## Honors Theses

2021

#### Document Type

Honors Thesis (Open Access)

#### Department

Colby College. Mathematics and Statistics Dept.

Tamar Friedmann

Fernando Gouvêa

#### Abstract

Given a compact exceptional group $G$ and $m,s\in\mathbb{N}$, let $N(G,m)$ be the number of conjugacy classes of elements of order $m$ in $G$, and $N(G,m,s)$ the number of such classes whose elements have $s$ distinct eigenvalues. In string theory, the problem of enumerating certain classes of vacua in the string landscape can be rephrased in terms of the study of these quantities. We develop unified combinatorial algorithms based on Burnside's Lemma that can be used to compute both quantities for each of the five compact exceptional groups. Thus, we provide a combinatorial, alternative method to that of Djoković and extend previous results obtained by Friedmann-Stanley for the compact classical groups.

#### Keywords

Conjugacy classes, finite order, Lie groups, Burnside's Lemma, Möbius inversion

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