Absolute values and tensor powers of irreducible characters

TL;DR

This paper derives a simple formula for the expectation of a specific power of normalized irreducible characters of finite groups, linking it to character ratios and exploring asymptotic behavior related to tensor powers.

Contribution

It introduces a new, straightforward formula for expectations involving irreducible characters and discusses its asymptotic implications for tensor powers of representations.

Findings

Derived a formula for the expectation of $(| ext{chi}|/ ext{chi}(1))^{t}$

Connected the formula to character ratios and tensor power growth

Explored asymptotic properties of the formula

Abstract

Let $ \chi $ be a character of a complex irreducible representation of a finite group $G$. We present a simple formula for the expectation of the random variable $(|\chi|/\chi(1))^{t} $ in terms of character ratios $ (|\chi(g)|/\chi(1))^{t}, \; g \in G, \; t \geq 0 $. As a follow up we briefly discuss asymptotic properties of the formula and its relation to the growth of dimensions of isotypic components in (virtual) tensor powers of irreducible representations