ASYMPTOTIC BOUNDS FOR THE SIZE OF Hom(A, GLn(q))

Fix an arbitrary finite group A of order a, and let X(n, q) denote the set of homomorphisms from A to the finite general linear group GLn(q). The size of X(n, q) is a polynomial in q. In this note, it is shown that generically this polynomial has degree n2(1 – a−1) − εr and leading coefficient mr, where εr and mr are constants depending only on r := n mod a. We also present an algorithm for explicitly determining these constants.

Twisted Dickson–Mui invariants and the Steinberg module multiplicity

We determine the invariants, with arbitrary determinant twists, of the parabolic subgroups of the finite general linear group GLn(q) acting on the tensor product of the symmetric algebra S•(V) and the exterior algebra ∧•(V) of the natural GLn(q)-module V. In addition, we obtain the graded multiplicity of the Steinberg module of GLn(q) in S•(V) ⊗ ∧•(V), twisted by an arbitrary determinant power.

THE LOEWY SERIES OF THE STEINBERG-PIM OF FINITE GENERAL LINEAR GROUPS

In this paper we calculate the Loewy series of the projective indecomposable module of the unipotent block contained in the Gelfand–Graev module of the finite general linear group in the case of non-describing characteristic and Abelian defect group.

The Cuspidal Modules of the Finite General Linear Groups

In this paper we prove a conjecture due to R. Carter [2], concerning the action of the finite general linear group GLn(q) on a cuspidal module. As an application of this result, we work out the caseGL4(q).