A Frobenius-Type Formula for Compact Lie Groups

TL;DR

This paper extends Frobenius's formula for counting elements in fibers of the commutator map from finite groups to compact Lie groups and explores its relation to commutator probability.

Contribution

It generalizes Frobenius's formula to compact Lie groups and links it to the groups' commutator probability, broadening the understanding of group structure.

Findings

Derived a Frobenius-type formula for compact Lie groups

Connected the formula to the commutator probability in these groups

Provided a new tool for analyzing group commutators

Abstract

Let $G$ be a group and $\alpha: G \times G \to G$ denote the commutator map. In the case of finite groups, Frobenius gave the formula to compute the cardinalities of the fibres $\alpha^{-1}(g)$ in terms of the character values $\chi(g)$ for irreducible characters $\chi$ of $G$. We generalise this formula to compact Lie groups. Further, we connect this generalised formula to the commutator probability of the concerned groups.