Conjugacy classes in unitriangular matrices

Abstract

Let G n= G n(q) be the group of the upper unitriangular matrices of size n× n over F q , the finite field with q= p t elements. Higman has conjectured that, for each n, the number of conjugacy classes of elements of G n is a polynomial expression in q. In this paper we develop the algorithm given in [J. Algebra 177 (1995) 899] introducing new ideas and theoretical properties which lead us to get the conjugacy vector of G n for n⩽13. These vectors for n⩽5 (resp. for n=6,7,8) were given in [J. Algebra 152 (1) (1992) 1] (resp. [J. Algebra 177 (1995) 899]). In particular, we conclude that, for these values, the number of conjugacy classes of G n is a polynomial in q with integral coefficients and degree [ n( n+6)/12]. Thus, Higman’s conjecture holds for n⩽13. On the other hand, for each positive integer n, we find explicitly a set of q [ n( n+6)/12] different conjugacy classes.