THE MODULO 2 STRUCTURE OF RANK 3

PERMUTATION MODULES FOR ODD CHARACTERISTIC SYMPLECTIC GROUPS

J.M.LATAILLE, PETER SIN, AND PHAM HUU TIEP

Abstract. This paper studies the permutation representation of the symplectic group Sp(2m, 𝔽_{q}), where q is odd, on the 1-spaces of its natural module. The complete submodule lattice for the modulo ℓ reduction of this permutation module is known for all odd primes ℓ not dividing q. In this paper we determine the complete submodule lattice for the mod 2 reduction. Similar results are then obtained for the orthogonal group O(5, 𝔽_{q}).

1. Introduction

Let p be an odd prime and let q = p^{f}. Throughout the following V will be a 2m-dimensional 𝔽_{q}-vector space equipped with a non-singular alternating bilinear form ( , ). We shall assume m ⩾ 2 to avoid trivial exceptions. For 1 ⩽ r ⩽ m, let L_{r} denote the set of r-dimensional isotropic subspaces of V. Then L_{1} is the set of all 1-dimensional subspaces of V, and L_{m} is the set of all maximal isotropic subspaces of V. The group G := Sp(2m, 𝔽_{q}) acts transitively with rank 3 on L_{1}.

Let 𝔽 be any field of characteristic coprime to p, and let 𝔽^{L_{r}} denote the 𝔽G-permutation module on L_{r}. We ask for its submodule lattice in the case where r = 1. This has been determined in [9] by Liebeck in all cases except when char 𝔽 = 2. In [2] Bagchi et al have conjectured the submodule structure of 𝔽_{2}^{L_{1}} for the special case where m = 2.

In this paper we determine the complete 𝔽G-submodule lattice of 𝔽^{L_{1}}, where 𝔽 is a field of characteristic 2. Our approach is to first restrict the action of G to that of a maximal parabolic subgroup. The composition factors of this restricted action are determined and using a recent result [5], we are then able to determine the composition factors for the action of the full group. This puts us in a position to obtain the submodule lattice (see Theorem 2.13). Taking m = 2 in our work, we see that the conjecture in [2] is correct only if q ≡ ±3 mod 8.

In fact, for the case m = 2, Bagchi et al also conjectured in [2] the submodule structure of 𝔽_{2}^{L_{2}}. We are able to use our results along with results [10] of White and [9] of Liebeck to show that this conjecture is true only if q ≡ ±3 mod 8 (see Theorem 3.1 and its corollary).

-The second author was supported by NSF grant number DMS 0071060.

-The third author was supported by NSF grant number DMS 0070647.

For simplicity, we will always work over an algebraically closed field of characteristic 2, which we denote by k.

We mention here that other rank 3 permutation modules for finite classical groups will be treated in a forthcoming paper.

2. the submodule structure of k^{L_{1}}

2.1. Restriction to a maximal parabolic subgroup. Fix a symplectic basis e_{1}, …, e_{m}, f_{1}, …, f_{m} for V over 𝔽_{q}, so that

(e_{i}, e_{j}) = (f_{i}, f_{j}) = 0 and (e_{i}, f_{j}) = −(f_{j}, e_{i}) = \begin{cases}1 & if i = j \\ 0 & if i ≠ j \end{cases}

Let M := ⟨e_{1}, …, e_{m}⟩ and P := ⟨f_{1}, …, f_{m}⟩ be maximal isotropic subspaces of V.

Let G_{M} denote the set-wise stabilizer of M in G. Then

G_{M} = S ⋊ L (1)

where

S = \{(\begin{matrix}I & A \\ 0 & I \end{matrix}) | A = A^{t}, A ∈ Hom(P, M)\} (2)

and

L = \{(\begin{matrix}g & 0 \\ 0 & g^{−t} \end{matrix}) | g ∈ GL(M)\} . (3)

Here I is the m × m identity matrix and 0 is the m × m zero matrix.

To determine the kG-composition factors of k^{L_{1}} we will first need to determine the composition factors of Res_{G_{M}}^{G} k^{L_{1}}. We start by noting that G_{M} has two orbits on L_{1} :

O_{1} := {ω ∈ L_{1} | ω ⊂ M}

and

O_{2} := {ω ∈ L_{1} | ω ⊈ M} .

Now for any subset X ⊆ L_{1}, we will let k^{X} denote the k-span of the elements of X. Then we have the following decomposition of Res_{G_{M}}^{G} k^{L_{1}} as a direct sum of kG_{M}-submodules:

Res_{G_{M}}^{G} k^{L_{1}} = k^{O_{1}} ⊕ k^{O_{2}}. (4)

Thus, to determine the composition factors of Res_{G_{M}}^{G} k^{L_{1}} we may separately study the summands in (4).

The first summand is easily handled:

For v ∈ V, write v = (\begin{matrix}x \\ y \end{matrix}) , where x ∈ M and y ∈ P. Then

(5) ⟨v⟩ ∈ O_{2} if and only if y ≠ 0.

With this notation, the computation

(\begin{matrix}I & A \\ 0 & I \end{matrix}) (\begin{matrix}x \\ y \end{matrix}) = (\begin{matrix}x + Ay \\ y \end{matrix}) (6)

shows that S acts trivially on O_{1}, i.e. S acts trivially on k^{O_{1}}. From (1), (2), and (3) we see that the induced action of G_{M}/S ≃ GL(M) on k^{O_{1}} is the usual action of GL(M) on the 1-spaces of M. Thus, the kG_{M}-submodule lattice of k^{O_{1}} is known from [8]. Explicitly, if we put

1_{O_{1}} := ∑_{ω∈O_{1}} ω ∈ k^{O_{1}}

and

K := ⟨ω − α | ω, α ∈ O_{1}⟩_{k} ,

where ⟨ ⟩_{k} denotes k-span, then we have

Lemma 2.1. (a) If m is odd, then K is simple and

k^{O_{1}} = ⟨1_{O_{1}}⟩_{k} ⊕ K

(b) If m is even, then k^{O_{1}} is uniserial with composition series

k^{O_{1}}
|
K
|
⟨1_{O_{1}}⟩_{k}
|
{0}

In the situation of Lemma 2.1.b, put K′ := K/ ⟨1_{O_{1}}⟩_{k} .

We will indicate the composition factors of k^{O_{1}} informally by writing

(7) k^{O_{1}} = \begin{cases}k + K & if m is odd \\ (2)k + K′ & if m is even \end{cases}

Of course, here k denotes the simple trivial module.

To determine the kG_{M}-composition factors of the second summand in (4), we will once again begin by restricting the action of G_{M} to that of its normal subgroup S, i.e. first we will determine the composition factors of Res_{S}^{G_{M}} k^{O_{2}}. We will then use Clifford’s theorem along with a result of Higman’s (see [6]) to recover the G_{M}-composition factors.

2.2. The composition factors of Res_{S}^{G_{M}} k^{O_{2}}. Using elementary linear algebra we see that given any non-zero y ∈ P and any z ∈ M we can always find a symmetric transformation A ∈ Hom(P, M) which sends y to z. Therefore, it follows from (6) that the S-orbits on O_{2} are indexed by the

1-spaces in P. Explicitly, let ⟨y_{1}⟩ , …, ⟨y_{\frac{q^{m}−1}{q−1}} ⟩ be a list of the 1-spaces in

P. Then the S-orbits on O_{2} are the sets

O_{⟨y_{i}⟩} := \{⟨(\begin{matrix}x \\ y_{i} \end{matrix})⟩ | x ∈ M, \} .

Thus, we have the following decomposition of Res_{S}^{G_{M}} k^{O_{2}} as a direct sum of kS-submodules:

Res_{S}^{G_{M}} k^{O_{2}} = ⨁_{i=1}^{\frac{q^{m}−1}{q−1}} k^{O_{⟨y_{i}⟩}} (8)

Let S_{y_{i}} ⩽ S be the stabilizer of ⟨(\begin{matrix}0 \\ y_{i} \end{matrix})⟩ ∈ O_{⟨y_{i}⟩}. So

S_{y_{i}} = \{(\begin{matrix}I & A \\ 0 & I \end{matrix}) ∈ S | Ay_{i} = 0\} .

Then

k^{O_{⟨y_{i}⟩}} = Ind_{S_{y_{i}}}^{S} k,

so that by (8) we may write

Res_{S}^{G_{M}} k^{O_{2}} = ⨁_{i=1}^{\frac{q^{m}−1}{q−1}} Ind_{S_{y_{i}}}^{S} k. (9)

We now pause to establish a correspondence between the irreducible kScharacters and the symmetric bilinear forms on M. This correspondence will be the key to determining the composition factors of Res_{S}^{G_{M}} k^{O_{2}}.

We start by noting that

P ≃ V/M = V/M^{⊥} ≃ M^{∗},

(where M^{∗} denotes the dual space of M) so we may identify P with M^{∗}. If we also identify M with (M^{∗})^{∗}, then we can identify Hom(P, M) with Hom(M^{∗}, (M^{∗})^{∗}), i.e. we may regard Hom(P, M) as the set of all bilinear forms on M^{∗}.

The correspondence

(\begin{matrix}I & A \\ 0 & I \end{matrix}) ↦ A (10)

then identifies S with the set of symmetric bilinear forms on M^{∗}. Under this identification S_{y_{i}} corresponds to the set of all symmetric bilinear forms on M^{∗} which have y_{i} in their radical, i.e. S_{y_{i}} corresponds to the symmetric bilinear forms on (Ker y_{i})^{∗}.

Now let ζ be a primitive p-th root of unity in algebraically closed k. The corresondence

f(·) ↦ ζ^{T race_{𝔽q/𝔽p}(f(·))} (11)

allows us to identify the linear functionals on the 𝔽_{q}-vector space S with the irreducible k-characters of the elementary abelian p-group S. Since S is the set of symmetric bilinear forms on M^{∗}, we see that S^{∗} is the set of symmetric bilinear forms on M. Thus, we may identify the irreducible characters of S with the symmetric bilinear forms on M.

Remark 2.2. Let N be an irreducible submodule of Ind_{S_{y_{i}}}^{S} k and let f ∈ S^{∗} be the linear functional which corresponds under (11) to the character of N. By Frobenius reciprocity, we know that S_{y_{i}} acts trivially on N. This means that T race_{𝔽_{q}/𝔽_{p}}(f(A)) = 0 for every A ∈ S_{y_{i}}, from which it follows that f(A) = 0 for all A ∈ S_{y_{i}}. But as S_{y_{i}} is the set of symmetric bilinear forms on (Ker y_{i})^{∗}, this means that the symmetric bilinear form on M which corresponds to f must be isotropic on the hyperplane Ker y_{i} ⊂ M.

Thus, the irreducible characters in Ind_{S_{y_{i}}}^{S} k are the symmetric bilinear forms on M which are isotropic on Ker y_{i}. Again using Frobenius reciprocity, we see that each such form occurs with multiplicity one. In particular, the zero form (which corresponds to the trivial character) occurs exactly once in each Ind_{S_{y_{i}}}^{S} k. In fact, it is easily seen that the unique trivial submodule of Ind_{S_{y_{i}}}^{S} k is

T_{i} := ⟨1_{O_{⟨y_{i}⟩}}⟩_{k} (12)

where

1_{O_{⟨y_{i}⟩}} := ∑_{ω∈O_{⟨y_{i}⟩}} ω ∈ k^{O_{⟨y_{i}⟩}}. (13)

Now let B be a non-zero symmetric bilinear form on M which has an isotropic hyperplane. Then B has either rank 1 or 2. If B has rank 1, then the radical of B, denoted by Rad B, is the unique isotropic hyperplane for B. If B has rank 2 then M/Rad B is hyperbolic and therefore has precisely two isotropic lines for the form induced from B, i.e. M has precisely two isotropic hyperplanes for B.

In light of (9), the above then gives us all of the composition factors of Res_{S}^{G_{M}} k^{O_{2}}. We record this information as

Lemma 2.3. Under the identification in (11), Res_{S}^{G_{M}} k^{O_{2}} has the following composition factors:

(a) The zero form, i.e. the trivial character, which occurs with multiplicity \frac{q^{m}−1}{q−1}.

(b) The rank 1 symmetric bilinear forms, where each occurs with multiplicity 1.

(c) The rank 2 symmetric bilinear forms having isotropic hyperplane, where each occurs with multiplicty 2.

2.3. The kG_{M}-composition factors of k^{O_{2}}. We start by examining the S-fixed points of k^{O_{2}}. Define

T := ⨁_{i=1}^{\frac{q^{m}−1}{q−1}} T_{i} (14)

where the T_{i} are as in (12). Now it is easily seen from (1) that G_{M}/S ≃ GL(M) permutes the vectors in (13) in the usual way that GL(M) acts on the 1-spaces of M^{∗}, i.e. in the usual way that GL(M) acts on the hyperplanes of M. Thus, if we let L_{m−1} denote the set of hyperplanes in M, then the kG_{M}-module T can be naturally identified with the kGL(M)-module k^{L_{m−1}}.

It is well-known that the permutation modules on the 1-spaces and the hyperplanes, respectively, of M are isomorphic over a field of characteristic zero. Therefore, from a general principle of modular representation theory (see [4], Theorem 17.7) we know that k^{L_{m−1}} and k^{O_{1}} have the same composition factors. Therefore, it follows from (7) that

(15) T = \begin{cases}k + K & if m is odd \\ (2)k + K′ & if m is even \end{cases}

We remark here that it can actually be shown that k^{L_{m−1}} and k^{O_{1}} are isomorphic for G.

To find the remaining composition factors, we now consider the action of G_{M} on the irreducible characters of S. We start by observing that as S acts trivially on its characters, we need only consider the induced action of G_{M}/S ≃ GL(M). Now GL(M) acts by congruence transformations on S. Therefore, if we view the elements of S^{∗} as symmetric matrices, then the action of GL(M) is again by congruence transformations. We then see that under correspondence (11), GL(M) acts by congruence transformations on the characters of S.

There are two GL(M) congruence classes of rank 1 symmetric bilinear forms, represented by

diag(1, 0, …, 0) (16)

and

diag(α, 0, …, 0) (17) where α is a non-square in 𝔽_{q}^{×} (see [1]). The stabilizer of both classes is

\{(\begin{matrix}±1 & 0 \\ ∗ & ∗ \end{matrix})\}

which has index \frac{q^{m}−1}{2} in GL(M).

Let B_{1} denote the congruence class of (16) and B_{α} denote the congruence class of (17). Let W_{1} denote the external direct sum of the S-characters which correspond to the forms in B_{1}, and let W_{2} denote the external direct sum of the S-characters which correspond to the forms in B_{α}. Then it follows from Lemma 2.3.b and Clifford’s Theorem [4] that k^{O_{2}} has composition factors, call them W_{1} and W_{2}, which when restricted to S are isomorphic to W_{1} and W_{2}, respectively. Note that

dim_{k} W_{1} = dim_{k} W_{2} = \frac{q^{m} − 1}{2} (18)

but W_{1} ≇ W_{2}.

Now, there is one congruence class of rank 2 symmetric bilinear forms having isotropic hyperplane, represented by

(\begin{matrix}( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & 0 \\ 0 & 0 \end{matrix}) . (19)

The stabilizer in GL(M) of this class is

\{(\begin{matrix}C & 0 \\ ∗ & ∗ \end{matrix})\}

where C is a 2×2 monomial matrix. This subgroup has index \frac{q(q^{m}−1)(q^{m−1}−1)}{2(q−1)} in GL(M).

Let D denote the external direct sum of the S-characters which correspond to these forms. Note that dim_{k} D = \frac{q(q^{m}−1)(q^{m−1}−1)}{2(q−1)}. It then follows from Lemma 2.3.c and Clifford’s Theorem that exactly one of the following cases holds for k^{O_{2}} :

Case A: k^{O_{2}} has precisely two composition factors, call them D_{1} and D_{−1}, which when restricted to S are isomorphic to D.

Case B: k^{O_{2}} has a single composition factor, call it D_{0}, which when restricted to S is isomorphic to D ⊕ D.

We now show that the former is true. We start by establishing some notation which we will use throughout the remainder of the paper:

For any field 𝔽, we let 𝔽^{L_{r}} denote, as usual, the 𝔽G-permutation module on L_{r}. Let

η_{r,s} : 𝔽^{L_{r}} → 𝔽^{L_{s}}

be the 𝔽G-module homomorphism which sends each isotropic r-space to the (formal) sum of the isotropic s-spaces which are incident with it.

Define

1 := ∑_{ω∈L_{1}} ω ∈ 𝔽^{L_{1}}.

For ω ∈ L_{1} put

Δ(ω) := {α ∈ L_{1} | α ⊈ ω^{⊥}},

and define an element s_{Δ(ω)} ∈ 𝔽^{L_{1}} as follows:

s_{Δ(ω)} := ∑_{α∈Δ(ω)} α. (20)

Define a non-singular symmetric bilinear form [−, −]_{𝔽} by demanding that

the elements of L_{1} form an orthonormal basis. For any subset S ⊆ 𝔽^{L_{1}} put

S^{⊥} := {v ∈ k^{L_{1}} | [v, s]_{𝔽} = 0, for all s ∈ S}

Note that we have used the same notation for orthogonal complements in V, but no confusion should arise. Note also that if S is a 𝔽G-submodule of 𝔽^{L_{1}}, then so is S^{⊥}.

Now let ℚ_{2} denote the field of 2-adic numbers and let \overline{ℚ_{2}} be its algebraic closure. Then F will be the maximal unramified extension of ℚ_{2} in \overline{ℚ_{2}} (see [7], pg.37), and R will be the valuation ring of F. Note that F has residue field k. From [6] we have that

F^{L_{1}} = ⟨1⟩_{F} ⊕ M_{−1} ⊕ M_{1}, (21)

where M_{±1} are irreducible FG-submodules with

dim_{F} M_{−1} = \frac{q(q^{m} − 1)(q^{m−1} + 1)}{2(q − 1)} (22)

and

dim_{F} M_{1} = \frac{q(q^{m} + 1)(q^{m−1} − 1)}{2(q − 1)}. (23)

Let \overline{M_{±1}} be the reductions modulo 2 of M_{±1}. Their restrictions to G_{M} must be collections of the composition factors described above. By (22) and (23) the dimensions of the composition factors of \overline{M_{±1}} add up to \frac{q(q^{m}±1)(q^{m−1}∓1)}{2(q−1)}. Assume now that (m, q) ≠ (2, 3). Then

\frac{q(q^{m} ± 1)(q^{m−1} ∓ 1)}{2(q − 1)} < 2(dim_{k} D). (24)

So it cannot be that either Res_{G_{M}}^{G} \overline{M_{±1}} contains a composition factor which when restricted to S is isomorphic D⊕D. Thus, we deduce that Case A holds. If (m, q) = (2, 3), then dim_{F} M_{−1} = 2(dim_{k} D). However, it is easy to see (e.g. by considering degrees) that M_{−1} is the unique non-trivial composition factor which is common to both F^{L_{1}} and F^{L_{2}}. Since F^{L_{2}} = Ind_{G_{M}}^{G} F, it then follows from Frobenius reciprocity that G_{M} (and hence S) has a non-zero fixed point on M_{−1}. This then implies that Res_{G_{M}}^{G} \overline{M_{−1}} contains a trivial composition factor. Since S has no fixed points on D ⊕ D, we deduce that Case A holds in this case as well. We mention here that it will be shown in §2.4 (see (32)) that D_{1} ≃ D_{−1}.

Hence, we have found all the kG_{M}-composition factors of k^{O_{2}}. Combining this information with Lemma 2.1 and using the informal notation of (7) and (15), we may now state

Lemma 2.4. (a) If m is odd, then

Res_{G_{M}}^{G} k^{L_{1}} = (2)k + (2)K + W_{1} + W_{2} + D_{1} + D_{−1}.

(b) If m is even, then

Res_{G_{M}}^{G} k^{L_{1}} = (4)k + (2)K′ + W_{1} + W_{2} + D_{1} + D_{−1}.

2.4. The kG-composition factors of k^{L_{1}}. Let F, R, M_{±1}, and \overline{M_{±1}} be as in §2.3. It follows from (24) and the remarks following it that each of D_{±1} occurs in exactly one of Res_{G_{M}}^{G} \overline{M_{±1}}, and that the D_{±1} do not occur together. Thus, we may assume that our notation is chosen so that D_{±1} is a composition factor of Res_{G_{M}}^{G} \overline{M_{±1}}. Also, since

dim_{k} \overline{M_{1}} − dim_{k} D_{1} = \frac{q(q^{m} + 1)(q^{m−1} − 1)}{2(q − 1)} − \frac{q(q^{m} − 1)(q^{m−1} − 1)}{2(q − 1)}

= \frac{q^{m} − 1}{q − 1} − 1

< \frac{q^{m} − 1}{2}

= dim_{k} W_{1} = dim_{k} W_{2},

we deduce upon inspection of Lemma 2.4 that

(25) Res_{G_{M}}^{G} \overline{M_{1}} = \begin{cases}K + D_{1} & (m odd) \\ k + K′ + D_{1} & (m even) \end{cases}

Suppose for the sake of contradiction that \overline{M_{1}} has a kG-composition factor, call it K, which when restricted to G_{M} is isomorphic to K. Since S acts trivially on K, it is contained in the kernel, call it J, of the representation of G on K. Since S is not contained in the center of G, and since the center of G is the only non-trivial normal subgroup of G, we deduce that J must be all of G. But G_{M} acts non-trivially on K, a contradiction. It follows that \overline{M_{1}} has no such composition factor for G, and therefore \overline{M_{1}} is irreducible if m is odd. If m is even, then similar reasoning allows us to conclude that either \overline{M_{1}} is irreducible or else \overline{M_{1}} = k + X, with X irreducible. We now show that the latter is true.

Using the notation in (20), we define a kG-module homomorphism

φ : k^{L_{1}} → k^{L_{1}} (26)

by

ω ↦ ω + s_{Δ(ω)},

where ω ∈ L_{1}. Put

U := Im φ. (27)

Now define

U′ := ⟨u_{1} + u_{2} | u_{1}, u_{2} ∈ U⟩_{k} . (28)

From [9] we have the following non-trivial fact:

Lemma 2.5. (Theorem 1.1 of [9]) Every kG-submodule of k^{L_{1}} not contained in ⟨1⟩_{k} must contain U′.

It is easily seen that

M_{1} ∩ R^{L_{1}}

is an R-form of M_{1} and a pure RG-submodule of R^{L_{1}}. Therefore,

\overline{M_{1} ∩ R^{L_{1}}}

is a mod 2 reduction of M_{1} as well as a kG-submodule of k^{L_{1}}. Since \overline{M_{1} ∩ R^{L_{1}}} is certainly not contained in ⟨1⟩_{k} , we see from Lemma 2.5 that

U′ ⊆ \overline{M_{1} ∩ R^{L_{1}}},

and therefore \overline{M_{1}} contains the composition factors of U′.

We require the following result:

Lemma 2.6. If m is even, then ⟨1⟩_{k} ⊂ U′.

Proof. Let M ∈ L_{m}. Then an easy computation shows that

∑_{\begin{matrix}ω⊆M \\ ω∈L_{1} \end{matrix}} ω + s_{Δ(ω)} = 1. (29)

Now the number of 1-spaces in M is \frac{q^{m}−1}{q−1}, which is an even number since m is even. Thus we may group the summands in the left-hand side of (29) into pairs. The result then follows from the definition (28) of U′.

Since \overline{M_{1} ∩ R^{L_{1}}} has at most 2 composition factors, it follows from Lemma 2.6 that

\overline{M_{1} ∩ R^{L_{1}}} = U′.

We may summarize the above as Lemma 2.7. \overline{M_{1}} and U′ have the same composition factors.

(a) If m is odd, then U′ is simple.

(b) If m is even, then U′ is uniserial with composition series

\begin{matrix}U′ \\ | \\ ⟨1⟩ & k \\ | \\ \{0\} \end{matrix}

In the situation of Lemma 2.7.b, we put U′/ ⟨1⟩_{k} := X. In the situation of Lemma 2.7.a, we will use X′ to denote the composition factor isomorphic to U′.

In view of Lemma 2.7, we have only to determine the composition factors of \overline{M_{−1}}. We do this now:

A simple matrix computation shows that

φ^{2} = 0. (30)

Thus,

U ⊆ Ker φ, (31)

where U is as in (27).

Since φ is symmetric, we see that Ker φ = U^{⊥}. It then follows that

U ≃ k^{L_{1}}/Ker φ = k^{L_{1}}/U^{⊥} ≃ U^{∗},

i.e. U is self-dual. So from the structure of U′ given in Lemma 2.7 we deduce

Lemma 2.8. (a) If m is odd, then

U = ⟨1⟩_{k} ⊕ U′.

(b) If m is even, then U is uniserial with composition series

U
|
U′
|
⟨1⟩_{k}
|
{0}

We next observe that \overline{M_{−1}} has the same composition factors as k^{L_{1}}/U. But since U ⊂ U^{⊥}, and since k^{L_{1}}/U^{⊥} ≃ U, we see that k^{L_{1}}/U contains the composition factors of U. We pause now to note that this implies that D_{1} is a composition factor of Res_{G_{M}}^{G} \overline{M_{−1}}. Since D_{−1} is the only composition factor of Res_{G_{M}}^{G} \overline{M_{−1}} of the same dimension as D_{1}, we deduce that

D_{1} ≃ D_{−1}, (32)

as was promised in §2.3.

It remains to determine the kG-composition factors of U^{⊥}/U. By inspecting Lemma 2.4, we see that

Res_{G_{M}}^{G} U^{⊥}/U = W_{1} + W_{2}.

We now show that \overline{M_{−1}} has kG-composition factors, call them W_{1} and W_{2}, which when restricted to G_{M} are isomorphic to W_{1} and W_{2}. We will need to consider the conformal symplectic group

CSp(2m, q) := {T ∈ GL(V) | ∃ α ∈ 𝔽_{q}^{×} so that (T v, T w) = α(v, w), ∀v, w ∈ V } .

For brevity, we put Γ := CSp(2m, q). Then Γ ≃ G ⋊ 𝔽_{q}^{×} and it is easy to see that U is a module for Γ. Therefore, U^{⊥}/U is also a kΓ-module.

We claim that U^{⊥}/U is simple for Γ. Suppose not. Then it follows that U^{⊥}/U has kΓ-composition factors, call them Ŵ_{1} and Ŵ_{2}, which when restricted to G_{M} are isomorphic to W_{1} and W_{2}, respectively, and (hence) when restricted to S are isomorphic to W_{1} and W_{2}, respectively. Since W_{1} and W_{2} are not isomorphic as kS-modules, we see that the following result then leads to a contradiction:

Lemma 2.9. The kS-modules W_{1} and W_{2} are conjugate for Γ.

Proof. Let β ∈ 𝔽_{q} be a non-square and consider the element g̃ ∈ Γ whose matrix representation with respect to the basis in §2.1 is

g̃ := (\begin{matrix}βI & 0 \\ 0 & I \end{matrix}) ∈ N_{Γ}(S),

where N_{Γ}(S) denotes the normalizer of S in Γ. If h := (\begin{matrix}I & A \\ 0 & I \end{matrix}) ∈ S, then

an easy computation shows that

g̃hg̃^{−1} = (\begin{matrix}I & βA \\ 0 & I \end{matrix}) ,

i.e. g̃ acts as multiplication by β on S. It then follows that g̃ acts (on the left) on S^{∗} as multiplication by β^{−1}. Under the identification in (11), this means that g̃ acts as multiplication by β^{−1} on the characters of S. Taking β = α, where α is as in (17), it is now easy to see that the conjugate by g̃ of the form in (17) is the form in (16). The result now follows from the construction of the W′_{i}s in §2.3.

Thus, U^{⊥}/U is simple for Γ, and it follows from Clifford’s theorem that U^{⊥}/U is semi-simple for G. Now, either U^{⊥}/U is a simple kG-module, or else U^{⊥}/U ≃ W_{1} ⊕ W_{2}, where W_{1} and W_{2} are simple kG-modules which when restricted to G_{M} are isomorphic to W_{1} and W_{2}, respectively.

Consider the following result from [5]:

Lemma 2.10. (Theorem 2.1, [5]) Any irreducible kG-module of dimension less than \frac{(q^{m}−1)(q^{m}−q)}{2(q+1)} is either the

trivial module, or a Weil module of dimension \frac{(q^{m}±1)}{2}.

This result shows that we must have U^{⊥}/U ≃ W_{1} ⊕ W_{2}, where W_{1} and W_{2} are irreducible Weil modules of dimension \frac{q^{m}−1}{2}. Hence, we now have all of the kG-composition factors of k^{L_{1}}.

If we let V_{1} and V_{2} be submodules such that U ⊂ V_{1}, V_{2} ⊂ U^{⊥} and V_{1}/U ≃ W_{1} and V_{2}/U ≃ W_{2}, then the above arguments yield the following filtration of k^{L_{1}} :

\begin{matrix}k^{L_{1}} \\ | \\ U^{⊥} \\ ⧸⧹ \\ V_{1} V_{2} \\ ⧹⧸ \\ U \\ | \\ \{0\} \end{matrix} (33)

2.5. The kG-submodule lattice of k^{L_{1}}. By the minimality of U′ (see Lemma 2.5) it suffices to determine the submodule structure of (U′)^{⊥}/U′. We start by defining submodules C and C^{+} as follows:

C := Im η_{m,1}

and

C^{+} := ⟨x + y | x, y ∈ C⟩_{k} .

We will need the following results:

Lemma 2.11. (a) C^{+} ⊊ C

(b) Hom_{kG}(k^{L_{m}}, W_{i}) = {0}, for i = 1, 2.

(c) C has no quotient isomorphic to W_{i}, for i = 1, 2.

(d) Hom_{kG}(k^{L_{m}}, k) ≃ k

(e) C^{+} is the unique maximal submodule of C.

Proof. An easy computation shows that

[ω + s_{Δ(ω)}, η_{m,1}(M)]_{k} = 1,

for all ω ∈ L_{1} and all M ∈ L_{m}. From the definition of C^{+}, we then deduce that C^{⊥} ⊊ (C^{+})^{⊥}. Now (a) follows by taking orthogonal complements.

We have Res_{S}^{G} W_{i} ≃ W_{i}, for i = 1, 2. But from our construction of the W_{i} in §2.3, we know that they are fixed point free for S. Therefore, G_{M} has no fixed points on the Res_{G_{M}}^{G} W_{i} ≃ W_{i}, i.e.

Hom_{kG_{M}} (k, Res_{G_{M}}^{G} W_{i}) = {0},

for i = 1, 2. Since

k^{L_{m}} = Ind_{G_{M}}^{G} k,

the assertions in (b) follow from Frobenius reciprocity. Since C is a homomorphic image of k^{L_{m}}, we see that (c) is an immediate consequence of (b).

Again by Frobenius reciprocity, we have

Hom_{kG}(k^{L_{m}}, k) ≃ Hom_{kG_{M}} (k, k) ≃ k.

This proves part (d).

It follows from (a) and (d) that C^{+} is the unique maximal submodule of C with trivial quotient. From Lemma 2.5, we have U′ ⊆ C. Using the inner product computation at the start of the proof, we have C ⊆ (U′)^{⊥}. Thus,

U′ ⊆ C ⊆ (U′)^{⊥}.

Since

(U′)^{⊥}/(U′) = k + k + W_{1} + W_{2},

we know that any maximal submodule of C with non-trivial quotient must have quotient W_{1} or W_{2}, which is impossible by (c). Then (e) follows.

Since C^{+} is not orthogonal to C, we get

C ∩ C^{⊥} ⊊ C,

and hence

C ∩ C^{⊥} ⊊ C^{+}.

by Lemma 2.11.e. Thus, the quotient C/(C ∩C^{⊥}) has at least 2 composition factors. Furthermore, C/(C∩C^{⊥}) has a unique maximal submodule, namely C^{+}/(C ∩ C^{⊥}).

Lemma 2.12. (a) C/(C ∩ C^{⊥}) is self-dual.

(b) C/(C ∩ C^{⊥}) has a unique maximal submodule and a unique simple submodule. Both the head and socle of C/(C ∩ C^{⊥}) are trivial.

Proof. The form induced by [−, −]_{k} on the quotient C/(C ∩ C^{⊥}) is nonsingular and therefore induces an isomorphism between C/(C ∩ C^{⊥}) and its dual. Since the form is G-invariant, this is actually a kG-isomorphism, and (a) follows. Part (b) then follows immediately from the remarks following Lemma 2.11.

In light of Lemma 2.9, it follows from Clifford’s theorem that any kΓmodule having at least one of the W_{i} as a composition factor for G must have the other as well. Since C and C^{⊥} are modules for Γ, we deduce from Lemma 2.12 that either C/(C ∩ C^{⊥}) = k + k (34)

or

C/(C ∩ C^{⊥}) = k + k + W_{1} + W_{2}. (35)

Suppose by way of contradiction that (34) holds. By Lemma 2.12.b it must then be the case that C/(C ∩ C^{⊥}) is uniserial. But as G is perfect, it has no module which is a non-split extension of the simple trivial module by itself. So (35) holds and it follows that C = (U′)^{⊥} and C^{+} = U^{⊥}.

We may now state our main result:

Theorem 2.13. Using the above notation, k^{L_{1}} has the following submodule lattice:

[figure: \underline{m even}; k^{L_{1}}; k; _{⟨1⟩_{⊥}}; X; C; k; C^{+}; W_{2}; W_{1}; V_{1}; W_{1}; V_{2}; W_{2}; (C^{+})^{⊥}; k; C^{⊥}; X; ⟨1⟩; k; \{0\}; \underline{m odd}; k^{L_{1}}; X′; k; C; k; _{⟨1⟩_{⊥}}; X′; C^{+}; W_{2}; W_{1}; V_{1}; W_{1}; V_{2}; W_{2}; (C^{+})^{⊥}; X′; k; ⟨1⟩; k; C^{⊥}; X′; \{0\}]

Proof. Let N be a kG-submodule of k^{L_{1}}. Assume N ≠ {0} or ⟨1⟩. Then we know from Lemma 2.5 that U′ ⊆ N. If we assume that N ≠ k^{L_{1}} or ⟨1⟩^{⊥}, then we have that N^{⊥} ≠ {0} or ⟨1⟩. But then from Lemma 2.5 we have U′ ⊆ N^{⊥}, i.e. N ⊆ (U′)^{⊥}. Thus,

U′ ⊆ N ⊆ (U′)^{⊥}.

From the remarks immediately following Lemma 2.12, we know that U′ = C^{⊥} and U^{⊥} = C^{+}. Thus, if N ≠ U′ or (U′)^{⊥}, it follows from Lemma 2.12.b that

U ⊆ N ⊆ U^{⊥}.

But as U^{⊥}/U ≃ W_{1} ⊕ W_{2}, and since W_{1} ≇ W_{2}, we see that V_{1} and V_{2} are the only kG-submodules between U and U^{⊥}, i.e. N = V_{1} or V_{2}.

Although the dimensions of the submodules pictured above have been given earlier, for convenience we recall here that

dim_{k} C = 1 + \frac{q(q^{m} − 1)(q^{m−1} + 1)}{2(q − 1)}

and

dim_{k} V_{1} = dim_{k} V_{2} = \frac{q^{2m} − 1}{2(q − 1)}.

It has already been noted (see the comments immediately following Lemma 2.9) that C^{+}/(C^{+})^{⊥} is a simple kΓ-module. In the sequel, we shall denote this simple quotient by W. Since all of the kG-submodules of k^{L_{1}} except for the V_{i} are also kΓ-submodules, we then have

Corollary 2.14. The pictures in Theorem 2.13 are the Hasse diagrams for Γ = CSp(2m, q), except that the quotient W = C^{+}/(C^{+})^{⊥} is irreducible.

By abuse of notation, we shall also denote by X and X′ the Γ-composition factors which when restricted to G are isomorphic to the composition factors X and X′, respectively, which are mentioned above. However, we caution the reader that these G-modules need not have unique extensions to Γ-modules.

Remark 2.15. From [5] we know that the Weil modules can be realized over 𝔽_{2} if and only if q ≡ ±1 mod 8. If q ≡ ±3 mod 8, then the smallest field of definition for the Weil modules is 𝔽_{4}. With this insight, we may deduce from Theorem 2.13 the complete kG-submodule lattice of 𝔽^{L_{1}} for any field 𝔽 of characteristic 2. Explicitly, if q ≡ ±1 mod 8 and 𝔽 is arbitrary, or if q ≡ ±3 mod 8 and 𝔽_{4} ⊆ 𝔽, then the submodule lattice of 𝔽^{L_{1}} is as pictured in Theorem 2.13. However, if q ≡ ±3 mod 8 and 𝔽_{4} ⊈ 𝔽, then the submodule lattice is as pictured in Theorem 2.13 except that the quotient U^{⊥}/U is irreducible.

Remark 2.16. In [2] (see pg. 353), Bagchi et al conjectured the submodule lattice of 𝔽_{2}^{L_{1}} for G. We now see from Theorem 2.13 and the preceding remark that their conjectured structure is correct in all cases except when q ≡ ±1 mod 8, in which case the Weil modules have been neglected. However, their structure is correct for the conformal group Γ.

3. the Sp(4, q)-submodule structure of k^{L_{2}}

Throughout this section we take m = 2, i.e. V is a 4-dimensional nonsingular symplectic 𝔽_{q}-vector space and G = Sp(4, q). So L_{1} is the set of 1-spaces in V, and L_{2} is the set of maximal isotropic subspaces in V. As usual, k is an algebraically closed field of characteristic 2.

Let M ∈ L_{2} and define

Φ(M) := {N ∈ L_{2} | dim_{𝔽_{q}} M ∩ N = 1} .

Now let

s_{Φ(M)} := ∑_{N∈Φ(M)} N ∈ k^{L_{2}}.

Define submodules C and P as follows:

C := ⟨M | M ∈ L_{2}⟩_{k}

and

P := ⟨s_{Φ(M)} | M ∈ L_{2}⟩_{k} .

Now put

C^{+} := ⟨M + N | M, N ∈ L_{2}⟩_{k}

and

P^{+} := ⟨P_{1} + P_{1} | P_{1}, P_{2} ∈ P⟩_{k}

Finally, denote Y := C^{⊥}/P^{+}. We now prove

Theorem 3.1. Using the above notation, k^{L_{2}} has the following submodule lattice for Γ = CSp(2m, q) :

[figure: k^{L_{2}}; k; _{⟨1⟩{⊥ \atop k}}; W; P^{+⊥}; Y; k; C; k; P^{⊥}; Y; X; C^{+}; X; C^{+⊥}; Y; k; P; k; C^{⊥}; Y; P^{+}; W; _{⟨1⟩_{k}}; k; \{0\}]

Proof. That all the containments pictured above actually hold has been proven in [2] by Bagchi et al. Furthermore, in the same paper the authors have determined the dimensions of all the submodules pictured above, and we will freely use that information here.

The incidence map η_{1,2} : k^{L_{1}} → k^{L_{2}} induces an isomorphism k^{L_{1}}/C^{⊥} ≃ C. By Corollary 2.14, we then have that C is uniserial with composition factors as indicated in the picture above. The incidence map η_{2,1} : k^{L_{2}} → k^{L_{1}} induces an isomorphism k^{L_{2}}/C^{⊥} ≃ C. By Corollary 2.14, we then have that k^{L_{2}}/C^{⊥} is uniserial with composition factors as indicated in the picture above. We now see that Res_{G}^{Γ} k^{L_{2}} has the Weil modules, W_{1} and W_{2}, as composition factors, and that each occurs with multiplicity at least 2. Now from [10] we know that k^{L_{2}} has composition length 10 for G. It then follows that Y (= C^{⊥}/P^{+}) is simple for G, and hence simple for Γ.

Now from Theorem 2.2 of [9], we know that every submodule of k^{L_{2}} which is not contained in ⟨1⟩_{k} must contain P^{+}. Thus, to verify the conjectured structure, it suffices to prove that P^{+⊥}/P^{+} is as pictured.

Since P ∩ C^{⊥} = P^{+} we see that

P^{+⊥}/P^{+} = (C/P^{+}) ⊕ (C^{⊥}/P^{+}), (36)

i.e. the quotient P^{+⊥}/P^{+} is the direct sum of a uniserial module and a simple module.

It is determined in [2] that dim_{k} (C^{⊥}/P^{+}) = \frac{q(q−1)^{2}}{2} , so that C^{⊥}/P^{+} is non-trivial. Since C/P^{+} has trivial socle and trivial head, we see that we have have found all the submodules of P^{+⊥}/P^{+}. The result follows.

In the isomorphism k^{L_{1}}/C^{⊥} ≃ C, let V_{1} and V_{2} denote the images in C of V_{1}/C^{⊥} and V_{2}/C^{⊥}, respectively. Thus,

⟨1⟩_{k} ⊂ V_{1}, V_{2} ⊂ P^{+}

and

P^{+}/ ⟨1⟩_{k} ≃ W_{1} ⊕ W_{2}.

Let V_{3} := V_{1}^{⊥} and V_{4} := V_{2}^{⊥}. Thus,

⟨1⟩_{k}^{⊥} ⊃ V_{3}, V_{4} ⊃ P^{+⊥}

and

⟨1⟩_{k}^{⊥} /P^{+⊥} ≃ W_{1} ⊕ W_{2}.

Corollary 3.2. The picture in Theorem 3.1 shows every kG-submodule of k^{L_{2}}, except for the modules V_{i}, for i = 1 to 4.

Proof. By Theorem 2.2 of [9], we know that every submodule of k^{L_{2}} which is not contained in ⟨1⟩_{k} must contain one of V_{1} or V_{2}. Therefore, in light of Theorem 3.1, it suffices to show that any submodule which properly contains either V_{1} or V_{2} must contain the other one as well, i.e. must contain P^{+}.

Let N be a kG-submodule of k^{L_{2}} which properly contains V_{1} but does not contain V_{2}. Assume that N is chosen minimal with respect to this property, i.e. assume no submodule of N has this property also. Denote by Ñ the Γ-module generated by N.

Assume first that N ⊆ P^{+⊥}. Since N properly contains V_{1}, we have P^{+} ⊊ N + P^{+}. Thus,

P^{+} ⊊ N + P^{+} ⊆ P^{+⊥}.

But we know that, in general, any kG-submodule A such that P^{+} ⊆ A ⊆ P^{+⊥} is actually a kΓ−submodule. Hence, N +P^{+} is a kΓ-submodule. Since N + P^{+} ⊆ Ñ, we deduce

N + P^{+} = Ñ.

We have

(N + P^{+})/N ≃ P^{+}/(P^{+} ∩ N)

= P^{+}/V_{1} ≃ W_{2}.

Therefore,

Hom_{kG}(Res_{G}^{Γ} N + P^{+}, Res_{G}^{Γ} W) ≠ {0}.

Let H be the subgroup (of index 2) of Γ which is generated by G along with all of the scalar matrices. We then have

Hom_{kH}(Res_{H}^{Γ} N + P^{+}, Res_{H}^{Γ} W) ≠ {0}.

Now H acts trivially on Hom_{kH}(Res_{H}^{Γ} N+P^{+}, Res_{H}^{Γ} W), so we may consider the induced action of the 2-group Γ/H on Hom_{kH}(Res_{H}^{Γ} N + P^{+}, Res_{H}^{Γ} W). But the action of a 2-group on a vector space over a field of characteristic 2 always has a fixed point. Thus,

Hom_{kΓ}(N + P^{+}, W) ≠ {0},

i.e. N +P^{+} has a quotient isomorphic to W. But from Theorem 3.1 we know that no submodule of P^{+⊥} which contains P^{+} has such a quotient. Thus, we have obtained a contradiction.

Now assume that N ⊈ P^{+⊥}. Then (N + P^{+}) ⊈ P^{+⊥} either. But

(N + P^{+})^{⊥} = N^{⊥} ∩ P^{+⊥}

is, of course, a kG-submodule of P^{+⊥}. From the first part, we know all such submodules. Obviously, we cannot have P^{+} ⊆ (N + P^{+})^{⊥}, so it must be the case that (N + P^{+})^{⊥} = {0}, ⟨1⟩_{k}, V_{1}, or V_{2}.

Consider the following composition series for N + P^{+} :

{0} ⊂ ⟨1⟩_{k} ⊂ V_{1} ⊂ N ⊂ N + P^{+}.

Note that the simplicity of Q := N/V_{1} is a consequence of the minimality of N. We know all of the kG-composition factors of k^{L_{2}}, but we do not know the isomorphism class of Q. However, by considering the various possibilities for Q, we may see that regardless of its isomorphism class, we always have that the dimension of (N + P^{+})^{⊥} is strictly greater than the dimensions of {0}, ⟨1⟩_{k}, V_{1}, and V_{2}. Thus, we have reached a contradiction and the assertion has been established.

Remark 3.3. In view of Remark 2.15, we see that we may deduce from Theorem 3.1 and its corollary the submodule structure of 𝔽^{L_{2}} where 𝔽 is any field of characteristic 2.

Remark 3.4. In [2] (see pg. 352), Bagchi et al conjectured the submodule lattice of 𝔽_{2}^{L_{2}} for G. We now see from Corollary 3.2 and the preceding remark that their conjectured structure is correct in all cases except when q ≡ ±1 mod 8, in which case the Weil modules have been neglected. However, their structure is correct for the conformal group Γ.

Remark 3.5. Let E be a 5-dimensional vector space over 𝔽_{q} which is endowed with a non-singular orthogonal geometry. Denote by O(5, q) the corresponding full orthogonal group and by Ω(5, q) its derived subgroup. We will write L_{1}(E) for the set of isotropic 1-spaces in E. There is a natural identification (see [3]) of the elements of L_{2} with the elements of L_{1}(E). This identification carries Γ onto O(5, q) and G onto Ω(5, q). Thus, we see that the above results give us the submodule structure of 𝔽^{L_{1}(E)}, where 𝔽 is any field of characteristic 2.

References

[1] E.Artin, Geometric Algebra, New York, 1957.

[2] B.Bagchi, A.E.Brouwer, H.A.Wilbrink, Notes on Binary Codes Related to the O(5, q) Generalized Quadrangle for Odd q, Geometriae Dedicata 39 (1991), 339-355.

[3] C.T.Benson, On the Structure of Generalized Quadrangles, Journal of Algebra 15 (1970), 443-454.

[4] W.Feit, The Representation Theory of Finite Groups, North Holland, Amsterdam, 1982.

[5] R.M.Guralnick, K.Magaard, J.Saxl, P.H.Tiep, Cross Characteristic Representations of Odd Characteristic Symplectic Groups and Unitary Groups, (submitted).

[6] D.G.Higman, Finite Permutation Groups of Rank 3, Math Z. 86 (1964), 145-156.

[7] K.Iwasawa, Local Class Field Theory, Oxford, 1986.

[8] M.Klemm, Über die Reduktion von Permutation Moduln, Math Z. 143 (1975), 113-117.

[9] M.W.Liebeck, Permutation Modules for Rank 3 Symplectic and Orthogonal Groups, Journal of Algebra 92 (1985), 9-15.

[10] D.White, The 2-Decomposition Numbers of Sp(4, q), q Odd, Journal of Algebra 131 (1990), 703-725.

Department of Mathematics, University of Florida, Gainesville, FL 32611

  E-mail address: lmj@math.ufl.edu

Department of Mathematics, University of Florida, Gainesville, FL 32611

  E-mail address: sin@math.ufl.edu

Department of Mathematics, University of Florida, Gainesville, FL 32611

  E-mail address: tiep@math.ufl.edu