Quasi-permutation Representations of Borel and Parabolic Subgroups of Steinbergʹʹs triality groups

If G is a finite linear group of degree n , that is , a finite group of automorphisms of an n-dimensional complex vector space , or equivalently , a finite group of non-singular matrices of order n with complex coefficients , we shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace.Thus every permutation matrix over C is a quasi-permutation matrix.For a given finite group G , let c(G) denotes the minimal degree of a faithful representation of G by quasi-permutation matrices over the complex numbers and let r(G) denote the minimal degree of a faithful rational valued complex character of G. The purpose of this paper is to calculate c(G) and r(G) for the Borel and Parabolic Subgroups of Steinbergʹs triality groups.