Character ratios for finite groups of Lie type, and applications

Abstract

For a nite group G , a character ratio is a complex number of the form ( x ) (1) , where x 2 G and is an irreducible character of G . Upper bounds for absolute values of character ratios, particularly for simple groups, have long been of interest, for various reasons; these include applications to covering numbers, mixing times of random walks, and the study of word maps. In this article we shall survey some results on character ratios for nite groups of Lie type, and their applications. Character ratios for alternating and symmetric groups have been studied in great depth also { see for example [ 32, 33 ] { culminating in the de nitive results and applications to be found in [ 20 ]; but we shall not discuss these here. It is not hard to see the connections between character ratios and group struc- ture. Here are three well known, elementary results illustrating these connections. The rst two go back to Frobenius. Denote by Irr( G ) the set of irreducible charac- ters of G.