Branching rules and commuting probabilities for Triangular and Unitriangular matrices

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.