The 2-modular Permutation Modules on Fixed Point Free Involutions of Symmetric Groups

1.1 Let A = kG, the group algebra of some finite group where the characteristic of the field k divides |G|. In contrast to working over the complex field, the kG-modules are not usually semisimple. If a Sylow p-subgroup of G is not cyclic then there are infinitely many indecomposable kG-modules, and we usually enjoy little control over the category of such modules. It is therefore instructive to find classes of modules which may be expressed as a sum of a not very great number of indecomposables, and to understand the structure of these indecomposables. Permutation kG-modules, and their indecomposable summands (called p-permutation modules), provide one such class. In [2] we studied permutation modules for symmetric groups acting on conjugacy classes of fixed point free elements which are products of q-cycles, and where k has characteristic p. With each component (that is, indecomposable summand) we associated a fixed point set, and we obtained a general description for the fixed point sets. In this paper we shall complete the analysis of the case q = p = 2. Thus we shall determine the fixed point sets of the components of the permutation module of the symmetric group Sym(2n) on the set of its fixed point free involutions. At the same time we find the vertex and Brauer quotient for each component, and the ordinary character associated with each component.