Conjugacy classes in unitriangular matrices.
Let \(G_n=G_n(q)\) be the group of the upper unitriangular matrices of size \(n\times n\) over \(\text{GF}(q)\), 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 the paper under review, the authors develop the algorithm given by them [in J. Algebra 177, No. 3, 899-925 (1995; Zbl 0839.20061)] and introduce new ideas and theoretical properties which lead to get the conjugacy vector of \(G_n\) for \(n\leq 13\). In particular, they 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 \(n\leq 13\).
scientific article; zbMATH DE number 1970612