A Probabilistic Approach Toward Conjugacy Classes in the Finite General Linear and Unitary Groups

Abstract

The conjugacy classes of the finite general linear and unitary groups are used to define probability measures on the set of all partitions of all natural numbers. Probabilistic algorithms for growing random partitions according to these measures are obtained. These algorithms are applied to prove group theoretic results which are typically proved by techniques such as character theory and Moebius inversion. Among the theorems studied are Steinberg's count of unipotent elements, Rudvalis' and Shinoda's work on the fixed space of a random matrix, and Lusztig's count of nilpotent matrices of a given rank. Generalizations of these algorithms based on Macdonald's symmetric functions are given.