Branching rules and commuting probabilities for Triangular and Unitriangular matrices

Abstract

This paper concerns the enumeration of simultaneous conjugacy classes of k-tuples of commuting matrices in the upper triangular group Tn(Fq) and unitriangular group UTm(Fq) over the finite field Fq of odd characteristic. This is done for n=2,3,4 and m=3,4,5, by computing the branching rules. Further, using the branching matrix thus computed, we explicitly get the commuting probabilities cpk for k≤5 in each case.