The subring of group cohomology constructed by permutation representations^{∗}

David J. Green  Ian J. Leary^{†} Björn Schuster

21 April 1999

Abstract

Each permutation representation of a finite group G can be used to pull cohomology classes back from a symmetric group to G. We study the ring generated by all classes that arise in this fashion, describing its variety in terms of the subgroup structure of G.

We also investigate the effect of restricting to special types of permutation representations, such as GL_{n}(𝔽_{p}) acting on flags of subspaces.

Introduction

Each action of a finite group G on a set X can be used to pull back cohomology classes from the cohomology of the symmetric group on X to the cohomology of G. For example, the characteristic classes of Segal and Stretch [6] arise in this way.

We shall study the cohomology classes that come from all actions of a fixed group G by taking the ring S_{h} they generate and investigating its variety. In Theorem 1.5 we obtain a description of this variety in terms of the group structure of G. Typically the inclusion of S_{h} in the cohomology ring is not an inseparable isogeny; but it does always induce a bijection of irreducible components. Equivalently, distinct minimal prime ideals in the cohomology ring have distinct intersections with S_{h}. The idea of studying the variety of the cohomology ring originates in Quillen’s paper [5]. Our results rely on work in [4], where two of the current authors suggest an extension of Quillen’s results to certain subrings of the cohomology ring.

We also investigate what happens when we impose conditions on the G-sets by putting restrictions on the point stabilizers. In particular we show that, for large values of n, the GL_{2n}(𝔽_{p}) actions with parabolic stabilizers give rise to a strictly smaller subring than the subring for arbitrary actions, which in turn is strictly smaller than the whole cohomology ring.

∗1991 Mathematics Subject Classification 20J06.

†Partially funded by the Nuffield Foundation, grant no. SCI/180/96/127

Throughout this paper, G will be a finite group and p a prime number. We write H^{∗}(G) for the mod-p cohomology H^{∗}(G, 𝔽_{p}) of G.

1 Definitions and our main theorem

First we describe the object of study precisely.

Definition 1.1 A non-empty family F of subgroups of G will be called admissible if it is closed under conjugation in G, and the subgroup ⋂_{H∈F} H of G is a p′-group. A G-set X will be called an F-set if each point stabilizer belongs to F.

In particular, the family F_{h} consisting of all subgroups of G is admissible, and all G-sets are F_{h}-sets.

Definition 1.2 Each finite G-set X induces a homomorphism ρ_{X}: G → Σ_{n}, where n is |X|. This induces in turn a ring homomorphism ρ_{X}^{∗}: H^{∗}(Σ_{n}) → H^{∗}(G). Define S_{F} as the subring of H^{∗}(G) generated by all Im(ρ_{X}^{∗}) with X an F-set.

We shall now determine the variety of this ring S_{F}. The following definition is needed to state the result.

Definition 1.3 Denote by A_{F} the category whose objects are the elementary abelian p-subgroups of G, with A_{F}(V, W) the set of injective group homomorphisms f: V → W satisfying: for every H ∈ F the V -sets f^{!}(G/H) and G/H are isomorphic. Here f^{!}(G/H) means the following action of V on G/H:

k ∗ gH = f(k)gH .

Remark 1.4 The category A_{F_{h}} is identified in Lemma 2.2.

Recall that the variety var(R) of a connected graded commutative 𝔽_{p}-algebra R is the functor that assigns to each algebraically closed field k the topological space of ring homomorphisms from R to k with the Zariski topology.

Theorem 1.5 The cohomology ring H^{∗}(G) is finitely generated as a module over S_{F}. The restriction maps in cohomology induce a natural homeomorphism

colim_{V ∈A_{F}} var(H^{∗}(V)) ≅ var(S_{F}) .

Proof. Let H_{1}, …, H_{r} be a full set of class representatives for the conjugation action of G on F. Let X be the G-set (G/H_{1})⨿⋯⨿(G/H_{r}), and n = |X|. Then X is an F-set, and the kernel of the associated group homomorphism ρ: G → Σ_{n} is a p′-group by admissibility.

Now compose ρ with the regular representation reg_{Σ_{n}} of Σ_{n}. We obtain a degree n! representation of G, whose restriction to a Sylow p-subgroup P of G is a direct sum of copies of the regular representation. In particular, it is a faithful representation of P. The Chern classes of reg_{Σ_{n}} ∘ρ lie in S_{F} as they are images under ρ^{∗}. Hence by Venkov’s proof [7] of the Evens–Venkov theorem, H^{∗}(P) is finitely generated as a module over S_{F}. Therefore H^{∗}(G) is finitely generated too.

This representation reg_{Σ_{n}} ∘ρ also restricts to every elementary abelian psubgroup of G as a (non-zero) direct sum of copies of the regular representation, and so is p-regular in the sense of [4]. So S_{F} contains the Chern classes of a p-regular representation. Moreover, the ring S_{F} is clearly homogeneously generated and closed under the action of the Steenrod algebra. By Theorem 6.1 of [4] it follows firstly that var(S_{F}) is a colimit of the desired form over some category of elementary abelians; and secondly that Lemma 1.6 identifies this category as being A_{F}.

Lemma 1.6 Let V, W be elementary abelian subgroups of G, and f: V → W an injective group homomorphism. Then f lies in A_{F} if and only if for every x ∈ S_{F}, the class Res_{V}^{G}(x) − f^{∗} Res_{W}^{G} (x) lies in the nilradical of H^{∗}(V).

Proof. Suppose f ∈ A_{F}. Pick any F-set Y, and let ρ: G → Σ_{|Y |} be the associated group homomorphism. Since the V -sets Y and f^{!}(Y) are isomorphic, f induces a map ρ(V) → ρ(W), and this is conjugation by some σ ∈ Σ_{|Y |}. Hence Res_{V}^{G} −f^{∗} Res_{W}^{G} kills Im(ρ^{∗}).

Conversely, suppose that f ∉ A_{F}. Recall that in the proof of Theorem 1.5 we constructed an F-set X, such that the kernel of the associated group homomorphism ρ: G → Σ_{|X|} is a p′-group. By assumption on f there is some H ∈ F with f^{!}(G/H), G/H non-isomorphic as V -sets. Define Y by

Y = \begin{cases}X ⨿ (G/H) & if f^{!}(X), X isomorphic as V -sets \\ X & otherwise. \end{cases}

Then Y is an F-set and V acts faithfully on Y, f^{!}(Y), but these two V -sets are non-isomorphic.

We have thus constructed embeddings of V and W in Σ_{|Y |}, such that f is not induced by conjugation in Σ_{|Y |}. Therefore there is a class ξ ∈ H^{∗}(Σ_{|Y |}) such that Res_{V}^{Σ_{|Y |}}(ξ) − f^{∗} Res_{W}^{Σ_{|Y |}}(ξ) is not nilpotent (apply the results of [4, §9] to the group Σ_{|Y |}). Moreover, these embeddings of V, W in Σ_{|Y |} factor through G → Σ_{|Y |}. Pulling ξ back to H^{∗}(G), we get the desired class.

2 Examples

Definition 2.1 We define the hereditary category A_{h} of G to be A_{F_{h}}, where F_{h} is the admissible family of all subgroups of G. Write S_{h} for S_{F_{h}}.

Recall that ∼_{G} denotes the equivalence relation conjugacy in G.

Lemma 2.2 Let f: V → W be an injective group homomorphism between elementary abelian subgroups of G. Then f lies in A_{h} if and only if f(U) ∼_{G} U for every elementary abelian U ≤ V.

Let F be an admissible family containing all nontrivial elementary abelian p-subgroups of G. Then A_{F} = A_{h}.

Remark 2.3 This property of A_{h} is the reason for the name hereditary.

Proof. We prove the first part holds for any F satisfying the conditions of the second part, not just for F_{h}.

First suppose that U is a subgroup of V and f(U) ≁_{G} U. Then the V -set G/U has a point stabilized by U, but f^{!}(G/U) does not. Hence these two V -sets are not isomorphic, and so f does not lie in A_{F}.

For the if part, consider any H ∈ F and any U ≤ V. The coset gH is fixed by U if and only if U^{g} ≤ H. Since f(U) ∼_{G} U, the number of U-fixed points in f^{!}(G/H) is the same as for G/H. It follows that the V -sets f^{!}(G/H) and G/H are isomorphic.

Corollary 2.4 The category A_{h} is the unique largest category of elementary abelians which is closed in the sense of [4, §9], and in which objects are isomorphic if and only if they are conjugate as subgroups of G.

Proof. Closure means that all inclusion and conjugation maps are contained in A_{h}; that isomorphisms lie in A_{h} if and only if their inverses do; and that f_{|U}: U → f(U) lies in A_{h} for every f: V → W in A_{h} and every U ≤ V.

Remark 2.5 It follows that “intersection with S_{h}” induces a bijection from the minimal primes of H^{∗}(G) to those of S_{h}. Hence the irreducible components of var(H^{∗}(G)) and of var(S_{h}) are in natural one-to-one correspondence.

Definition 2.6 Let G be the general linear group GL_{n}(𝔽_{p}). We define the parabolic category A_{π} to be A_{F_{π}}, where F_{π} is the collection of all parabolic subgroups of G. Write S_{π} for S_{F_{π}}.

Proposition 2.7 The parabolic category is admissible. We have

var(S_{h}) ≅ colim_{V ∈A_{h}} var(H^{∗}(V)) and var(S_{π}) ≅ colim_{V ∈A_{π}} var(H^{∗}(V)) .

Proof. The upper triangular matrices constitute a parabolic subgroup, as do the lower triangular matrices. These two groups intersect in a p′-group, so F_{π} is admissible. Apply Theorem 1.5 for the admissible families F_{h} and F_{π}.

Define the Quillen category A to be the category whose objects are the elementary abelian p-subgroups of G, with morphisms induced by inclusion and conjugation. It is a well-known theorem of Quillen (see [2, §9.2]) that the restriction maps induce a natural isomorphism

colim_{V ∈A} var(H^{∗}(V)) ≅ var(H^{∗}(G)) .

It follows from [4] that the inclusion of S_{F} in H^{∗}(G) induces an isomorphism of varieties if and only if A_{F} = A, and that S_{F_{1}}, S_{F_{2}} have the same variety as each other if and only if A_{F_{1}} = A_{F_{2}}.

Example 2.8 Let p be an odd prime, and let 1 < q < p. For any finite group G and any elementary abelian V ≤ G, the automorphism v ↦ v^{q} of V lies in A_{h} by Lemma 2.2. But in general this map does not lie in A. An example is when G is abelian (and not a p′-group). For such groups, the inclusion of S_{h} in H^{∗}(G) in not an inseparable isogeny.

Example 2.9 In Corollary 3.4, we shall see that for n ≥ 3 and G the group GL_{2n}(𝔽_{p}), there is a rank two elementary abelian subgroup E of G such that not all automorphisms of E lie in A; and yet all nontrivial elements of E are conjugate in G, which means that all automorphisms of E lie in A_{h}. Hence the inclusion of S_{h} in H^{∗}(G) is not an inseparable isogeny.

Example 2.10 In Theorem 3.6, we shall see that for n ≥ 6 and G the group GL_{2n}(𝔽_{p}), there are non-conjugate rank two elementary abelian subgroups of G which are isomorphic in A_{π}. Hence the varieties of S_{π}, S_{h} and H^{∗}(G) are all distinct.

Example 2.11 The elementary abelian p-subgroups of G form an admissible family, as do all p-subgroups of G. If G has p-rank at least two, then we can omit the trivial subgroup in both families.

In all these cases, the category A_{F} is equal to A_{h} by Lemma 2.2. Hence inclusion of S_{F} in S_{h} is an inseparable isogeny.

Example 2.12 Following Alperin [1], we define a subgroup H of an abstract finite group G to be parabolic if H = N_{G}(O_{p}(H)). That is, the parabolics are the normalizers of the p-stubborn subgroups. For G = GL_{n}(𝔽_{p}), this coincides with the normal definition of parabolic subgroup.

If O_{p}(G) = 1 then the parabolic subgroups and the p-stubborn subgroups each form admissible families, since Sylow p-subgroups are p-stubborn and O_{p}(G) is the intersection of all Sylow p-subgroups.

For p = 11 the sporadic finite simple group J_{4} has the trivial intersection property: distinct Sylow p-subgroups intersect trivially. Hence the parabolic subgroups are the admissible family consisting of J_{4} itself and the Sylow normalizers. The action of any order p cyclic subgroup on cosets of a Sylow normalizer has one fixed point, with the remaining orbits having length p. As there are two distinct conjugacy classes of order p cyclics, the parabolic category is larger than the hereditary category. The cohomology of J_{4} at the prime 11 was computed in [3].

Example 2.13 In general the subring S_{h} is far larger than the subring generated by Chern classes of permutation representations: i.e., the subring generated by all images of H^{∗}(BU(n)) under homomorphisms G → Σ_{n} → U(n), where Σ_{n} is embedded in U(n) as the permutation matrices.

In [4] it was shown that the variety for this subring is the colimit over the category A_{P}, where f: V → W lies in A_{P} if and only if f(U) ∼_{G} U for every cyclic subgroup U of V. This category is in general far larger than A_{h}. For example, there are elementary abelian p-groups of rank two in GL_{3}(𝔽_{p}) that are not conjugate (and hence not isomorphic in A_{h}), but are isomorphic in A_{P}.

3 An extended example

Fred Cohen asked the third author about the subring of H^{∗}(GL_{n}(𝔽_{p})) generated by the permutation representations on flags. In our language, the question concerns the subring S_{π}. This question provided the starting point for the current paper. We provide a partial answer to this question by comparing the varieties for H^{∗}(GL_{n}(𝔽_{p})), S_{h} and S_{π}, which is equivalent to comparing the categories A, A_{h} and A_{π}. Recall that there are inclusions

A ⊆ A_{h} ⊆ A_{π} .

Let G be the general linear group GL_{2n}(𝔽_{p}). We show that all three categories are distinct for n ≥ 6. The most time consuming part is showing that A_{π} differs from A_{h} for such n. By Corollary 2.4 it suffices to show that there are elementary abelian p-subgroups of G which are isomorphic in A_{π} but not conjugate in G. We shall find rank 2 examples using modular representation theory.

Let p be a prime number, and let A, B be generators for the rank 2 elementary abelian p-group V ≅ C_{p}×C_{p}. To each matrix J ∈ GL_{n}(𝔽_{p}), there is an associated representation ρ_{J}: V → GL_{2n}(𝔽_{p}) defined by

A ⟼^{ρ_{J}} (\begin{matrix}I & I \\ 0 & I \end{matrix}) B ⟼^{ρ_{J}} (\begin{matrix}I & J \\ 0 & I \end{matrix}) ,

where I ∈ GL_{n}(𝔽_{p}) is the identity matrix. The following lemma is well-known in the modular representation theory of V.

Lemma 3.1 Let J, J′ ∈ GL_{n}(𝔽_{p}). Then the representations ρ_{J}, ρ_{J′} are isomorphic if and only if J, J′ are conjugate in GL_{n}(𝔽_{p}).

Proof. The centralizer of ( \begin{matrix}I & I \\ 0 & I \end{matrix}) consists of all matrices of the form ( \begin{matrix}A & B \\ 0 & A \end{matrix}). The conjugate of ( \begin{matrix}I & J \\ 0 & I \end{matrix}) under such a matrix is ( \begin{matrix}I & J′ \\ 0 & I \end{matrix}) with J′ = AJA^{−1}.

Lemma 3.2 For any matrix M ∈ GL_{n}(𝔽_{p}), the matrix ( \begin{matrix}I & M \\ 0 & I \end{matrix}) is conjugate in

GL_{2n}(𝔽_{p}) to ( \begin{matrix}I & I \\ 0 & I \end{matrix}).

Proof. Conjugate on the right by ( \begin{matrix}M & 0 \\ 0 & I \end{matrix}).

First we compare the categories A_{h} and A.

Lemma 3.3 Suppose there is a primitive element θ ∈ 𝔽_{p^{n}}/𝔽_{p} with minimal polynomial f such that θ + 1 is not a root of f. Then the Quillen category A for G = GL_{2n}(𝔽_{p}) is strictly smaller than the hereditary category A_{h}.

Proof. Let J ∈ GL_{n}(𝔽_{p}) be the matrix in rational canonical form with characteristic polynomial f. Since f is irreducible, J has no eigenvalues in 𝔽_{p}. In particular, this means that I + J lies in GL_{n}(𝔽_{p}). The condition on the roots of f means that J and I + J have distinct characteristic polynomials, and so are non-conjugate in GL_{n}(𝔽_{p}).

Let E be Im(ρ_{J}), the rank 2 elementary abelian generated by a = ρ_{J}(A) and b = ρ_{J}(B). Hence

a = (\begin{matrix}I & I \\ 0 & I \end{matrix})  b = (\begin{matrix}I & J \\ 0 & I \end{matrix})  ab = (\begin{matrix}I & I + J \\ 0 & I \end{matrix}) .

Let ϕ be the automorphism of E which fixes a and sends b to ab. By the proof of Lemma 3.1 we see that ϕ ∉ A, since J and I + J are not conjugate. To see that ϕ ∈ A_{h}, it suffices by Lemma 2.2 to show that e, ϕ(e) are conjugate in G = GL_{2n}(𝔽_{p}) for each nontrivial e ∈ E. But this follows from Lemma 3.2.

Corollary 3.4 Set n_{0} = 2 for p ≥ 3 and n_{0} = 3 for p = 2. For G = GL_{2n}(𝔽_{p}) and n ≥ n_{0}, the Quillen category A is strictly smaller than the hereditary category A_{h}.

Proof. We show that there is a θ satisfying the conditions of Lemma 3.3. The Galois group of 𝔽_{p^{n}}/𝔽_{p} is cyclic of order n, generated by the Frobenius automorphism. Hence θ ∈ 𝔽_{p^{n}} has the same minimal polynomial as θ + 1 if and only if θ is a root of x^{p^{m}} − x − 1 for some m < n. Therefore there are at least p^{n} − p^{n−1} − p^{n−2} − ⋯ − p elements θ of 𝔽_{p^{n}} such that θ, θ + 1 do not have the same minimal polynomial. If p ≥ 3 and n ≥ 2 then this exceeds p^{n−1}, and there are at most p^{n−1} non-primitive elements of 𝔽_{p^{n}}/𝔽_{p}: hence there exists a θ of the required form.

Now suppose that p is 2. The roots of x^{2^{m}} − x − 1 all lie in 𝔽_{2^{2m}}, and so can only be primitive elements of 𝔽_{2^{n}}/𝔽_{2} if n | 2m. Since m < n, this can only happen if n = 2m. So the number of θ ∈ 𝔽_{2^{n}}/𝔽_{2} such that θ, θ + 1 have distinct minimal polynomials exceeds 2^{n−1} provided n > 2, and there are at most 2^{n−1} non-primitives. Again, the required θ exists.

Now we compare the categories A_{π} and A_{h}. To each irreducible degree n monic polynomial f ∈ 𝔽_{p}[x] there is an associated (n×n)-matrix J_{f} in rational canonical form. Define the representation ρ_{f}: V → GL_{2n}(𝔽_{p}) to be ρ_{J_{f}}. By Lemma 3.1, distinct f give rise to non-isomorphic representations.

Proposition 3.5 Let H be a parabolic subgroup of GL_{2n}(𝔽_{p}), and let f be an irreducible degree n polynomial. The embedding ρ_{f} turns G/H into a V -set. The isomorphism type of this V -set does not depend on f.

Theorem 3.6 Set n_{0} = 5 for p ≥ 5 and n_{0} = 6 for p = 2, 3. For G = GL_{2n}(𝔽_{p}) and n ≥ n_{0}, there are rank two elementary abelian subgroups of G which are isomorphic in the parabolic category A_{π} without being conjugate in G.

Proof. For any pair f, g of irreducible degree n monic polynomials over 𝔽_{p}, the isomorphism

ρ_{g} ∘ ρ_{f}^{−1} : Im(ρ_{f}) ⟶ Im(ρ_{g})

lies in A_{π} by Proposition 3.5. As distinct irreducible polynomials give rise to non-isomorphic representations, the number of irreducible g such that Im(ρ_{g}) is conjugate to a given Im(ρ_{f}) cannot exceed | Aut(V)| = (p^{2} − 1)(p^{2} − p). But for n ≥ n_{0} there are always more irreducibles than this. For the total number of irreducibles is equal to π_{n}/n, where π_{n} is the number of primitive elements in 𝔽_{p^{n}}/𝔽_{p}. We have π_{5} = p^{5} − p, π_{6} = p^{6} − p^{3} − p^{2} + p and π_{n} ≥ p^{n} − p^{n−2} for n ≥ 7. It is then straightforward to check that π_{n}/n > (p^{2} − 1)(p^{2} − p) for n ≥ n_{0}.

We now derive some results needed in the proof of Proposition 3.5. We take f to be a degree n irreducible polynomial over 𝔽_{p}, and J = J_{f} to be the associated matrix in rational canonical form.

Lemma 3.7 Let W be a proper subspace of 𝔽_{p}^{n}. Define m, r by m = dim(W) and m + r = dim(W + JW). Then there is partition λ = (λ_{1}, …, λ_{r}) of m with length r (so λ_{r} ≥ 1) and elements w_{1}, …, w_{r} of W, such that

1. The J^{a}w_{i} for 1 ≤ i ≤ r and 0 ≤ a ≤ λ_{i} − 1 are a basis for W, and

2. The J^{a}w_{i} for 1 ≤ i ≤ r and 0 ≤ a ≤ λ_{i} are a basis for W + JW.

We call such an r-tuple w_{1}, …, w_{r} a (J, λ)-base for W.

Furthermore, λ is uniquely determined by J, W; and the number of (J, λ)-bases for W depends solely on λ.

Observe that m + r ≤ n and that r ≤ m. Since J is the rational canonical form associated to an irreducible polynomial, there are no J-invariant subspaces other than 0 and 𝔽_{p}^{n}. Hence r = 0 if and only if m = 0.

Proof. The proof is by induction on m. The case m = 0 is clear. Now suppose that m > 0 and the result has been proved for dim(W) ≤ m − 1. Set W′ = W ∩ J^{−1}W, so dim(W′) = m − r. Define r′ by r′ = dim(W′ + JW′) − dim(W′).

As m > 0 we have m−r ≤ m−1, so can apply the result to W′. Thus we obtain a length r′ partition λ′ = (λ′_{1}, …, λ′_{r′}) of m − r and an r′-tuple w′_{1}, …, w′_{r′} ∈ W′. For 1 ≤ i ≤ r′ set λ_{i} = λ′_{i} + 1 and w_{i} = w′_{i}. Observe that

dim(W) − dim(W′ + JW′) = r − r′ .

Pick a basis w_{r′+1}, …, w_{r} for any complement of W′ + JW′ in W, and set λ_{i} = 1 for r′ < i ≤ r. Then λ is a length r partition of n, and the J^{a}w_{i} for 1 ≤ i ≤ r and 0 ≤ a ≤ λ_{i} − 1 are a basis for W.

Moreover, the J^{λ′_{i}}w′_{i} for 1 ≤ i ≤ r′ are a basis for a complement of W′ in W′ + JW′; and w_{r′+1}, …, w_{r} are a basis for a complement of W′ + JW′ in W. Hence the J^{λ_{i}−1}w_{i} for 1 ≤ i ≤ r are a basis for a complement of W′ in W. By definition of W′, this means that the J^{λ_{i}}w_{i} for 1 ≤ i ≤ r are a basis for a complement of W in W + JW. So the w_{i} constitute a (J, λ)-base.

Conversely, suppose that μ ⊣ m has length r, and that v_{1}, …, v_{r} is a (J, μ)- base for W. The elements J^{a}v_{i} for 0 ≤ a ≤ μ_{i} − 2 are a basis for W′, the J^{μ_{i}−1}v_{i} with μ_{i} ≥ 2 extend this to a basis for W′ + JW′, and the v_{i} with μ_{i} = 1 extend this to a basis for W. Hence the number of i with μ_{i} = 1 is equal to dim(W) − dim(W′ + JW′). Passing to W′, we deduce by induction that λ and μ are equal; and that λ alone determines the number of (J, λ)-bases w_{1}, …, w_{r}.

Lemma 3.8 Fix J and fix partitions λ, λ′. For any proper W ⊂ 𝔽_{p}^{n} with partition λ, the number of subspaces W′ of W with partition λ′ depends solely on λ, λ′.

Proof. Denote by w_{i}, w′_{i} the elements of a (J, λ)-base for W, W′ respectively. Set m = dim(W) and r = dim(W + JW) − m, as in Lemma 3.7.

Construct a basis b_{1}, …, b_{n} for 𝔽_{p}^{n} as follows:

• b_{1}, …, b_{m} is the the basis w_{1}, Jw_{1}, …, J^{λ_{1}−1}w_{1}, w_{2}, …, J^{λ_{r}−1}w_{r} for W given by Lemma 3.7;

• b_{m+1}, …, b_{m+r} is the corresponding extension J^{λ_{1}}w_{1}, …, J^{λ_{r}}w_{r} to a basis for W + JW;

• b_{m+r+1}, …, b_{n} is any extension to a basis for 𝔽_{p}^{n}.

Consider the matrix of J for this basis: the first m columns describe the action on W, and depend solely on λ. Hence the number of (J, λ′)-bases giving rise to a subspace of W with partition λ′ is independent of J. Moreover, the number of (J, λ′)-bases for any such W′ depends solely on λ′, by Lemma 3.7.

Corollary 3.9 Let λ be a partition of m < n. The number of proper subspaces W of 𝔽_{p}^{n} with partition λ is independent of f.

Proof. The codimension 1 subspaces of 𝔽_{p}^{n} all have partition (n − 1): so by Lemma 3.8 each contains the same number of such W, and this number is independent of f.

Corollary 3.10 Fix 0 ≤ m_{0} < m_{1} < ⋯ < m_{s} and partitions λ^{i} ⊣ m_{i}. The number of flags W_{0} ⊂ W_{1} ⊂ ⋯ ⊂ W_{s} of proper subspaces of 𝔽_{p}^{n} in which W_{i} has partition λ^{i} is independent of f.

Proof. The case s = 1 is Corollary 3.9. The general case is by induction on s using Lemma 3.8.

Proof of Proposition 3.5. We must show that for each parabolic subgroup H ≤ G, the isomorphism class of the V -set structure induced on G/H by ρ_{f} does not depend on f. Now, two finite V -sets X, Y are isomorphic if and only if for each subgroup U of V, the sets X^{U}, Y^{U} have the same cardinality.

The case U = 1 is clear. For the cyclic subgroups, observe that since J has no invariant subspaces and therefore no eigenvectors, the matrix λI + μJ is invertible for all (λ, μ) ∈ 𝔽_{p}^{2} \ {0}. Therefore by Lemma 3.2, all nontrivial elements of Im(ρ_{f}) are conjugate in GL_{2n}(𝔽_{p}) to each other, and so the number of fixed cosets is independent of f.

Only the hardest case remains to be proved: that the number of cosets fixed by V itself is independent of f. Recall that the parabolic subgroups in GL_{2n} are the flag stabilizers. Define the type of a flag

X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{t}

of subspaces of 𝔽_{p}^{2n} to be the (t + 1)-tuple (dim(X_{0}), …, dim(X_{t})). The flags of any given type are permuted transitively by GL_{2n}(𝔽_{p}). Our task is to show that the number of V -invariant flags of any given type does not depend on the choice of irreducible polynomial f.

Associated to the block matrices is a splitting of 𝔽_{p}^{2n} as 𝔽_{p}^{n} ⊕ 𝔽_{p}^{n}. Let i: 𝔽_{p}^{n} → 𝔽_{p}^{2n} be inclusion as the first factor, and j: 𝔽_{p}^{2n} → 𝔽_{p}^{n} projection onto the second factor. Let X be an invariant subspace of 𝔽_{p}^{2n}, and set W = j(X), Z = i^{−1}(X). Then

(\begin{matrix}I & I \\ 0 & I \end{matrix}) (\begin{matrix}z \\ w \end{matrix}) = (\begin{matrix}z + w \\ w \end{matrix}) (\begin{matrix}I & J \\ 0 & I \end{matrix}) (\begin{matrix}z \\ w \end{matrix}) = (\begin{matrix}z + Jw \\ w \end{matrix}) .

We deduce that X is invariant if and only if W + JW ⊆ Z. In particular, the only invariant subspace with W equal to 𝔽_{p}^{n} is 𝔽_{p}^{2n}.

Clearly we may restrict our attention to invariant flags of proper subspaces. Based on Lemma 3.7, we define the fine type of an invariant flag X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{t} of proper subspaces to be (d_{0}, …, d_{t}; λ^{0}, …, λ^{t}), where d_{i} = dim(X_{i}), and λ^{i} is the partition associated to W_{i}. Of course, the fine type of a flag determines its type. But by Lemma 3.11, the number of flags of a given fine type is independent of f.

Lemma 3.11 The number of invariant flags X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{t} of proper subspaces with given fine type (d_{0}, …, d_{t}; λ^{0}, …, λ^{t}) does not depend on f.

Proof. An invariant subspace X determines W, Z and a linear map α: W → 𝔽_{p}^{n}/Z defined by w + α(w) ⊆ X ⊆ 𝔽_{p}^{2n} = 𝔽_{p}^{n} ⊕ 𝔽_{p}^{n}. Conversely, any such triple W, Z, α with W + JW ⊆ Z determines an invariant X. For an invariant flag we also require that W_{i} ⊆ W_{j} and Z_{i} ⊆ Z_{j} for i < j; and that α_{i}(w) + Z_{j} = α_{j}(w) for all w ∈ W_{i}.

By Corollary 3.10, the number of flags W_{0} ⊆ W_{1} ⊆ ⋯ ⊆ W_{t} with partition type (λ^{0}, …, λ^{t}) is independent of f. The number of flags Z_{0} ⊆ ⋯ ⊆ Z_{t} in 𝔽_{p}^{n} such that W_{i} + JW_{i} ⊆ Z_{i} and dim(Z_{i}) = d_{i} − dim(W_{i}) does not depend on the flag W_{i} or on f: for the type τ of the flag W_{i} + JW_{i} is determined, and all flags of type τ are in the same orbit. Given flags W_{i} and Z_{i}, the number of choices for the α_{i} is independent of f: pick α_{1} first, and pick α_{i+1} to be any extension of α_{i}.

Remark 3.12 Theorem 3.6 can be interpreted in terms of prime ideals. For an elementary abelian p-group V ≤ G, the classes in H^{∗}(G) with nilpotent restriction to V constitute a prime ideal 𝔭_{V}. Let V, W be elementary abelian subgroups of G which are isomorphic in A_{π} but not conjugate in G. Then 𝔭_{V} ∩ S_{h} and 𝔭_{W} ∩ S_{h} are distinct prime ideals in S_{h}, but 𝔭_{V} ∩ S_{π} and 𝔭_{W} ∩ S_{π} are the same prime ideal of S_{π}. In the specific case constructed, V, W have p-rank 2 and lie in an elementary abelian subgroup of rank n^{2}, the p-rank of G. Hence 𝔭_{V} and 𝔭_{W} have height n^{2} − 2.

References

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[2] L. Evens, The Cohomology of Groups (Clarendon Press, Oxford, 1991).

[3] D. J. Green, ‘On the cohomology of the sporadic simple group J_{4}’, Math. Proc. Cambridge Philos. Soc. 113 (1993) 253–266.

[4] D. J. Green and I. J. Leary, ‘The spectrum of the Chern subring’, Comm. Math. Helv. 73 (1998) 406–426.

[5] D. Quillen, ‘The spectrum of an equivariant cohomology ring: II’, Ann. of Math. 94 (1971) 573–602.

[6] G. B. Segal and C. T. Stretch, ‘Characteristic classes for permutation representations’, Math. Proc. Cambridge Philos. Soc. 90 (1981) 265–272.

[7] B. B. Venkov, ‘Cohomology algebras for some classifying spaces’, Dokl. Akad. Nauk SSSR 127 (1959) 943–944.

Green & Schuster Fachbereich 7 Mathematik Bergische Universität GH Wuppertal D–42097 Wuppertal Germany Leary Faculty of Mathematical Studies University of Southampton Southampton SO17 1BJ U.K.