Word-Induced Measures on Compact Groups

TL;DR

This paper links measures induced by group words on compact groups to the topology of associated cell complexes, revealing new connections with quantum gauge theory, representation theory, and group properties.

Contribution

It introduces a topological framework for understanding word-induced measures on compact groups, connecting algebraic, geometric, and representation-theoretic aspects.

Findings

Measures are determined by the topology of associated cell complexes.

Rediscovery of Witten's formulas relating to quantum gauge theory.

Elementary proof that the dimension of irreducible representations divides group order.

Abstract

Consider a group word w in n letters. For a compact group G, w induces a map G^n \rightarrow G$ and thus a pushforward measure {\mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and prove in Theorem 2.5 that {\mu}_w is determined by the topology of X(w). The proof makes use of non-abelian cohomology and Nielsen's classification of automorphisms of free groups [Nie24]. Focusing on the case when X(w) is a surface, we rediscover representation-theoretic formulas for {\mu}_w that were derived by Witten in the context of quantum gauge theory [Wit91]. These formulas generalize a result of Erd\H{o}s and Tur\'an on the probability that two random elements of a finite group commute [ET68]. As another corollary, we give an elementary proof that the dimension of an irreducible complex representation of a finite group divides the order of the group; the only ingredients are Schur's lemma, basic counting, and a divisibility argument.