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 T-n(F-q) and unitriangular group UTm(F-q) over the finite field F-q 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 cp(k) for k <= 5 in each case.