Branching rules and commuting probabilities for Triangular and Unitriangular matrices

TL;DR

This paper computes the enumeration of conjugacy classes and commuting probabilities for specific upper triangular and unitriangular matrix groups over finite fields, providing explicit formulas for small dimensions.

Contribution

It introduces new branching rules and explicit commuting probability calculations for small-dimensional triangular matrix groups over finite fields.

Findings

Branching rules for $GT_n(F_q)$ with n=2,3,4

Branching rules for $UT_m(F_q)$ with m=3,4,5

Explicit commuting probabilities for k ≤ 5

Abstract

This paper concerns the enumeration of simultaneous conjugacy classes of $k$-tuples of commuting matrices in the upper triangular group $GT_n(\mathbf F_q)$ and unitriangular group $UT_m(\mathbf F_q)$ over the finite field $\mathbf F_q$ 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 $k\leq 5$ in each case.