Polynomial Properties in Unitriangular Matrices

✍ Scribed by Antonio Vera-López; J.M Arregi

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 92 KB
• Volume: 244
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.8896

✦ Synopsis

Let n = n q be the group of the upper unitriangular matrices of size n over q , the finite field of q = p t 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 q s , with s ≤ n -3, is a polynomial in q -1, with non-negative integral coefficients, f s q -1 , of degree less than or equal to the integer part of √ 2s + 1. In addition, f s q -1 depends only on s and not on n. We determine these polynomials arguing with the methods we gave previously (1995, J. Algebra 177, 899-925). In fact, the coefficients of these polynomials are obtained by certain generating functions.