Polynomial properties in unitriangular matrices. II
Antonio Vera López, J.M. Arregi · Journal of Group Theory · 2005
Let = ( q ) be the group of all upper unitriangular matrices of size n over , the field with q = p t elements. We show that the computation of many canonical representatives of the conjugacy classes of can be reduced to similar calculations for certain matrices of smaller size that we call condensed matrices. Our methods make use of combinatorial techniques including (polynomial) generating functions, and they greatly increase the efficiency of the calculations of the conjugacy vector of . They also allow us to obtain the number of conjugacy classes of size q z for any z ≤ 2 n − 8. These numbers are polynomial functions of q , in accordance with a well-known conjecture of G. Higman.