Polynomial Properties in Unitriangular Matrices,

Let n = n(q) be the group of the upper unitriangular matrices of size n over q, the finite field of q = pt elements. G. Higman has conjectured that, for each n, the number of conjugacy classes of elements of n is a polynomial expression in q. In this paper we prove that the number of conjugacy classes of n of cardinality qs, with s ≤ n − 3, is a polynomial in q − 1, with non-negative integral coefficients, fs(q − 1), of degree less than or equal to the integer part of . In addition, fs(q − 1) depends only on s and not on n. We determine these polynomials arguing with the methods we gave previously (1995, J. Algebra177, 899–925). In fact, the coefficients of these polynomials are obtained by certain generating functions.