On the GL(V)–module structure of K(n)^{∗}(BV)

By IAN J. LEARY

Max-Planck Institut für Mathematik, 53225 Bonn, Germany

and BJÖRN SCHUSTER

Centre de Recerca Matemàtica, E–08193 Bellaterra (Barcelona), Spain

(January 22, 1996)

Abstract

We study the question of whether the Morava K-theory of the classifying space of an elementary abelian group V is a permutation module (in either of two distinct senses, defined below) for the automorphism group of V. We use Brauer characters and computer calculations. Our algorithm for finding permutation submodules of modules for p-groups may be of independent interest.

1. Introduction

Let p be a prime, let K(n)^{∗} denote the nth Morava K-theory, V an elementary abelian p-group, or equivalently an F_{p}-vector space, and GL(V) the group of automorphisms of V. Then GL(V) acts naturally on the classifying space BV of V and hence on h^{∗}(BV) for any cohomology theory h. In the case when h = K(n), K(n)^{∗}(BV) is a finitely generated free module over the coefficient ring K(n)^{∗} whose structure is known [9], and it is natural to ask what may be said about its structure as a module for the group ring K(n)^{∗}[GL(V)]. The Morava K-theory of arbitrary finite groups is not known, and there is no direct construction of Morava K-theory itself. We hope that a better understanding of K(n)^{∗}(BV) may lead to progress with these questions.

For any ring R, and finite group G, we say that an R-free R[G]-module M is a permutation module if there is an R-basis for M which is permuted by the action of G. Call such an R-basis a permutation basis for M. If S is a G-set, write R[S] for the permutation module with permutation basis S. If M is a graded module for the graded ring R[G] (where elements of G are given grading zero), we call M a graded permutation module if it is a permutation module with a permutation basis consisting of homogeneous elements.

If M is a graded module for K(n)^{∗}[G], and G_{p} is a Sylow p-subgroup of G, the following four conditions on M are progressively weaker, in the sense that each is implied by the previous one.

(1) M is a graded permutation module;

(2) M is a permutation module;

(3) M is a direct summand of a permutation module;

(4) as a K(n)^{∗}[G_{p}]-module, M is a graded permutation module. The implications (1) ⟹ (2) ⟹ (3) are obvious, and hold for M a graded R[G]-module for any R. The implication (3) ⟹ (4) is explained below in Sections 2 and 7.

One might hope for K(n)^{∗}(BV) to satisfy condition (1) for G = GL(V), i.e., for K(n)^{∗}(BV) to be a graded permutation module for the group ring K(n)^{∗}[GL(V)]. This would be useful for the following reason: The ordinary cohomology of a group with coefficients in any permutation module is determined by the Eckmann-Shapiro lemma. Hence if K(n)^{∗}(BV) is a graded permutation module for GL(V), and H is a group expressed as an extension with kernel V, the E_{2} page of the Atiyah-Hirzebruch spectral sequence converging to K(n)^{∗}(BH) is easily computable. Even condition (4) would be very useful, as it would facilitate the computation of the E_{2}-page of the Atiyah-Hirzebruch spectral sequence for any p-group H expressed as an extension with kernel V.

The recent work of I. Kriz on Morava K-theory, including his dramatic discovery of a 3-group G such that K(2)^{∗}(BG) is not concentrated in even degrees, has emphasised the importance of studying the Aut(H)-module structure of K(n)^{∗}(BH) [6]. For example, Kriz has shown that for any prime p and any cyclic p-subgroup C of GL(V), K(n)^{∗}(BV) is a (graded) permutation module for C. He uses this result to deduce that for p odd and G a split extension with kernel V and quotient C, K(n)^{∗}(BG) is concentrated in even degrees. N. Yagita has another proof of this result [17].

It should be noted that if the dimension of V is at least three, there are infinitely many indecomposable graded K(n)^{∗}[GL(V)]-modules, of which only finitely many occur as summands of modules satisfying condition (4), which suggests that a ‘random’ module will not satisfy any of the conditions. On the other hand, work of Hopkins, Kuhn and Ravenel [4] shows that for certain generalized cohomology theories h^{∗}, h^{∗}(BV) is a permutation module for h^{∗}[GL(V)]. Amongst these h^{∗} are theories closely related to K(n)^{∗}, albeit that their coefficient rings are torsion-free and contain an inverse for p.

A result due to Kuhn [7] shows that K(n)^{∗}(BV) has the same Brauer character as the permutation module K(n)^{∗}[Hom(V, (F_{p})^{n})] for GL(V). We shall show however that in general K(n)^{∗}(BV) does not have the same Brauer character as a graded permutation module for K(n)^{∗}[GL(V)]. Note that Brauer characters give no information whatsoever concerning the structure of K(n)^{∗}(BV) as a module for the Sylow p-subgroup of GL(V). We give an algorithm to determine, for any pgroup G, whether an F_{p}[G]-module is a permutation module, and use this algorithm and computer calculations to determine in some cases whether K(n)^{∗}(BV) satisfies condition (4) above.

The main results of this paper are summarised in the following four statements. Before making them, we fix some notation.

Definition. Throughout the paper, let p be a prime, K(n)^{∗} the nth Morava Ktheory (at the prime p), and let V be a vector space over the field of p elements of dimension d. Let GL(V) act on the right of V, which will have the advantage that the modules we consider will be left modules. Let U(V) be a Sylow p-subgroup of GL(V).

Theorem 1.1. Let p be a prime, let V be a vector space of dimension d over F_{p}, and let K(n)^{∗} stand for the nth Morava K-theory.

(a) If p is odd, then K(n)^{∗}(BV) is not a graded permutation module for GL(V).

(b) If p = 2 and n = 1, then for any d, K(n)^{∗}(BV) is a graded permutation module for GL(V).

(c) For p = 2, n > 1 and d ≥ 4, K(n)^{∗}(BV) is not a graded permutation module for GL(V) if d is greater or equal to the smallest prime divisor of n.

(d) For p = 2 and d = 3, K(n)^{∗}(BV) is not a graded permutation module if n is a multiple of three, or if n is 2, 4, or 5.

(e) For p = 2 and d = 2, K(n)^{∗}(BV) is a graded permutation module for GL(V) if and only if n is odd.

Theorem 1.2. The K(n)^{∗}[GL(V)]-modules K(n)^{∗}(BV) and K(n)^{∗}[Hom(V, (F_{p})^{n})] are (ungraded) isomorphic in the following cases:

(a) For n = 1, for any p and d.

(b) For d = 2, p = 2, and any n. And are isomorphic as K(n)^{∗}[SL(V)]-modules in the case:

(c) d = 2, p = 3, n = 1, 2 or 3.

Theorem 1.3. K(n)^{∗}(BV) is not a permutation module for K(n)^{∗}[U(V)] in the following cases:

(a) d = 3, p = 3, n = 2,

(b) d = 3, p = 5, n = 2. In the following cases, as well as those implied by Theorems 1.1 and 1.2, K(n)^{∗}(BV) is a graded permutation module for K(n)^{∗}[U(V)]:

(c) d = 3, p = 2, n = 2, 3 or 4.

Work of Kriz [6] shows that for any V, K(n)^{∗}(BV) is a (graded) permutation module for any subgroup of GL(V) of order p. In the cases covered by Theorem 1.3, the group U(V) has order p^{3}, and for p > 2 it contains no element of order p^{2}. The gap between Theorem 1.3 and a special case of Kriz’s result is filled by:

Theorem 1.4. Let d = 3, let p = 3 or 5, and let H be any subgroup of GL(V) of order p^{2}. Then K(2)^{∗}(BV) is not a permutation module for H.

Statements 1.1(b) and 1.2(a) are corollaries of Kuhn’s description of the mod-p K-theory of BG [8]. Our interest in these questions was aroused by [2], in which it is shown that in the case V = (Z/2)^{2}, K(n)^{∗}(BV) is a graded permutation module for n = 3 but is not a graded permutation module for n = 2, i.e., the cases n = 2 and n = 3 of 1.1(e).

The remaining sections of the paper are organised as follows. In Section 2 we describe K(n)^{∗}(BV), the action of GL(V), and the process of reduction to questions concerning finite-dimensional F_{p}-vector spaces. This material is well-known to many topologists, but we hope that its inclusion will make the rest of the paper accessible to the reader who knows nothing about Morava K-theory. We also include some remarks concerning K(n)^{∗}[Hom(V, (F_{p})^{n})]. In Section 3 we prove those of our results that require only Brauer character methods. In Section 4 we deduce 1.1(b) and 1.2(a) from Kuhn’s work on mod-p K-theory and give a second proof of 1.1(b). Section 5 studies F_{2}[GL_{2}(F_{2})]-modules, and contains proofs of 1.1(e) and 1.2(b). In Section 6 we describe how to decompose F_{3}[SL_{2}(F_{3})]-modules, and outline the proof of 1.2(c). In Section 7 we describe our algorithm for determining when a module for a pgroup is a permutation module, and outline the proofs of the rest of the results we have obtained using computer calculations. The algorithm of Section 7 may be of independent interest. Section 8 contains the tables of computer output relevant to Sections 6 and 7, together with some final remarks.

2. Preliminaries

Fix a prime p. The nth Morava K-theory, K(n)^{∗} (which depends on p as well as on the positive integer n), is a generalized cohomology theory whose coefficient ring is the ring F_{p}[v_{n}, v_{n}^{−1}] of Laurent polynomials in v_{n}, which has degree −2(p^{n} − 1). All graded modules for this ring are free, which implies that there is a good Künneth theorem for K(n)^{∗}. For any graded K(n)^{∗}-module M, let \overline{M} be the quotient M/(1 − v_{n})M. Then \overline{M} is an F_{p}-vector space, naturally graded by the cyclic group Z/2(p^{n} − 1). If M is a graded K(n)^{∗}[G]-module for some finite group G, then \overline{M} is naturally a Z/2(p^{n} − 1)-graded F_{p}[G]-module, and M is determined up to isomorphism by \overline{M}. It is easy to see that M is a (graded) permutation module for K(n)^{∗}[G] if and only if \overline{M} is a (Z/2(p^{n} − 1)-graded) permutation module for F_{p}[G].

For C a cyclic group of order p^{m}, it may be shown [9] that the Morava K-theory of BC is a truncated polynomial ring on a generator of degree two:

K(n)^{∗}(BC) = K(n)^{∗}[x]/(x^{p^{mn}}).

The generator x is a Chern class in the sense that it is the image of a certain element of K(n)^{2}(BU(1)) under the map induced by an inclusion of C in the unitary group U(1). From the Künneth theorem mentioned above it follows that if V is an elementary abelian p-group of rank d, then

K(n)^{∗}(BV) = K(n)^{∗}[x_{1}, …, x_{d}]/(x_{1}^{p^{n}}, …, x_{d}^{p^{n}}),

where x_{1}, …, x_{d} are Chern classes of d 1-dimensional representations of V whose kernels intersect trivially. The Chern class of a representation is natural, and the d representations taken above must generate the representation ring of V. Thus the action of GL(V) on K(n)^{∗}(BV) may be computed from its action on Hom(V, U(1)) together with an expression for the Chern class of a tensor product ρ ⊗ θ of two 1-dimensional representations in terms of the Chern classes of ρ and θ.

For any generalized cohomology theory h^{∗} such that h^{∗}(BU(1)) is a power series ring h^{∗}[[x]] (Morava K-theory has this property), Chern classes may be defined, and there is a power series x+_{F} y ∈ h^{∗}[[x, y]] expressing the Chern class of a tensor product of line bundles in terms of the two Chern classes. This power series is called the formal group law for h^{∗}, because it satisfies the axioms for a 1-dimensional commutative formal group law over the ring h^{∗}. Since each Chern class in K(n)^{∗}(BV) is nilpotent of class p^{n}, we need only determine x+_{F} y modulo (x^{p^{n}}, y^{p^{n}}). This is the content of the following proposition, which is well-known, but for which we can find no reference.

Proposition 2.1. Modulo the ideal generated by x^{p^{n}} and y^{p^{n}}, the formal sum x+_{F} y for K(n)^{∗} is

x+_{F} y = x + y − v_{n}(∑_{i=1}^{p−1} \frac{1}{p}\binom{p}{i}x^{ip^{n−1}}y^{(p−i)p^{n−1}}).

Sketch proof. First we recall the formal sum for BP^{∗}, Brown-Peterson cohomology [15]. Let l be the power series

l(x) = ∑_{i≥0} m_{i}x^{p^{i}},

where m_{0} = 1, but the remaining m_{i}’s are viewed as indeterminates, and let e(x) be the compositional inverse to l, i.e., a power series such that e(l(x)) = l(e(x)) = x. The BP^{∗} formal sum is the power series e(l(x) + l(y)). The K(n)^{∗} formal sum may be obtained as follows: Take the BP^{∗} formal sum, replace the indeterminates m_{i} by indeterminates v_{i} using the relation

v_{j} = pm_{j} − ∑_{i=1}^{j−1} m_{i}v_{j−i}^{p^{i}},

set v_{i} = 0 for i ≠ n, by which point all the coefficients lie in Z_{(p)}, and take the reduction modulo p. To calculate the K(n)^{∗} formal sum, it is helpful to set v_{i} = 0 for i ≠ n as early as possible, and one may as well set v_{n} = 1, since every term in x+_{F} y has degree 2. Solving for the m_{i}’s in terms of the v_{i}’s gives

m_{i} = 0 if n does not divide i,
m_{ni} = 1/p^{i}.

Thus to compute x+_{F} y, let e′(x) be the compositional inverse to

l′(x) = ∑_{i≥0} x^{p^{ni}}/p^{i},

and then x+_{F} y is the mod-p reduction of e′(l′(x) + l′(y)). It is easy to see that

e′(x) ≡ x − x^{p^{n}}/p modulo x^{2p^{n}},

and so

x+_{F} y ≡ x + y − (x + y)^{p^{n}}/p modulo (x^{p^{n}}, y^{p^{n}}, p).

The claimed result follows.

Using the reduction M ↦ \overline{M} as at the start of this section and Proposition 2.1, the study of the graded K(n)^{∗}[GL(V)]-module structure of K(n)^{∗}(BV) reduces to the study of the Z/(p^{n} − 1)-graded F_{p}[GL_{d}(F_{p})]-module K_{n,d}^{∗} defined below.

As an F_{p}-algebra,

K_{n,d}^{∗} ≅ F_{p}[x_{1}, …, x_{d}]/(x_{1}^{p^{n}}, …, x_{d}^{p^{n}}).

Each x_{i} has degree 1, and the GL_{d}(F_{p})-action is compatible with the product. The action of the matrix (a_{ij}) ∈ GL_{d}(F_{p}) is given by

x_{j} ↦ e′(∑_{i} l′(x_{i})a_{ij}),

where e′ and l′ are as in the proof of Proposition 2.1. (Recall that for any V we take GL(V) to act on the right of V, and hence obtain a left K(n)^{∗}[GL(V)]-module structure on K(n)^{∗}(BV).)

Note that we have halved the original degrees because K(n)^{∗}(BV) is concentrated in even degrees. Until recently it was an open problem whether a similar statement holds for arbitrary finite groups, although some cases had been verified [9,5,13,14,11]. Kriz has recently announced that this is not the case [6].

If we are only interested in Brauer characters, or equivalently composition factors, then a further simplification may be made, see [7]. Let L_{n,d}^{∗} denote the algebra of polynomial functions on (F_{p})^{d}, modulo the ideal of p^{n}th powers of elements of positive degree. Grade L_{n,d}^{∗} by Z/(p^{n} − 1), and let GL_{d}(F_{p}) act on L_{n,d}^{∗} by its natural action on the polynomial functions. Thus L_{n,d}^{∗} is a truncated polynomial algebra F_{p}[x_{1}, …, x_{d}]/(x_{1}^{p^{n}}, …, x_{d}^{p^{n}}), cyclically graded, and having the standard action of GL_{d}(F_{p}).

Lemma 2.2. K_{n,d}^{∗} has a series of (graded) submodules such that the direct sum of the corresponding quotients is isomorphic to L_{n,d}^{∗}. In particular, K_{n,d}^{∗} and L_{n,d}^{∗} have the same composition factors (as graded modules).

Proof. For each degree k, take the basis consisting of monomials of length congruent to k modulo p^{n} − 1, and arrange them in blocks with respect to length. For any g, the matrix of its action on L_{n,d}^{k} with respect to this basis consists of square blocks along the diagonal, whereas the corresponding matrix for the action on K_{n,d}^{k} has some extra entries below the blocks.

The permutation module K(n)^{∗}[Hom(V, (F_{p})^{n})] occurs in the statement of Theorem 1.2, so we complete this preliminary section with some remarks concerning this module. If ϕ is a homomorphism from V to F_{p}^{n}, then g ∈ GL(V) acts by composition, i.e.,

gϕ(v) = ϕ(vg).

Since we view GL(V) as acting on the right of V, this makes Hom(V, (F_{p})^{n}) into a left GL(V)-set. The GL(V)-orbits in Hom(V, (F_{p})^{n}) may be described as follows. For W a subspace of V, let H(W) ≤ GL(V) be

H(W) = {g ∈ GL(V) : vg − v ∈ W ∀v ∈ V }.

For example, H({0}) = {1}, and H(V) = GL(V). For 0 ≤ i ≤ dim(V), let H_{i} be H(W_{i}) for some W_{i} of dimension i. Thus H_{i} is defined only up to conjugacy, but this suffices to determine the isomorphism type of the GL(V)-set GL(V)/H_{i}. Now let ϕ be an element of Hom(V, (F_{p})^{n}). The stabilizer of ϕ in GL(V) is the subgroup H(ker(ϕ)), and the orbit of ϕ consists of all ϕ′ such that Im(ϕ′) = Im(ϕ). It follows that as GL(V)-sets,

Hom(V, (F_{p})^{n}) ≅ ∐_{0≤i≤dim(V)} m(n, i) · G/H_{i},

where m(n, i) is the number of subspaces of (F_{p})^{n} of dimension i. Thus to decompose the module F_{p}[Hom(V, (F_{p})^{n})], it suffices to decompose each F_{p}[GL(V)/H_{i}].

3. Brauer characters

In this section we shall prove most of the negative results of Theorem 1.1. Firstly, we describe how to compute the values of the modular characters afforded by the modules L_{n,d}^{k}. As a general reference, see [3], in particular §17. Fix an embedding of the multiplicative group of the algebraic closure of F_{p} in the group of roots of 1 in C. Let g be a p-regular element of GL(V), i.e., an element whose order is coprime to p, and let λ_{1}, λ_{2}, …, λ_{d} denote the images in C of the eigenvalues of its action on V^{∗}. Then the Brauer character of g is

χ_{V^{∗}}(g) = λ_{1} + λ_{2} + ⋯ + λ_{d} .

Two F_{p}[G]-modules have the same Brauer character if and only if they have the same composition factors. To compute the character of L_{n,d}^{k} we proceed as follows: an argument similar to the one used to prove Molien’s theorem (see e.g. [3], p. 329) shows that the character of a truncated polynomial algebra has a generating function

f_{g}(t) = ∏_{i=1}^{d}(\frac{1 − (λ_{i}t)^{p^{n}}}{1 − λ_{i}t} ) .

Then the character of L_{n,d}^{∗} evaluated at g is simply f_{g}(1), whereas for each degree k (recall that we are grading cyclically) one has

χ_{L_{n,d}^{k}}(g) = \frac{1}{p^{n} − 1} ∑_{τ} τ^{−k}f_{g}(τ) ,  (3.1)

where the sum ranges over all (p^{n} − 1)-st roots of unity—to see this, recall that the sum, over all mth roots of unity λ, of λ^{k} is equal to zero if m does not divide k, and equal to m if m does divide k.

Proof of 1.1(a). Let D be the subgroup of diagonal matrices in GL_{d}(F_{p}), so that D is isomorphic to a direct product of d cyclic groups of order p − 1. In K_{n,d}^{∗}, each monomial in x_{1}, …, x_{d} is an eigenvector for D, and the monomials fixed by D are those in which the exponent of each x_{i} is divisible by p − 1. Hence if p − 1 does not divide k, then K_{n,d}^{k} cannot be a permutation module for D because it contains no D-fixed point.

Proof of 1.1(c). As already said above, this is done by computing the character values on certain 2-regular elements of GL(V). We shall first look at the case where d equals a prime divisor q of the fixed number n. Consider an element, g_{q} say, of GL_{q}(F_{2}) which permutes the 2^{q} − 1 nontrivial elements of (F_{2})^{q} cyclically. (To see that there is always such an element consider the action of the multiplicative group of F_{2^{q}} on the additive group of F_{2^{q}}.) The set of eigenvalues of g_{q} contains a primitive (2^{q} − 1)st root of unity, and is closed under the action of the Galois group Gal(F_{2^{q}}/F_{2}). Hence the Brauer lifts of the eigenvalues of g_{q} are λ, λ^{2}, …, λ^{2^{q−1}} for some primitive (2^{q} − 1)-st root of unity λ ∈ C. Consequently, the generating function for the character afforded by L_{n,q}^{∗} is given by

f_{g_{q}}(t) = ∏_{i=0}^{q−1}(\frac{1 − (λ^{2^{i}}t)^{2^{n}}}{1 − λ^{2^{i}}t} ) .

If τ is a (2^{n} − 1)-st root of unity, one gets

f_{g_{q}}(τ) = \{ \begin{matrix}2^{n} & if τ ∈ \{λ^{−2^{i}}, i = 0, 1, … q − 1\} \\ 1 & otherwise. \end{matrix}

Thus evaluating the formula (3.1) for the character afforded by L_{n,q}^{k} yields

χ_{L_{n,q}^{k}}(g_{q}) = \frac{1}{2^{n} − 1} ∑_{τ≠λ^{−2^{i}}} τ^{k} + \frac{2^{n}}{2^{n} − 1} ∑_{i=0}^{q−1} λ^{2^{i}k}

which is equal to

\{ \begin{matrix}q + 1 & for k = 0 \\ ∑_{i=0}^{q−1} λ^{2^{i}k} & for k ≠ 0. \end{matrix}

Specializing to the case k = 1, this sum is never zero, since the powers λ^{i} for i coprime to 2^{q} − 1 form a Q-basis for Q[λ]. For q > 2 the sum is not a rational, because it is not fixed by the whole Galois group Gal(Q[λ]/Q). In the case q = 2, one obtains −1 (the sum of the two primitive third roots of unity). Since permutation modules have positive integer character values, this shows that K(n)^{∗}(BV_{q}) is not a graded GL_{q}(F_{2})-permutation module. To proceed with vector spaces of dimension bigger than q we consider the cases q > 2 and q = 2 separately. In the first case we use the following lemma to conclude that the character still takes non-integer values on certain elements of GL(V). Let g be an (arbitrary) 2-regular element of GL_{d}(F_{2}) and I_{r} the r × r identity matrix. If we denote by g × I_{r} the element of GL_{d+r}(F_{2}) which acts like g on the first d generators of L_{n,d+r}^{∗} and trivially on the last r, one has

Lemma 3.2. χ_{L_{n,d+r}^{k}}(g × I_{r}) = χ_{L_{n,d}^{k}}(g) + ((2^{nr} − 1)/(2^{n} − 1))χ_{L}(g) .

Proof. The generating function for g × I_{r} is obtained from the one for g as the product with r factors (1 + t + t^{2} + … + t^{2^{n}−1}), thus

χ_{L_{n,d+r}^{k}}(g × I_{r}) = \frac{1}{2^{n} − 1} ∑_{τ} τ^{−k}f_{g×I_{r}}(τ)

= \frac{1}{2^{n} − 1} ∑_{τ≠1} τ^{−k}f_{g}(τ)(\frac{1 − τ^{2^{n}}}{1 − τ} )^{r} + \frac{2^{nr}}{2^{n} − 1}f_{g}(1)

= χ_{L_{n,d}^{k}}(g) + (\frac{2^{nr} − 1}{2^{n} − 1})f_{g}(1) .

Thus g_{q} × I_{r} will do the trick when V has rank q + r. This fails for q = 2, whence we choose the element g′ which consists of d/2 copies of g_{2} = \binom{0 1}{1 1} arranged along the diagonal if d is even, and add an extra diagonal entry 1 if d is odd. Then a computation similar to the one carried out in the previous lemma shows that for k ≢ 0 mod 3,

χ_{L_{n,d}^{k}}(g′) = \{ \begin{matrix}−\frac{2^{nd/2} − 1}{2^{n} − 1} & if d is even \\ −\frac{2^{n(d−1)/2} − 1}{2^{n} − 1} + 1 & if d is odd \end{matrix}

For d > 3 these numbers are negative.

The other parts of Theorem 1.1 that may be proved using Brauer characters are some cases of 1.1(d) and both implications of 1.1(e). The details are similar to the above proof so we shall not give them. In the case when V has rank 3, evaluation of the Brauer character of an element of GL(V) of order 7 shows that L_{n,3}^{1} is not a GL(V)-permutation module if 3 divides n. Similarly, when V has rank 2, the Brauer character of an element of GL(V) of order 3 on L_{n,2}^{1} is negative if n is even. When V has rank 2 and n is odd, it may be shown that for each k, any GL(V)-module having the same Brauer character as L_{n,2}^{k} is a permutation module. This shows that for n odd, K_{n,2}^{k} is a permutation module, but does not specify which one. In Section 5 we shall describe the isomorphism type of K_{n,2}^{k} and L_{n,2}^{k} for all n and k, giving an alternative proof of 1.1(e).

4. On K(1)

Here we describe how those parts of Theorems 1.1 and 1.2 that concern K(1)^{∗} (i.e., 1.1(b) and 1.2(a)) follow from Kuhn’s description of the mod-p K-theory of finite groups [8]. An ‘elementary’ proof, working directly with the description of K_{1,d}^{∗} in the previous section, would be more in keeping with the rest of the paper. We give such a proof in the case p = 2.

Proof of 1.1(b) and 1.2(a). First, note that for p = 2, v_{1} has degree −2, so that the ‘cyclically graded’ modules K_{1,d}^{∗} are in fact concentrated in a single degree. Hence 1.2(a) implies 1.1(b). To prove 1.2(a) recall [15] that the spectrum representing mod-p K-theory splits as a wedge of one copy of each of the 0th, 2nd, …, (2p − 4)th suspensions of the spectrum representing K(1)^{∗}. Since K(1)^{∗}(BV) is concentrated in even degrees it follows that K_{1,d}^{∗} is naturally isomorphic to K^{0}(BV; F_{p}). In [8] it is shown that for any p-group G, K^{0}(BG; F_{p}) is naturally isomorphic to F_{p} ⊗ R(G), where R(G) is the (complex) representation ring of G. The case G = V gives 1.2(a), because as a GL(V)-module, F_{p} ⊗ R(V) is isomorphic to F_{p}[Hom(V, F_{p})].

Alternative proof, p = 2. In this case, K_{1,d}^{∗} is isomorphic to an exterior algebra Λ[x_{1}, …, x_{d}] = F_{2}[x_{1}, …, x_{d}]/(x_{i}^{2}). The monomial 1 generates a trivial GL(V)- summand. Let H be the subgroup of GL(V) fixing x_{1}. Then H is the subgroup of GL(V) stabilizing some hyperplane W and inducing the identity map on the quotient V/W. There is a GL(V)-set isomorphism

Hom(V, F_{2}) ≅ GL(V)/GL(V) ⨿ GL(V)/H,

so it will suffice to show that the submodule M generated by x_{1} contains each monomial in Λ[x_{1}, …, x_{d}] of strictly positive length. The permutation matrices permute the monomials of any given length transitively. Assume that M contains all the monomials of length i (this holds for i = 1), and let g ∈ GL(V) be such that

gx_{1} = x_{1}, …, gx_{i−1} = x_{i−1}, gx_{i} = x_{i}+_{F} x_{i+1}.

Then

g(x_{1} … x_{i}) + x_{1} … x_{i} + x_{1} … x_{i−1}x_{i+1} = x_{1} … x_{i}x_{i+1} ∈ M,

so M contains all monomials of length i + 1.

It should be possible to give an ‘elementary’ proof of 1.2(a) for p > 2 by considering the element x_{1} + x_{1}^{2} + ⋯ + x_{1}^{p−1}, but we have not done so.

5. When V has order four

Here we shall prove Theorems 1.1(e) and 1.2(b), which concern K(n)^{∗}(BV) for V of dimension two over F_{2}. We determine the structure of L_{n,2}^{k} as a GL_{2}(F_{2})-module, and deduce that K_{n,2}^{k} and L_{n,2}^{k} are isomorphic. Note that it is also possible to prove 1.1(e) using the methods of Section 3 without determining the isomorphism type of K_{n,2}^{k}. Throughout this section, let V = (F_{2})^{2}.

There are three isomorphism types of indecomposable F_{2}[GL(V)]-modules: the 1-dimensional trivial module T; the natural module V which is both simple and projective (and is the Steinberg module for GL(V)); and a module N expressible as a non-split extension of T by T, which is the projective cover of T. Each module is self-dual. There are four conjugacy classes of subgroups of GL(V). The transitive permutation modules are the following four modules:

T,  N,  T ⊕ V,  N ⊕ 2V.

Let S^{∗}[V^{∗}] stand for the algebra of polynomial functions on V as a graded GL(V)- module.

Proposition 5.1. The generating functions P_{T}, P_{N} and P_{V} for the number of each indecomposable GL(V)-summand of S^{∗}[V^{∗}] are the following power series:

P_{T} (t) = \frac{1}{1 − t^{2}} ,  P_{N}(t) = \frac{t^{3}}{(1 − t^{2})(1 − t^{3})},  P_{V} (t) = \frac{t}{(1 − t)(1 − t^{3})}.

Proof. Recall that the ring of invariants S^{∗}[V^{∗}]^{GL(V)} is a free polynomial ring on two generators of degrees two and three (see [16]). The Poincaré series for S^{∗}[V^{∗}], and the ring of invariants, together with the generating function for the Brauer character of an element of GL(V) of order three give the three equations below, whose solution is as claimed.

\begin{aligned}P_{T} + 2P_{N} + 2P_{V} & = \frac{1}{(1 − t)^{2}} \\ P_{T} + P_{N} & = \frac{1}{(1 − t^{2})(1 − t^{3})} \\ P_{T} + 2P_{N} − P_{V} & = \frac{1 − t}{1 − t^{3}} \end{aligned}

Proposition 5.2. Let k be an element of Z/(2^{n} −1). The direct sum decomposition for the module L_{n,2}^{k} is:

2T ⊕ (2^{n} − 2)/6 N ⊕ (2^{n} + 1)/3 V for n odd, k = 0,
T ⊕ (2^{n} − 2)/6 N ⊕ (2^{n} + 1)/3 V for n odd, k ≠ 0,
2T ⊕ (2^{n} + 2)/6 N ⊕ (2^{n} − 1)/3 V for n even, k = 0,
T ⊕ (2^{n} + 2)/6 N ⊕ (2^{n} − 1)/3 V for n even, k ≠ 0, k ≡ 0 mod 3,
T ⊕ (2^{n} − 4)/6 N ⊕ (2^{n} + 2)/3 V for n even, k ≢ 0 mod 3.

Proof. Let L̃^{∗} be the truncated symmetric algebra L_{n,2}^{∗}, but graded over the integers rather than over the integers modulo 2^{n}−1. Then L̃^{k} = {0} for k > 2(2^{n}−1), and for k = 2(2^{n} − 1), L̃^{k} is isomorphic to T, generated by x_{1}^{2^{n}−1}x_{2}^{2^{n}−1}. For 0 < k < 2^{n} − 1, viewing k as either an integer or an integer modulo 2^{n} − 1 as appropriate, L_{n,2}^{k} is isomorphic to L̃^{k} ⊕ L̃^{2^{n}−1+k}, while L_{n,2}^{0} is isomorphic to L̃^{0} ⊕ L̃^{2^{n}−1} ⊕ L̃^{2(2^{n}−1)} ≅ 2T ⊕ L̃^{2^{n}−1}. For 0 ≤ k ≤ 2^{n} − 1, L̃^{k} is isomorphic to S^{k}[V^{∗}]. The product structure on L̃^{∗} gives a duality pairing

L̃^{k} × L̃^{2(2^{n}−1)−k} → L̃^{2(2^{n}−1)} ≅ T,

and since all GL(V)-modules are self-dual it follows that for 2^{n} ≤ k ≤ 2(2^{n} − 1), L̃^{k} ≅ S^{2(2^{n}−1)−k}[V^{∗}]. The claimed description of L_{n,2}^{k} follows from Proposition 5.1.

Corollary 5.3. For each n and each k ∈ Z/(2^{n}−1), K_{n,2}^{k} and L_{n,2}^{k} are isomorphic.

Proof. For k ≠ 0 K_{n,2}^{k} has odd dimension, so must contain at least one direct summand isomorphic to T. It is easy to see that 1 generates a summand of K_{n,2}^{0}, and the same dimension argument applied to a complement of this summand shows that K_{n,2}^{0} contains at least two summands isomorphic to T. On the other hand, V and N are projective, and (Lemma 2.2) K_{n,2}^{k} has a filtration such that the sum of the factors is isomorphic to L_{n,2}^{k}. Hence K_{n,2}^{k} has at least as many N summands and V summands as L_{n,2}^{k}. This accounts for all the summands of K_{n,2}^{k}.

The proof of 1.1(e) follows easily from the given description of K_{n,2}^{∗}. For 1.2(b), recall from the end of Section 2 that

Hom(V, (F_{2})^{n}) = GL(V)/GL(V)
⨿ (2^{n} − 1) · GL(V)/H_{1}
⨿ (2^{n} − 1)(2^{n} − 2)/6 · GL(V)/{1},

where H_{1} is a subgroup of GL(V) of order two, and that

F_{2}[GL(V)/H_{1}] ≅ T ⊕ V, F_{2}[GL(V)/{1}] ≅ N ⊕ 2V.

The argument used in the proof of Corollary 5.3 shows that for any p, n, and k such that L_{n,2}^{k} contains at most one non-projective summand, K_{n,2}^{k} ≅ L_{n,2}^{k}. For odd primes this does not always occur however. If 0 < k < p^{n} − 1 then L_{n,2}^{k} splits as a direct sum of submodules of dimensions k + 1 and p^{n} − k − 2 coming from the standard Z-grading on the truncated polynomial algebra. If k is not congruent to either −1 or −2 modulo p, the dimensions of these summands are not divisible by p, and hence L_{n,2}^{k} contains at least two non-projective indecomposable summands. The calculations described in the next section show that for p = 3, n = 2, 3 and 0 < k < 3^{n} − 1, the module K_{n,2}^{k} has exactly one non-projective indecomposable summand. It follows that for p = 3, K_{n,2}^{k} and L_{n,2}^{k} are not necessarily isomorphic.

6. When V has order nine

Our results concerning the SL_{2}(F_{3})-module structure of K_{n,2}^{∗} in the case when p = 3 were obtained by computer. We wrote a Maple program to generate matrices representing the action of a pair of generators for SL_{2}(F_{3}) on K_{n,2}^{k}. These matrices were fed to a GAP [10] program which, given a matrix representation of SL_{2}(F_{3}), outputs a list of its indecomposable summands. In fact the output from the Maple program needed a little editing before being read into the GAP program. This was done by a third program, although it could equally have been done by hand.

Using standard techniques of representation theory [1,3], the following facts may be verified. For V = (F_{3})^{2}, there are three simple F_{3}[SL(V)]-modules: the trivial module T, the natural module V, and a simple projective module P = S^{2}(V) of dimension three. There are three blocks. The blocks containing T and V each contain three indecomposable modules, each of which is uniserial. This data may be summarised as follows:

block of T = I_{1} :  T ↣ I_{2} ↠ T,  T ↣ I_{3} ↠ I_{2},
block of V = I_{4} :  V ↣ I_{5} ↠ V,  V ↣ I_{6} ↠ I_{5},
block of P = I_{7} :  contains no other indecomposables.

Letting τ stand for the element of order two in SL(V) and σ for the sum of the six elements of SL(V) of order four, the block idempotents are

b_{T} = 2 + 2τ + 2σ,  b_{V} = 2 + τ,  b_{P} = σ.

The modules in any single block are distinguishable by their restrictions to a cyclic subgroup of SL(V) of order three. Thus if α is an element of SL(V) of order three, and M is an SL(V)-module, the direct summands of M are determined by the ranks of the elements of End(M) representing the actions of the following seven elements of F_{3}[SL(V)]:

b_{T} , (1 − α)b_{T} , (1 − α)^{2}b_{T} , b_{V} , (1 − α)b_{V} , (1 − α)^{2}b_{V} , b_{P} .

More precisely, if the seven ranks are r_{1}, …, r_{7}, and n_{i} stands for the number of factors of M isomorphic to I_{i}, then

n_{1} = r_{1} − 2r_{2} + r_{3}, n_{2} = r_{2} − 2r_{3}, n_{3} = r_{3}, n_{4} = r_{5} − 2r_{6},
n_{5} = r_{4} − 2r_{5} + r_{6}, n_{6} = (2r_{5} − r_{4})/2, n_{7} = r_{7}/3.

Our GAP program reads in matrices representing the action on M of a certain pair of generators for SL(V), and calculates n_{1}, …, n_{7} by first finding r_{1}, …, r_{7} as above.

Recall from the end of Section 2 that there is an isomorphism of (right) GL(V)- sets:

Hom(V, (F_{3})^{n}) =GL(V)/GL(V)
⨿ (3^{n} − 1)/2 · GL(V)/H_{1}
⨿ (3^{n} − 1)(3^{n} − 3)/48 · GL(V)/{1},

where H_{1} is the subgroup stabilizing a line L in V and acting trivially on V/L. As SL(V)-modules, it may be checked that

\begin{aligned}F_{3}[GL(V)/GL(V)] & ≅ I_{1}, \\ F_{3}[GL(V)/L(V)] & ≅ I_{1} ⊕ I_{5} ⊕ I_{7}, \\ F_{3}[GL(V)/\{1\}] & ≅ 2I_{3} ⊕ 4I_{6} ⊕ 6I_{7}. \end{aligned}

From this information together with the results given in Table 8.1 it is easy to check the claim of Theorem 1.2(c).

There are fourteen indecomposable GL(V)-modules in four blocks, two of which contain a single simple projective module. The six indecomposables in the block containing V are comparatively hard to distinguish, which is the reason why we considered only SL(V).

7. Permutation modules for p-groups

In this section we shall describe the computer programs used in the proofs of Theorem 1.3, Theorem 1.4, and the cases n = 2, 4, and 5 of 1.1(d). In Sections 5 and 6 our programs made use of the fact that there were only finitely many indecomposable modules. If G is a group whose Sylow p-subgroup is not cyclic, then F_{p}[G] has infinitely many indecomposable modules, so the same sort of methods cannot work. Here we shall describe an algorithm which may be used to determine, for any p-group G, whether an F_{p}[G]-module is a permutation module, and if so to decompose it. (For a precise statement, see Proposition 7.1 below.) As before, we use a Maple program to generate matrices representing the action of a Sylow psubgroup of GL_{d}(F_{p}) on K_{n,d}^{k} for various p, k, n, and d, and we use a GAP program working with our algorithm to decompose these modules.

Our algorithm relies on the following fact [3]: For G a p-group, any transitive permutation module for F_{p}[G] has a unique minimal submodule, which is the trivial module generated by the sum of the elements of a permutation basis. This implies that any transitive permutation module is indecomposable. Note that the Krull-Schmidt theorem and the indecomposability of transitive permutation modules together imply that if a graded F_{p}[G]-module is a permutation module, then it is also a graded permutation module.

Proposition 7.1. Let G_{1}, …, G_{n} be subgroups of a p-group G, where the order of G_{i+1} is at least the order of G_{i}, and let M be a (finitely generated) F_{p}[G]-module. Let m_{1}, …, m_{n} be the integers whose calculation is described below. Then M contains a submodule M′, where

M′ ≅ m_{1}F_{p}[G/G_{1}] ⊕ ⋯ ⊕ m_{n}F_{p}[G/G_{n}],

and M′ has maximal dimension among all submodules of M isomorphic to a direct sum of copies of the F_{p}[G/G_{i}].

To compute m_{i}, proceed as follows. Let M_{0} be the zero submodule of M. If M_{i−1} has been defined, let

M_{i} = M_{i−1} + Im(( ∑_{g∈G/G_{i}} g) : M^{G_{i}} → M),

where the sum ranges over a transversal to G_{i} in G, M^{G_{i}} denotes the G_{i}-fixed points of M, and the sum is an element of F_{p}[G] viewed as an element of End(M). Now define

m_{i} = dim M_{i} − dim M_{i−1}.

Without loss of generality, it may be assumed that no two of G_{1}, …, G_{n} are conjugate. The dimension of M′ is equal to the sum ∑_{i} m_{i}|G : G_{i}|. If G_{1}, …, G_{n} contains a representative of each conjugacy class of subgroups of G, then dim M′ = dim M if and only if M is a permutation module.

Proof. First, recall that the socle, Soc(N), of a module N is the smallest submodule of N containing every minimal submodule. The following statement is easy to prove, and will be useful below. If L is a submodule of M, and f : N → M is a module homomorphism, then f is injective if and only if its restriction to Soc(N) is injective. If f is injective, then Soc(f(N)) = f(Soc(N)), and the sum L + f(N) in M is direct if and only if the sum Soc(L) + f(Soc(N)) is direct.

Module homomorphisms from F_{p}[G/G_{i}] to M are naturally bijective with elements of M^{G_{i}}, where the element x corresponds to the homomorphism θ_{x} sending 1·G_{i} to x. The socle of F_{p}[G/G_{i}] is a trivial submodule generated by ∑_{g∈G/G_{i}} g·G_{i}, so its image under θ_{x} is generated by ∑_{g∈G/G_{i}} g · x. It follows that any submodule of M isomorphic to a direct sum of copies of the modules F_{p}[G/G_{1}], …, F_{p}[G/G_{i}] has socle contained in M_{i}, and in particular consists of at most dim M_{i} summands. This shows that any submodule of M isomorphic to a direct sum of F_{p}[G/G_{i}]’s has dimension less than or equal to ∑_{i} m_{i}|G : G_{i}|, but it remains to exhibit a submodule M′ having this dimension.

Define M′_{0} to be the zero submodule of M, and assume that for some j with 1 ≤ j ≤ n we have constructed a submodule M′_{j−1} of M with

M′_{j−1} ≅ m_{1}F_{p}[G/G_{1}] ⊕ ⋯ ⊕ m_{j−1}F_{p}[G/G_{j−1}].

Let x_{1}, …, x_{m_{j}} ∈ M^{G_{j}} be such that the images ∑_{g∈G/G_{j}} g · x_{i} form a basis for a complement to M_{j−1} in M_{j}. Taking L = M_{j−1}, N = m_{j}F_{p}[G/G_{j}], and f : N → M the map sending the elements (0, …, 1 · G_{j}, …, 0) to the x_{i}’s, the statements in the first paragraph of the proof show that f is injective, and that M′_{j} defined as the submodule of M spanned by M_{j−1} and the x_{i}’s is isomorphic to M′_{j−1}⊕m_{j}F_{p}[G/G_{j}]. Now M′ may be taken to be M′_{n}.

The data in Tables 8.2–8.6 of the next section were obtained using the algorithm described above.

8. Tables and final remarks

For each p, n, and d, let K̃_{n,d}^{∗} be the direct summand of K_{n,d}^{∗} corresponding to the reduced Morava K-theory K̃(n)^{∗}(B(F_{p})^{d}). Thus K̃_{n,d}^{k} = K_{n,d}^{k} for k ≠ 0, and K_{n,d}^{0} = K̃_{n,d}^{0} ⊕ T, where T is the trivial F_{p}[GL(V)]-submodule of dimension one spanned by the monomial 1. The F_{p}-dimension of K̃_{n,d}^{k} is (p^{nd} − 1)/(p^{n} − 1). Table 8.1 describes the SL_{2}(F_{3})-module structure of K̃_{n,2}^{k} (for p = 3) in terms of the indecomposable modules I_{1}, …, I_{7} as described in Section 6.

Table 8.1: The SL_{2}(F_{3})-summands of K̃_{n,2}^{k}.

\begin{matrix}n & k & I_{1} & I_{2} & I_{3} & I_{4} & I_{5} & I_{6} & I_{7} \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 2 & 0,4 & 1 & 0 & 1 & 0 & 0 & 0 & 2 \\ 2 & 1,3,5,7 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 2 & 2,6 & 1 & 0 & 0 & 0 & 0 & 0 & 3 \\ 3 & even & 1 & 0 & 2 & 0 & 0 & 0 & 7 \\ 3 & odd & 0 & 0 & 0 & 0 & 1 & 4 & 0 \end{matrix}

Let V have dimension d = 3 over F_{p}. Let l be a line in V, and let π be a plane in V containing l. The group GL(V) acts on the set of all such pairs, and the stabilizer of the pair (l, π) contains a unique Sylow p-subgroup U(V) of GL(V) (and is in fact equal to the normalizer of U(V)). Let C be a generator for the centre of U(V), which is cyclic of order p. Let A be a non-central element of U(V) stabilizing every line in π, and let B be a non-central element of U(V) stabilizing every plane containing l. Then A and B generate U(V), and after replacing C by a power if necessary, the commutator of A and B is equal to C. If we identify V with (F_{p})^{3}, and take U(V) to be the upper triangular matrices, then we may take

A = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) B = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) .

Table 8.2: The D_{8}-summands of K̃_{n,3}^{k}.

\begin{matrix}n & k & P_{1} & P_{2} & P_{3} & P_{4} & P_{5} & P_{6} & P_{7} & P_{8} \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 2 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 0 & 1 \\ 2 & 1,2 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 \\ 3 & 0 & 4 & 4 & 4 & 2 & 0 & 0 & 0 & 1 \\ 3 & 1,6 & 6 & 1 & 3 & 0 & 0 & 3 & 1 & 1 \\ 3 & 2,5 & 5 & 3 & 3 & 1 & 0 & 1 & 1 & 1 \\ 3 & 3,4 & 5 & 2 & 4 & 1 & 0 & 2 & 0 & 1 \\ 4 & 0 & 24 & 8 & 8 & 4 & 0 & 0 & 0 & 1 \\ 4 & 1,14 & 28 & 1 & 7 & 0 & 0 & 7 & 1 & 1 \\ 4 & 2,13 & 25 & 7 & 7 & 3 & 0 & 1 & 1 & 1 \\ 4 & 3,12 & 27 & 2 & 8 & 1 & 0 & 6 & 0 & 1 \\ 4 & 4,11 & 25 & 6 & 8 & 3 & 0 & 2 & 0 & 1 \\ 4 & 5,10 & 27 & 3 & 7 & 1 & 0 & 5 & 1 & 1 \\ 4 & 6,9 & 26 & 5 & 7 & 2 & 0 & 3 & 1 & 1 \\ 4 & 7,8 & 26 & 4 & 8 & 2 & 0 & 4 & 0 & 1 \\ 5 & 0 & 112 & 16 & 16 & 8 & 0 & 0 & 0 & 1 \end{matrix}

Table 8.3: A maximal Syl_{3}(GL_{3}(F_{3}))-permutation submodule of K̃_{n,3}^{k}.

\begin{matrix}n & k & dim. M′ & P_{1} & P_{2} & P_{5} & P_{6} & P_{7} & P_{10} & P_{11} \\ 1 & 0,1 & 13 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 2 & 0 & 91 & 1 & 3 & 3 & 1 & 0 & 0 & 1 \\ 2 & 1 & 65 & 1 & 1 & 2 & 0 & 3 & 0 & 2 \\ 2 & 2 & 71 & 1 & 2 & 2 & 0 & 1 & 1 & 2 \\ 2 & 3 & 73 & 1 & 2 & 2 & 0 & 1 & 2 & 1 \\ 2 & 4 & 57 & 1 & 1 & 1 & 0 & 2 & 1 & 3 \\ 2 & 5 & 65 & 1 & 2 & 1 & 0 & 1 & 2 & 2 \\ 2 & 6 & 67 & 1 & 2 & 1 & 0 & 2 & 2 & 1 \\ 2 & 7 & 73 & 1 & 2 & 2 & 0 & 2 & 1 & 1 \end{matrix}

(Recall that V is to be viewed as the space of row vectors with a right GL(V)-action.) For p = 2 the group U(V) has 8 conjugacy classes of subgroups, which we list in the following order:

{1}, ⟨A⟩, ⟨B⟩, ⟨C⟩, ⟨AB⟩, ⟨A, C⟩, ⟨B, C⟩, U(V).

Let P_{1}, …, P_{8} be the corresponding transitive permutation modules, so that P_{1} is the free module and P_{8} is the trivial module. Similarly, for p > 2, U(V) has 2p + 5 conjugacy classes of subgroups, which we list as:

\begin{matrix}\{1\}, ⟨A⟩, ⟨AB⟩ & , …, ⟨AB^{p−1}⟩, ⟨B⟩, ⟨C⟩, \\ ⟨A, C⟩, ⟨AB, C⟩, …, ⟨AB^{p−1}, C⟩, ⟨B, C⟩, U(V). \end{matrix}

Again we let P_{1}, …, P_{2p+5} be the corresponding transitive permutation modules.

Table 8.4: A maximal Syl_{5}(GL_{3}(F_{5}))-permutation submodule of K̃_{n,3}^{k}.

\begin{matrix}n & k & dim. M′ & P_{1} & P_{2} & P_{7} & P_{8} & P_{9} & P_{14} & P_{15} \\ 1 & 0–3 & 31 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 2 & 0 & 651 & 3 & 5 & 5 & 1 & 0 & 0 & 1 \\ 2 & 1 & 527 & 3 & 1 & 4 & 0 & 5 & 0 & 2 \\ 2 & 2 & 447 & 2 & 3 & 4 & 0 & 3 & 1 & 2 \\ 2 & 3 & 467 & 2 & 4 & 4 & 0 & 2 & 1 & 2 \\ 2 & 4 & 587 & 3 & 4 & 4 & 0 & 1 & 1 & 2 \\ 2 & 5 & 591 & 3 & 4 & 4 & 0 & 1 & 2 & 1 \end{matrix}

Tables 8.2, 8.3, and 8.4 describe maximal U(V)-permutation submodules M′ of K̃_{n,3}^{k} in the cases p = 2, 3, and 5 respectively. These submodules were found using the algorithm of Proposition 7.1, with the conjugacy classes of subgroups of U(V) listed in the order given above. The permutation modules omitted from Tables 8.3 and 8.4 never arose as summands of any such M′. In Table 8.2 the dimension of M′ is omitted since in these cases M′ was always the whole of K̃_{n,3}^{k}, with dimension 2^{2n} + 2^{n} + 1. The dimensions of K̃_{1,3}^{k} and K̃_{2,3}^{k} are 13 and 91 respectively for p = 3, and 31 and 651 respectively for p = 5. Table 8.4 in the case n = 2 is incomplete in the sense that not all values of k have been considered. This is because each row required over 24 hours’ computing time.

Note that for fixed p and n, the modules K_{n,3}^{k} tend not to be isomorphic to each other, except when n = 1, and isomorphic in pairs when p = 2. For each p, the ring structure gives rise to a duality

K_{n,3}^{k} × K_{n,3}^{N−k} → F_{p} ⊆ K_{n,3}^{N},

where N = p^{n} − 1. This explains the observed fact that whenever K_{n,3}^{k} is a permutation module, then K_{n,3}^{N−k} ≅ K_{n,3}^{k}.

We find it intriguing that in the case p = 2 we have been unable to find pairs (n, k) such that K_{n,3}^{k} is not a U(V)-permutation module. Note also that for each n and p considered, K_{n,3}^{0} is a U(V)-permutation module, although it is easy to show that usually K_{n,3}^{0} cannot be a GL(V)-permutation module by comparing the information in the tables with the information given by Brauer characters. (This technique may be used to prove the cases n = 2, 4 and 5 of Theorem 1.1(d), which we leave as an exercise.)

Finally, in Tables 8.5 and 8.6 we give just enough information to prove Theorem 1.4, in the cases p = 3 and p = 5 respectively. That is, for each subgroup H of U(V) of order p^{2}, we give the dimension of a maximal H-permutation submodule M′′ of K_{2,3}^{1}. The dimension of K_{2,3}^{1} is 91 for p = 3 and 651 for p = 5. Only one of the subgroups ⟨AB, C⟩, …, ⟨AB^{p−1}, C⟩ is listed in these tables, because these subgroups are all conjugate in GL(V) and so give rise to M′′’s of the same dimension. The programs were run separately for each of these groups however, as a check.

Table 8.5: A maximal H-permutation submodule of K_{2,3}^{1} (p = 3).

Subgroup H  dim. M′′
⟨A, C⟩  69
⟨AB, C⟩  84
⟨B, C⟩  87

Table 8.6: A maximal H-permutation submodule of K_{2,3}^{1} (p = 5).

Subgroup H  dim. M′′
⟨A, C⟩  535
⟨AB, C⟩  628
⟨B, C⟩  643

Kriz’s example of a 3-group G such that K(2)^{∗}(BG) is not concentrated in even degrees is the Sylow 3-subgroup of GL_{4}(F_{3}) [6]. This group is expressible as the split extension with kernel (F_{3})^{3} and quotient the Sylow 3-subgroup of GL_{3}(F_{3}), with the natural action. There may be a connection between our result that K_{2,3}^{∗} is not a permutation module for the Sylow 3-subgroup of GL_{3}(F_{3}) and the fact that K(2)^{∗}(BG) is not entirely even. If so, then Theorem 1.4 suggests if H is a split extension with kernel (F_{3})^{3} and quotient a subgroup of GL_{3}(F_{3}) of order nine, then possibly K(2)^{∗}(BH) is not entirely even. Such H include the extraspecial group of order 3^{5} and exponent 3.

Acknowledgements. Throughout this project the second-named author was supported by an E.C. Leibniz Fellowship at the CRM. The first-named author was initially supported by a DGICYT Fellowship at the CRM and later by an E.C. Leibniz Fellowship at the MPI. Our computer programs were run on a network of Sun SPARC computers at the MPI. The authors gratefully acknowledge the hospitality of both the CRM and the MPI.

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Ian Leary

Max-Planck Institut für Mathematik 53225 Bonn Germany From Jan. 1996: Faculty of Math. Studies Univ. of Southampton Southampton SO17 1BJ England Björn Schuster Centre de Recerca Matemàtica E–08193 Bellaterra (Barcelona) Spain From June 1996: FB Mathematik Bergische Universität D–42097 Wuppertal Germany