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 $GT_n(\mathbf F_q)$andunitriangulargroup$UT_m(\mathbf F_q)$overthefinitefield$\mathbf F_q$ofoddcharacteristic\mathrm{.}Thisisdonefor$n=2,3,4$and$m=3,4,5$, by computing the branching rules. Further, using the branching matrix thus computed, we explicitly get the commuting probabilities $c{p}_k$ for $k\le 5$ in each case.