Permutation and quasi-permutation representations of the Chevalley groups

Abstract

Abstract A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G) . The minimal degree of a faithful representation of G by quasi-permutation matrices over the rationals and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G . In this paper p(G), q(G), c(G) and r(G) are calculated for the group G 2 ( q n ), q ≠ 3. AMS Classification: 20C15 Keywords: General linear group, Quasi-permutation. Introduction Let G be a finite linear group of degree n , that is, a finite group of automorphisms of an n -dimensional complex vector space. We shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n , its elements, considered as acting on the elements of a basis of an n -dimensional complex vector space V , induce automorphisms of V forming a group isomorphic to G . The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x and so is a non-negative integer.