O(q̂) with Sp(V). In the case of algebraic groups, the above diagram also shows that the Frobenius map (obtained by composing ρ with inverse-transpose) of O(q̂) factors through the noncentral infinitesimal isogeny σq̂.

4 Coordinates and filtrations

In this section we shall apply the results of the previous sections when the field K is either 𝔽_{2} or an algebraically closed extension k. We wish to view the 𝔽_{2} theory as embedded inside the theory for k, so we need first to clarify our notation. We will keep the notation V, U, V̂, Q̂, ζq̂, θ^{−1}(b), etc. to mean the objects with K = 𝔽_{2} and use parallel notation such as V_{k}, U_{k}, V̂_{k}, Q̂_{k}, ζq̂_{k} for the corresponding k-objects. Further, we shall consider the 𝔽_{2}-objects such as V, Q̂ as subsets of V_{k}, Q̂_{k}, etc.

4.1

We now fix some additional notation. We choose bases for V̂ and V and their duals as in Example 3.3 and use the same bases for the extensions to k. Then under our convention that V_{k}^{(2)} is the same abelian group as V_{k}, the elements e_{i}, f_{i} also form a basis for V_{k}^{(2)}, but because of the different scalar multiplication, we will denote these elements by capitals E_{i} and F_{i} when thinking of them this way. The dual basis will consist of the elements X_{i} = x_{i}^{2} and Y_{i} = y_{i}^{2}. We apply the same convention to V̂_{k}. Thus, our chosen basis for V̂_{k}^{(2)∗} is Ẑ, X̂_{i}, Ŷ_{i}, i = 1, …, n.

We now proceed to describe the G-orbits on θ^{−1}(b) ⊆ U_{k} in coordinates by means of Theorem 3.4. That theorem gives different isomorphisms ζq̂_{k} for different choices of q̂_{k} ∈ Q̂_{k}. Since π_{k}^{(2)∗}(X_{i}) = X̂_{i} and π_{k}^{(2)∗}(Y_{i}) = Ŷ_{i}, the commutativity of the diagram in Theorem 3.4 yields ζq̂_{k}(X̂_{i}) = X_{i} and ζq̂_{k}(Ŷ_{i}) = Y_{i} for all q̂_{k} ∈ Q̂_{k}. The image of Ẑ under ζq̂_{k} will depend on q̂_{k}. Let f ∈ θ^{−1}(b) and Q_{f} be its G-orbit. Theorem 3.4 shows that there is a unique choice of q̂_{f} ∈ Q̂_{k} such that ζq̂_{f} (Ẑ) = f and of course we have q̂_{f} ∈ Q̂.

Let Z_{f} be the G-orbit of Ẑ in V̂^{(2)∗} ⊆ V̂_{k}^{(2)∗} under the action defined by ζq̂_{f}. For v = ∑_{i=1}^{n}(α_{i}e_{i} + β_{i}f_{i}) ∈ V, α_{i}, β_{i} ∈ 𝔽_{2}, we have

b(−, v)^{2} = ∑_{i=1}^{n}(β_{i}X_{i} + α_{i}Y_{i}) ∈ V^{(2)∗} ⊆ V_{k}^{(2)∗}  (11)

so by (3)

ζ−1 ; q̂_{f}(θ^{−1}(b)) = {Ẑ + ∑_{i=1}^{n} β_{i}X̂_{i} + α_{i}Ŷ_{i} | α_{i}, β_{i} ∈ 𝔽_{2}}.  (12)

Since, by (5), f + b(−, v)^{2} ∈ Q_{f} if and only if v ∈ V and f(v) = 0, we have

Z_{f} = {Ẑ + ∑_{i=1}^{n} β_{i}X̂_{i} + α_{i}Ŷ_{i} | α_{i}, β_{i} ∈ 𝔽_{2}, f(∑_{i=1}^{n} α_{i}e_{i} + β_{i}f_{i}) = 0}. (13)

Thus, Z_{f} is the set of 𝔽_{2}-rational points of the affine subset of V̂_{k}^{(2)∗} given by the equations D̂ = 1, f(F̂_{i}, Ê_{i}) = 0. Here we have identified V̂_{k}^{(2)} with its double dual and f(F̂_{i}, Ê_{i}) means the quadratic form on V̂_{k}^{(2)∗} obtained by substituting F̂_{i} for x_{i} and Ê_{i} for y_{i} in f.

4.2

The action of Sp(V_{k}) on V̂_{k}^{(2)} extends to a homogeneous action of Sp(V_{k}) on the polynomial ring

R̂ = k[Ê_{1}, …, Ê_{n}, F̂_{1}, …, F̂_{n}, D̂].  (14)

Since Sp(V_{k}) preserves D̂, the subspaces

F_{t} = {h ∈ R̂ | degree of h in Ê_{i}, F̂_{i} is at most t}  (15)

define a filtration of R̂ by Sp(V_{k})-modules, as t ranges over the natural numbers. The filtrations for its quotient rings S considered below are the images of this filtration under natural surjections. We denote the image of F_{t} in S by F_{t}(S) and the quotient space F_{t}(S)/F_{t−1}(S) by gr_{t}(S).

Let I denote the ideal of R̂ generated by D̂, Ê_{1}F̂_{1} + ⋯ + Ê_{n}F̂_{n}, Ê_{i}^{2} ,F̂_{i}^{2}, i = 1,…,n. This ideal is stable under Sp(V_{k}). For f ∈ θ^{−1}(b), let I_{f} denote the ideal of R̂ generated by D̂ − 1, f(F̂_{i}, Ê_{i}), Ê_{i}^{2} − Ê_{i}, F̂_{i}^{2} − F̂_{i}, i = 1,…, n. This ideal is stable under the action of G. Let A = R̂/I and A_{f} = R̂/I_{f}.

Lemma 4.1 There exists a surjective kG-module homomorphism φ_{f} from gr(A) to gr(A_{f}).

Proof: Let t be a nonnegative integer and M_{t} denote the k-subspace k[Ê_{i}, F̂_{i}]_{i=1}^{n} ∩ F_{t} of R̂. Consider the diagram

\xymatrix{M_{t} \ar[dr]^{η_{t}∘i} \ar@{^{(}->}[r]^{i} & F_{t} \ar@{->>}[r]^{ψ_{t}} \ar@{->>}[d]^{η_{t}} & F_{t}(A_{f})/F_{t−1}(A_{f}) \\  & F_{t}(A) \ar@{->>}[ur]^{_{\overline{ψ}_{t}}}}  (16)

where i is the inclusion map and η_{t}, ψ_{t} are the natural surjections. Note that η_{t} ∘ i is onto with kernel M_{t} ∩ I. The map ψ_{t} ∘ i takes this kernel to zero because its elements can be written as ∑_{i=1}^{n}(Ê_{i}^{2}H_{i} + F̂_{i}^{2}K_{i}) + (∑_{i=1}^{n} Ê_{i}F̂_{i})L with H_{i}, K_{i}, L ∈ M_{t−2} and ψ_{t}∘i takes Ê_{i}^{2}, F̂_{i}^{2} and ∑_{i=1}^{n} Ê_{i}F̂_{i} and all multiples of them to zero. Therefore a k-linear surjective homomorphism \overline{ψ}_{t} exists and the diagram commutes. Since ψ_{t} and η_{t} are kG-module homomorphisms and η_{t} is surjective, it follows that \overline{ψ}_{t} also is a kG-module homomorphism. Since ψ_{t} is surjective, so is \overline{ψ}_{t}. Since F_{t−1} ⊆ Kerψ_{t}, it follows that \overline{ψ}_{t} maps F_{t−1}(A) to zero. Thus there exists a kG-module surjective homomorphism from gr_{t}(A) to gr_{t}(A_{f}).

5 The exterior algebra and the spin module

The results of this section are needed in the proof of Theorem 7.2 to determine the kernel of φ_{t} but may also be of general interest. They hold for any field K of characteristic two. Though Proposition 5.1 below can be extracted from [10], we present a proof of this because our approach is different and makes the presentation of Proposition 5.2 easier.

We consider b as an element of degree two in the exterior algebra ∧(V^{∗}). In characteristic two we have b^{2} = 0. So the map δ given by multiplication by b makes ∧(V^{∗}) into a complex. It is best for us to keep the standard grading on ∧(V^{∗}) so δ has degree two. Obviously, the complex decomposes into a direct sum ∧(V^{∗}) = ∧(V^{∗})^{even} ⨁ ∧(V^{∗})^{odd}. The symplectic group Sp(V) which preserves b acts on this complex, hence also on its homology groups.

Proposition 5.1 H^{i}(∧(V^{∗}), δ) = 0 unless i = n , in which case it affords an irreducible representation of Sp(V) of dimension 2^{n}.

Proof: First consider the case when n = 1. Here the even part is just V^{∗} concentrated in degree 1, while the odd part is δ : K → K. So the proposition is true in this case. For the general case, we consider the decomposition

V = W_{1} ⊕ W_{2} ⊕ ⋯ ⊕ W_{n},  b = b_{1} + b_{2} + ⋯ + b_{n}  (17)

into hyperbolic planes. We have the tensor factorization

∧(V^{∗}) = ⊗_{j=1}^{n} ∧ (W_{j}^{∗})  (18)

under which multiplication by b on the left becomes the map b_{1} ⊗ 1 ⊗ ⋯ ⊗ 1 + 1 ⊗ b_{2} ⊗ ⋯ ⊗ 1 + ⋯ + 1 ⊗ ⋯ ⊗ 1 ⊗ b_{n}. Therefore, (18) is an isomorphism of complexes. (There is no sign to worry about!)

In light of the case n = 1, we now see that the complex ∧(V^{∗}) is a direct sum of subcomplexes of the form M = M_{1} ⊗M_{2} ⊗⋯⊗M_{n}, where each M_{j} is either W_{j}^{∗} concentrated in degree 1, or δ_{j} : K → K. Now the tensor product of exact complexes is exact, as is the tensor product of an exact complex with one concentrated in a single degree, while the tensor product of two complexes each concentrated in a single degree results in one of the same kind. Thus, all subcomplexes of M above are exact, with the sole exception of the one in which all the M_{j} are taken to be W_{j}^{∗}, which is the complex ⊗_{j=1}^{n}W_{j}^{∗} concentrated in degree n. Thus, H^{i}(∧(V^{∗}), δ) = 0 if i ≠ n, while

H^{n}(∧(V^{∗}), δ) = ⊗_{j=1}^{n}W_{j}^{∗}.  (19)

This proves all statements about dimensions. The decomposition into subcomplexes M above is equivariant with respect to the subgroup SL(W_{1})× ⋯ × SL(W_{n}) of SL(V), and so (19) is an isomorphism of modules for this group, in which the right hand side is obviously simple. This completes the proof of the proposition.

If K is algebraically closed, then Sp(V) is a simple algebraic group (of type C_{n}) and the module L we have just constructed is a rational one. Its highest weight must be equal to the highest weight in the module ∧^{n}(V^{∗}), since this weight does not occur in any other exterior power. It is easy to see that this weight is the fundamental weight corresponding to the long fundamental root in the C_{n} root system. The module L is called the spin module for Sp(V).

With e_{i}, f_{i} , x_{i}, y_{i} as in 2.2, we have b = x_{1} ∧ y_{1} + ⋯ + x_{n} ∧ y_{n} and the space ∧^{n}(V^{∗}) has a basis consisting of elements y_{I}x_{J} , where I and J are subsets of N = {1, ⋯, n} such that |I| + |J| = n, x_{I} = ∏_{i∈I} x_{i} and y_{I} = ∏_{i∈I} y_{i}. Let \overline{I} denote the complement of I in N and [y_{I}x_{J}] denote the image of y_{I}x_{J} in L.

Proposition 5.2 The 2^{n} elements [y_{I}x\overline{I}] for I ⊆ N form a basis for the spin module L.

Proof: The elements [y_{I}x\overline{I}] are clearly annihilated by δ. It is not difficult to see they do not lie in δ(∧^{n−2}(V^{∗})) so they have nonzero images in L. To see that these images are linearly independent, there is no loss in assuming that K is algebraically closed. Then we see that they are simultaneous eigenvectors affording distinct characters of the maximal torus of Sp(V) which is the product of the diagonal subgroups of the SL(W_{j}) in the hyperbolic decomposition described in the proof of Proposition 5.1. This proves their linear independence and the proposition.

6 Good filtrations and Weyl filtrations

In this section we assume k to be an algebraically closed extension of the field K of the previous section. Later we will apply the results in the case K = 𝔽_{2}. Let V_{k} = k ⊗_{K} V. It is known [7, Appendix A] that as modules for the algebraic group Sp(V_{k}), the exterior powers have filtrations by Weyl modules ([12, p.251] and, by self-duality, also good filtrations (by duals of Weyl modules); see [12, p.238]).

Proposition 6.1 (i) ∧^{r}(V_{k})/Imδ ∩ ∧^{r}(V_{k}) has a good filtration for 0 ≤ r ≤ n and a Weyl filtration for n + 1 ≤ r ≤ 2n.

(ii) ∧^{r}(V_{k})/Kerδ ∩ ∧^{r}(V_{k}) has a good filtration for 0 ≤ r ≤ n − 1 and a Weyl filtration for n ≤ r ≤ 2n.

Proof: We shall use the general fact [12, II.4.17] that in a short exact sequence

0 → W′ → W → W′′ → 0  (20)

of rational modules for a reductive algebraic group, if W′ and W have good filtrations then so does W′′, together with the dual statement that if W′′ and W have Weyl filtrations then so does W′. This will allow us to argue by induction. It is clear that (i) holds for r = 0 and r = 1. By Lemma 5.1, for 2 ≤ r ≤ n, we have the exact sequence

0 → ∧^{r−2}(V_{k})/Imδ ∩ ∧^{r−2}(V_{k}) → ∧^{r}(V_{k}) → ∧^{r}(V_{k})/Imδ ∩ ∧^{r}(V_{k}) → 0. (21)

Then ∧^{r}(V_{k}) has a good filtration, as mentioned above, and ∧^{r−2}(V_{k})/Imδ ∩ ∧^{r−2}(V_{k}) has one by the inductive hypothesis. So by the general fact mentioned, we have proved (i) by induction. The remainder of the proof of (i) is the same argument using duality and reverse induction and (ii) is proved in the same way as (i).

6.1 Characters

Let ω_{i}, 1 ≤ i ≤ n be the fundamental dominant weights and let V (i) be the Weyl module with highest weight ω_{i}. We label the Dynkin diagram in the usual way so that V (1) is the natural module. We denote by H(i) and L(i) the corresponding dual Weyl module and simple module respectively. In addition, V (0), H(0) and L(0) all mean the trivial module.

By the independence of characters of a maximal torus, the multiplicities of the factors which occur in the good (or Weyl) filtration of ∧^{r}(V_{k}^{∗}) are the same as in characteristic zero; here it is a classical fact that the degree 2 map given by multiplication by the form is injective for degrees < n and that the dual Weyl module is its cokernel. Therefore, we have H(r) = ∧^{r}(V_{k}^{∗}) − ∧^{r−2}(V_{k}^{∗}) for 0 ≤ r ≤ n in the Grothendieck group of Sp(V_{k})-modules, where, as usual, we take modules to be zero if they are indexed by numbers outside the range [0, 2n]. Inverting this equation yields

∧^{r}(V_{k}^{∗}) = ∑_{t=0}^{⌊r/2⌋} H(r − 2t).  (22)

By the same token, the multiplicities of the good (and Weyl) filtration factors of Cokerδ and Imδ depend only on the (formal) characters. They can therefore be computed using (22) and Proposition 5.1.

Proposition 6.2 The Weyl modules and dual Weyl modules which occur in the filtrations of Proposition 6.1 each occur with multiplicity one and are as follows.

(i) For 0 ≤ r ≤ n, ∧^{r}(V_{k})/Imδ ∩ ∧^{r}(V_{k}) has factors H(r − 4t), with 0 ≤ t ≤ ⌊r/4⌋.

(ii) For n < r ≤ 2n, ∧^{r}(V_{k})/Imδ ∩ ∧^{r}(V_{k}) has factors V ((2n − r − 2) − 4t), with 0 ≤ t ≤ ⌊(2n − 2 − r)/4⌋.

(iii) For 0 ≤ r < n, ∧^{r}(V_{k})/Kerδ ∩ ∧^{r}(V_{k}) has factors H(r − 4t), with 0 ≤ t ≤ ⌊r/4⌋.

(iv) For n ≤ r ≤ 2n, ∧^{r}(V_{k})/Kerδ ∩∧^{r}(V_{k}) has factors V ((2n−r −2)−4t), with 0 ≤ t ≤ ⌊(2n − 2 − r)/4⌋.

(Here, ⌊x⌋ denotes the ‘integral part’ of x.)

Corollary 6.3 In the Grothendieck ring of Sp(V_{k})-modules,

(i)

∑_{r=0}^{n} Cokerδ_{r} = ∑_{t=0}^{⌊n/4⌋}(t + 1)[H(n − 4t) + H(n − 4t − 1)
+H(n − 4t − 2) + H(n − 4t − 3)];

(ii)

∑_{r=n+1}^{2n} Cokerδ_{r} = ∑_{t=0}^{⌊(n−3)/4⌋}(t + 1)[V (n − 4t − 3) + V (n − 4t − 4)
+V (n − 4t − 5) + V (n − 4t − 6)].

(iii) Cokerδ = L + Imδ.

As we have seen, it is an easy matter to determine which Weyl modules and dual Weyl modules appear in these filtrations. A harder question is to give the composition factors of the Weyl modules and harder still is the problem of describing the submodule lattices of the Weyl modules. Fortunately, these questions have been answered in the papers [2] and [3] and we will briefly describe the results here. The composition factors of the Weyl modules V (i) also have fundamental highest weights and their multiplicities |V (i) : L(j)| are either 0 or 1. The multiplicity is 1 if and only if i ≥ j, i − j is even and the binary digits of \frac{1}{2}(i − j) are among those of n − j + 1. The fact that the Weyl modules involved are multiplicity-free is of great help in describing their submodule structure. In general, this means that the submodule lattice is the latttice of ideals with respect to a natural ordering on the set of composition factors. The problem then is to give a combinatorial description of the ordering. For the Weyl modules with fundamental highest weights, this has been accomplished by Adamovich [3].

Finally we note that since all of the composition factors of the Weyl modules have fundamental, hence 2-restricted, highest weights, it follows from [1, Proposition 2.7] that when K = 𝔽_{2} the submodule structures of the Weyl modules are the same for the finite group Sp(V) as they are for the algebraic group Sp(V_{k}).

7 Functions on the Jordan-Steiner sets

We now resume the hypotheses and notations of §4 and work with 𝔽_{2} and its algebraically closed extension k.

7.1

Let f be one of the G-orbit representatives q^{+}, q^{−} in θ^{−1}(b) given in 2.2. We now consider the rings A and A_{f} from 4.2.

Then Sp(V_{k}) stabilizes the nondegenerate symplectic bilinear form ∑_{i=1}^{n} F_{i}∧ E_{i} on V_{k}^{(2)∗} and we have

A ≅ \frac{k[E_{1}, ⋯, E_{n}, F_{1}, ⋯, F_{n}]}{< E_{i}^{2}, F_{i}^{2}, ∑_{i=1}^{n} E_{i}F_{i} >} ≅ \frac{∧(V_{k}^{(2)})}{< ∑_{i=1}^{n} F_{i} ∧ E_{i} >}  (23)

as Sp(V_{k})-modules.

Let δ denote the Sp(V_{k})-module endomorphism of ∧(V_{k}^{(2)∗}) given by the left multiplication by the above symplectic bilinear form. Then (23) and Proposition 5.1 now yield the following lemma.

Lemma 7.1 (i) A ≅ Coker(δ) as Sp(V_{k})-modules;

(ii) A contains a simple Sp(V_{k})-submodule isomorphic to the spin module L = Kerδ/Imδ;

(iii) dim_{k} Im(δ) = 2^{n−1}(2^{n} − 1) and dim_{k}(Kerδ) = 2^{n−1}(2^{n} + 1); and (iv) dim_{k}(A) = 2^{n−1}(2^{n} + 1).

We view A_{f} as the space of all k-valued functions on V_{f}.

Theorem 7.2 Let φ_{t} be the kG-module homomorphism from gr_{t}(A) onto gr_{t}(A_{f}) defined in Lemma 4.1.

(i) If f = q^{+}, then φ_{t} is an isomorphism for all t.

(ii) If f = q^{−}, then φ_{t} is an isomorphism for degrees t with t ≠ n. The kernel of φ_{n} is the image in gr_{n}(A) of the simple kG-submodule of Lemma 7.1(ii).

Thus, we have gr(A_{q^{+}}) ≅ Cokerδ and gr(A_{q^{−}}) ≅ Imδ as kG-modules.

Proof: Since dim_{k} gr(A_{f}) = dim_{k} A_{f} = |V_{f}| = |{a ∈ V | f(a) = 0}| = 2^{n−1}(2^{n} + 1) or 2^{n−1}(2^{n} − 1) according as f = q^{+} or f = q^{−}, part (i) follows from the surjectivity of φ_{t} and (iv) of Lemma 7.1.

By Proposition 5.2 the simple Sp(V_{k})-submodule M of A given in Lemma 7.1(ii) which corresponds to Kerδ/Imδ in Cokerδ is generated as a kG-module by the image in A of the product Ê_{1} ⋯ Ê_{n} ∈ F_{n}. We must show that the image of Ê_{1} ⋯ Ê_{n} in gr_{n}(A_{q^{−}}) is zero. Consider the polynomial

h(Ê_{1}, ⋯, Ê_{n}) = Ê_{1} ∏_{i=2}^{n}(1 + Ê_{i})  (24)

in R̂. We will prove that this function vanishes on V_{q^{−}}. Then, on expanding h, it will be immediate that the image of Ê_{1} ⋯ Ê_{n} in A_{q^{−}} belongs to F_{n−1}(A_{q^{−}}). Let m ∈ V_{q^{−}}. Then m = Ẑ + ∑_{i=1}^{n}(β_{i}X̂_{i} + α_{i}Ŷ_{i}) with α_{i}, β_{i} ∈ F_{2} and q^{−}(∑_{i=1}^{n}(α_{i}e_{i} + β_{i}f_{i})) = 0. Then, ∏_{i=2}^{n}(1 + Ê_{i}) takes a nonzero value at m if and only if β_{i} = 0 for i ≥ 2. This happens if and only if β_{1}^{2} + α_{1}^{2} + β_{1}α_{1} is zero, equivalently, if and only if α_{1} = β_{1} = 0. So h vanishes on V_{q^{−}}.

Thus, Kerφ_{n} has the submodule M, of dimension 2^{n}. Since φ_{n} is surjective and its image gr(A_{q^{−}}) has dimension 2^{n−1}(2^{n} − 1), it follows that Kerφ_{n} = M. This proves (ii). The last statement follows from (i) for q^{+} and from (ii) for q^{−}, since the submodule M corresponds to Kerδ/Imδ.

Theorem 7.2 and Proposition 6.1 yield filtrations of gr(A_{q^{+}}) and gr(A_{q^{−}}) by kG-modules which are the restrictions to G of Weyl modules of Sp(V_{k}) and their duals. The multiplicites of the factors are given by Corollary 6.3.

Of course, since the permutation modules are self-dual, we can obtain additional filtrations simply by dualizing the above filtrations.

8 Module structure for O^{+}(2n, 2) and O^{−}(2n, 2)

The equation (5) shows that the restriction of the kG-module A_{f} to the orthogonal group O(V, f) ≅ O^{±}(2n, 2) is isomorphic to the permutation module on the set of zeroes of f in V. Our results therefore give us kO(V, f)-module filtrations of this permutation module by Weyl modules and dual Weyl modules of the algebraic group Sp(V_{k}). In order for these filtrations to be useful in studying the kO(V, f)-submodule structure of this permutation module, it is desirable to know the kO(V, f)-submodule structure of the Weyl modules involved. We have already seen at the end of § 6 that the submodule lattices of these Weyl modules remain unchanged when the group action is restricted from Sp(V_{k}) to Sp(V). The aim of this section is to show that the same holds when we restrict to O(V, f).

In this section we will need to consider the noncentral infinitesimal isogenies τ : Sp(V_{k}) → Spin(V̂_{k}) and σ : Spin(V̂_{k}) → Sp(V_{k}) such that the compositions τ ∘ σ and σ ∘ τ are the absolute Frobenius maps of the respective groups. These maps and their ramifications for the representation theory of the above and related groups have been studied in detail in [8].

Lemma 8.1 H^{1}(Sp(V_{k}), L(n)) = 0.

Proof: The module L(n) is the pullback along τ of the spin module for Spin(2n + 1, k), which we shall denote here by S. (In the notation of [8], the letter G is used for Spin(2ℓ + 1, k) and the spin module is L(ω_{ℓ}).) Then the two general reduction steps in [8, I.6.3] show that H^{1}(Sp(V_{k}), L(n)) ≅ H^{1}(Spin(V̂_{k}), S). Let G_{σ} be the (scheme-theoretic) kernel of σ. Then S is a simple injective module for G_{σ} and it follows immediately from the “inflationrestriction” sequence [8, I.6.2(3)] that H^{1}(Spin(V̂_{k}), S) = 0.

Proposition 8.2 Each Weyl module V (i) (i = 1,…,n) of the algebraic group Sp(V_{k}) satisfies the following properties. The restrictions of its composition factors to the subgroup O(V, f) remain simple and distinct as modules for this subgroup. Furthermore, the lattice of submodules remains the same.

That is, the groups Sp(V_{k}), O(V, f) leave invariant the same subspaces of each V (i).

Proof: The simple modules which appear as composition factors of the Weyl modules V (i) are the trivial module and the modules L(j). The highest weights of these modules are restricted weights (see [1, §1.]) for Sp(V_{k}) and so they are also restricted for the subgroup Ω(V_{k}, f), of type D_{n}, since they have the same maximal torus and the roots for Ω(V_{k}, f) are a subset of those for Sp(V_{k}, f). (Note however that ω_{n} is not 2-restricted for the covering group Spin(V_{k}, f).)

Moreover, for r ≠ n the weights ω_{r} are τ-restricted. (See [8, I.4.1].) This implies that the modules L(r) (r ≠ n) are simple for Ω(V_{k}, f). Then, by Steinberg’s theorem, they remain simple upon restriction to the finite subgroup Ω(V, f).

The restriction of the module L(n) to Ω(V_{k}) is isomorphic to the direct sum of two simple modules which as modules for the covering group Spin(V_{k}) are isomorphic to the first Frobenius twists of the half-spin modules. Thus, their highest weights are not 2-restricted with respect to Spin(V_{k}). However, these weights are 2-restricted with respect to Ω(V_{k}, f). Again by Steinberg’s theorem, the two simple modules remain simple for the finite groups Ω(V, f), for which they are isomorphic to the sum of the two halfspin modules, Therefore, their direct sum is simple for the slightly larger groups O(V, f), since these groups have elements which interchange the two types of maximal isotropic subspaces in the orthogonal space V_{k}. Thus, we have proved the first property.

In order to prove that the submodule structure is unchanged when we restrict to the finite groups O(V, f), we will show that any nonsplit extension

0 → L(i) → E → L(j) → 0  (25)

of simple Sp(V_{k})-modules with fundamental or zero highest weight remains nonsplit when restricted to O(V, f). By [12, II.2.12(1)], we can assume that i ≠ j, and by Lemma 8.1 and duality that {i, j} ≠ {0, n}. The latter assumption implies that the (scheme-theoretic) kernel G̃_{τ} of τ acts irreducibly and non-trivially on at least one of L(i) and L(j). This is because the modules L(r) for r ≠ n are τ-restricted. Since all the simple modules are self-dual and since (25) splits if and only the dual sequence does, we may assume without loss that L(j) is a nontrivial, simple module for G̃_{τ}, and that HomG̃_{τ} (L(j), L(i)) = 0. Then HomG̃_{τ} (L(j), E) has dimension ≤ 1 and so is trivial for any perfect group which acts on it. This shows first that

HomG̃_{τ} (L(j), E) = Hom_{Sp(V_{k})}(L(j), E) = 0.  (26)

Secondly, G̃_{τ} is also a normal subgroup of Ω(V_{k}, f). (See [8, I.3.1]; in the notation there Ω(V_{k}, f) is denoted by D̃.) So we have

Hom_{Ω(V_{k},f)}(L(j), E) = HomG̃_{τ} (L(j), E) = 0,  (27)

which shows that the extension (25) does not split for Ω(V_{k}, f). Now, as explained above, the composition factors of E are 2-restricted modules for Ω(V_{k}, f), so by [1, Proposition 2.7] E remains nonsplit on restriction to the finite groups Ω(V, f). It follows that for the slightly large groups O(V, f), the module E is a nonsplit extension of two simple modules (even in the case i = n when there are three composition factors for Ω(V, f)).

Remark 8.3 The assertion in the proof of Proposition 8.2 that the nonsplit extension (25) of Sp(V_{k}) modules remains nonsplit for O(V, f) would be false if we were to allow the composition factors of E to be Frobenius twists of L(i) and L(j), even though their restrictions to O(V, f) are isomorphic to L(i) and L(j). For example, the (2n + 1)-dimensional module V̂_{k} (with Sp(V_{k}) acting via τ) is a nonsplit extension of the first Frobenius twist of L(1) by L(0), which clearly splits when restricted to O(V, f). The example just given may appear at first sight to contradict the assertion. After all, the trivial composition factor can also be viewed as a Frobenius twist of itself, suggesting that V̂_{k} might be the Frobenius twist of an Sp(V_{k})-module E of the form in (25), contrary to the assertion. However, no such module really exists. The above observations are explained by the fact that while the first cohomology group of Sp(V_{k}) with values in V_{k} is trivial, the cohomology group with values in a Frobenius twist of V_{k} is nontrivial. This fact is a consequence of the exceptional non-vanishing [12, p. 371, Remark] of the first cohomology group of the Frobenius kernel of Sp(V_{k}) with values in k.

We end with some results about the socles of the Jordan-Steiner permutation modules and of the modules of functions on 𝔽_{2}-rational points of quadrics. We recall that the socle, socM, of a module M is the maximal semisimple submodule and that the radical, radM, is the intersection of all maximal submodules (whence the maximal semisimple quotient is M/radM). We also use the notation (soc^{2}/soc)M and (rad/rad^{2})M for the socle of M/socM and radM/rad(radM) respectively.

Proposition 8.4 Let k^{Q_{f}} denote the kG-permutation module on a Jordan-Steiner set and let k^{Z_{f}} denote the kO(V, f)-module of functions on the set of zeroes of f in V.

(i) soc(k^{Q_{f}}) ≅ k ≅ k^{Q_{f}} /rad(k^{Q_{f}}). We have soc(k^{Q_{f}}) ⊆ rad(k^{Q_{f}}).

(ii) (soc^{2}/soc)(k^{Q_{f}}) ≅ V ≅ (rad/rad^{2})(k^{Q_{f}}).

(iii) soc(k^{Z_{f}}) ≅ k ⊕ k ⊕ V ≅ k^{Z_{f}} /rad(k^{Z_{f}}).

Proof: By Frobenius reciprocity, the simple modules which appear as quotients of k^{Q_{f}} are those which have fixed points for kO(V, q). By Proposition 8.2 only the trivial module has this property. Since k^{Q_{f}} is a transitive permutation module, there is a unique trivial submodule. By the self-duality of permutation modules, the isomorphisms in (i) are proved. The last statement of (i) is merely the fact that the cardinality of the Jordan-Steiner set is even.

Next we consider (iii). Now k^{Z_{f}} is isomorphic to the direct sum of the functions supported at the origin and the permutation module on the onedimensional isotropic subspaces. By Frobenius reciprocity and self-duality, it is enough to show that the stabilizer of a one-dimensional isotropic subspace of V fixes no other vectors in V or in any other nontrivial simple module. This is well known and can be seen by explicit computation, or by [5, Theorem 6.13]. The latter implies that a transitive permutation module for a finite group of Lie type with maximal parabolic subgroups as stabilizers has a socle which is the direct sum of the trivial module and one nontrivial simple module. It remains to prove (ii). We have already seen (5) that the restriction of k^{Q_{f}} to O(V, f) is isomorphic to k^{Z_{f}}. Consider rad(k^{Q_{f}})/soc(k^{Q_{f}}). By (i) and (iii), its head and socle are isomorphic to V as kO(V, f)-modules, hence also as kG-modules, since by Proposition 8.2 we know that distinct simple kG-modules restrict to distinct simple kO(V, f)-modules.

9 Examples

To illustrate our results, we work out the case n = 3. In this case the dimensions of the simple modules L(r) for r = 0, 1, 2 and 3 are respectively 1, 6, 14 and 8.

Figure 1 shows the graded modules and in this small case, Propsition 8.4 is sufficient to recover the submodule structures of the Jordan-Steiner permutation modules. Figure 2 shows the submodule structures.

∧(V) : 6 1 ⊕ 6 ⊕ (1 ⊕ 14) ⊕ 8 ⊕ (14 ⊕ 1) ⊕ 6 ⊕ 1 6

Cokerδ :  6 1 ⊕ 6 ⊕ 14 ⊕  ⊕ 1 8

Figure 1:

k^{Q^{+}} :  [figure: 1; 6; 8; 14; 6; 1]  k^{Q^{−}} :  1 6 14 6 1

Figure 2: Submodule structures for Sp(6, 2) .

For n > 3, our results are not strong enough to recover the submodule structure from the structure of the graded modules. Figure 3 shows the structure of Cokerδ ≅ gr(k^{Q^{+}}) for n = 4 and n = 5. Again, the composition factors are indicated by their dimensions.

References

[1] H.H. Andersen, Extensions of simple modules for finite Chevalley groups, J. Algebra 111 (1987) 388–403. n = 4 :

[figure: 1; 26; 1 ⊕ 8 ⊕; 26; ⊕ 48 ⊕; 1; 16; ⊕ 8 ⊕ 1]

n = 5 :

[figure: 10; 1; 100; 1 ⊕ 10 ⊕ 44 ⊕; 100; ⊕; 164; ⊕; 10; 32; ⊕ 44 ⊕ 10 ⊕ 1; 1]

Figure 3: Submodule structure of gr(k^{Q^{+}}) for n = 4, 5 .

[2] A. M. Adamovich, Analogues of spaces of primitive forms over a field of positive characteristic, Moscow Univ. Math. Bull. 39 No. 1 (1984) 53–56.

[3] A. M. Adamovich, The submodule structure of Weyl modules for symplectic groups with fundamental highest weights, Moscow Univ. Math. Bull. 41 No 2 (1986) 6–9.

[4] M. Bardoe, P. Sin, The permutation modules for GL(n + 1, 𝔽_{q}) acting on ℙ^{n}(𝔽_{q}) and 𝔽_{q}^{n+1}, J. Lond. Math. Soc. 61 (2000) 58–80

[5] C. W. Curtis, Modular representations of finite groups with split BNpairs, in “Seminar on Algebraic Groups and Related Finite Groups” Lecture Notes in Mathematics 131, Springer, Berlin (1969) 57–95

[6] J. Dieudonne, “La Géométrie des groupes classiques” (3rd ed.) Springer Verlag, Berlin, 1971.

[7] S. Donkin, On tilting modules and invariants for Algebraic Groups, in “Finite Dimensional Algebras and related topics” ( V.Dlab and L.L.Scott, Eds.) Kluwer Academic Publishers, (1994) 59–77.

[8] M. Dowd, P. Sin, Representations of algebraic groups in characteristic 2, Comm. Algebra 24 (1996) 2597–2696.

[9] R. H. Dye, Interrelations of Symplectic and orthogonal groups in characteristic two, J. Algebra 59 (1979) 202–221.

[10] R. Gow, Construction of p − 1 irreducible modules with fundamental highest weight for the symplectic group in characteristic p, J. London Math. Soc. 58 (1998) 619–632.

[11] G. Hiss, On the incidence matrix of the Ree unital, Des. Codes Cryptogr. 10 (1997), no. 1, 57–62.

[12] J. C. Jantzen, “Representations of Algebraic Groups” Academic Press, London, 1987.

[13] C. Jordan, “Traité des Substitutions et des équations algébriques” Gauthier-Villars, Paris, 1870.

[14] W. M. Kantor, Symplectic groups, symmetric designs and line ovals, J. Algebra 33 (1975) 43–58.

[15] I. Kaplansky, “Linear Algebra and Geometry” Allyn and Bacon, Boston, 1969.

[16] T. Kasami, S. Lin, W. W. Peterson, Some results on cyclic codes which are invariant under the affine group and their applications, Information and Control 11 (1967) 475–496.

[17] F. J. MacWilliams, N. J. A. Sloane, “The theory of error-correcting codes” I, II, North-Holland Mathematical Library, Vol. 16. North-Holland, Amsterdam (1977).

[18] B. Mortimer, The modular permutation representations of the known doubly transitive groups, Proc. London Math.Soc. 41 (1980) 1–20

[19] D. E. Taylor, Regular 2-graphs, Proc. London Math. Soc. (3) 35 (1977), no. 2, 257–274.