Counting Conjugacy Classes of Elements of Finite Order in Lie Groups

TL;DR

This paper develops combinatorial methods to count conjugacy classes of finite order elements in classical Lie groups, addressing previously unexplored questions related to eigenvalue structures, with implications for string theory.

Contribution

It introduces novel combinatorial techniques to compute the number of conjugacy classes with specific eigenvalue properties in classical Lie groups, extending prior group-theoretic results.

Findings

Calculated $N(G,m)$ for unitary, orthogonal, and symplectic groups.

Determined $N(G,m,s)$ for these groups, a previously unasked question.

Provides formulas and methods applicable to string theory vacua enumeration.

Abstract

Using combinatorial techniques, we answer two questions about simple classical Lie groups. Define $N(G,m)$ to be the number of conjugacy classes of elements of finite order $m$ in a Lie group $G$, and $N(G,m,s)$ to be the number of such classes whose elements have $s$ distinct eigenvalues or conjugate pairs of eigenvalues. What is $N(G,m)$ for $G$ a unitary, orthogonal, or symplectic group? What is $N(G,m,s)$ for these groups? For some cases, the first question was answered a few decades ago via group-theoretic techniques. It appears that the second question has not been asked before; here it is inspired by questions related to enumeration of vacua in string theory. Our combinatorial methods allow us to answer both questions.