Counting conjugacy classes of elements of finite order in exceptional Lie groups

TL;DR

This paper develops algorithms to compute conjugacy class counts in exceptional Lie groups, providing explicit results and extending known data for classical groups to these complex structures.

Contribution

It introduces systematic algorithms for calculating conjugacy class counts in exceptional Lie groups and supplies explicit data, expanding the understanding beyond classical groups.

Findings

Explicit formulas for N(G,m) in exceptional groups

Computed N(G,m,s) for G2 and F4

Results align with known classical group data

Abstract

This paper continues the study of two numbers that are associated with Lie groups. The first number is $N(G,m)$, the number of conjugacy classes of elements in $G$ whose order divides $m$. The second number is $N(G,m,s)$, the number of conjugacy classes of elements in $G$ whose order divides $m$ and which have $s$ distinct eigenvalues, where we view $G$ as a matrix group in its smallest-degree faithful representation. We describe systematic algorithms for computing both numbers for $G$ a connected and simply-connected exceptional Lie group. We also provide explicit results for all of $N(G,m)$, $N(G_2,m,s)$, and $N(F_4,m,s)$. The numbers $N(G,m,s)$ were previously known only for the classical Lie groups; our results for $N(G,m)$ agree with those already in the literature but are obtained differently.