On the Fourier Expansion of Word Maps

TL;DR

This paper explores the Fourier expansion of word maps in finite groups, providing formulas for cases where some letters appear twice, leading to new insights and simplified proofs of classical results.

Contribution

It introduces new formulas for the Fourier expansion of specific word maps, enhancing understanding of their character-theoretic properties.

Findings

Formulas for Fourier expansion of words with repeated letters

Simplified proofs of classical results in group theory

New results on the structure of word maps

Abstract

Frobenius observed that the number of times an element of a finite group is obtained as a commutator is given by a specific combination of the irreducible characters of the group. More generally, for any word w the number of times an element is obtained by substitution in w is a class function. Thus, it has a presentation as a combination of irreducible characters, called its Fourier expansion. In this paper we present formulas regarding the Fourier expansion of words in which some letters appear twice. These formulas give simple proofs for classical results, as well as new ones.