Counting conjugacy classes of elements of finite order in exceptional Lie groups (2210.15737v2)

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.