UCLA

A classical conjecture of Graham Higman states that the number of conjugacy classes in U_n(q), the group of upper triangular (nxn)-matrices over F_q, is a polynomial function of q, for all n. This dissertation concerns itself with both enumerative and asymptotic results regarding the number of conjugacy classes in U_n(q). We present both positive and negative evidence for Higman’s conjecture, verifying the conjecture for n ≤ 16, and suggesting that it probably fails for n ≥ 59. The tools are both theoretical and computational. We introduce a new framework for testing Higman’s conjecture, which involves recurrence relations for the number of conjugacy classes in pattern groups. We prove these relations via the orbit method for finite nilpotent groups.

We also improve the best known asymptotic upper bound on the number of conjugacy classes in U_n(q), and introduce upper bounds on the number of conjugacy classes in groups in the lower central series for U_n(q). To do so, we introduce a technique involving a combinatorial structure called a gap array. Gap arrays encode properties of centralizers of Jordan forms. By proving asymptotic results on the structure of gap arrays we deduce asymptotic results about the number of conjugacy classes in U_n(q).