The Ordinary Characters of the Involution Module of SL<sub>2</sub>(2<i><sup>f</sup></i>)

Abstract

Let k be an algebraically closed field of characteristic two, and let G be a finite group. Furthermore let J be the set of involutions in G, that is, the set of elements in G of order two. Then J is a G-set under conjugation. In particular we get a fcG-permutation module fcJ, which is known as the Involution Module of G. The involution module has been studied in general by J. Murray in [5; 6; 7]. Also the author studied the involution module of the special linear group and general linear group in [10] and [11], respectively. The purpose of this paper is to investigate the ordinary characters of the involution module of the special linear group SL2(g), where q := 2-f, for some integer / > 1. We present our main result in Theorem 2.7. In the following we give a brief introduction to the idea of ordinary characters. Let (iž, F, k ) be a 2-modular system. That means R is a complete discrete valuation ring of characteristic 0, with field of fractions F, unique maximal ideal (7r) and residue field R/(7t) = k. Without loss of generality we assume that F is contained in the field C of complex numbers and furthermore that R contains all ( q l)-st roots of unity. Next let M be a &G-module. If there exists an i?G-module V such that Vo :=