Absolute values and tensor powers of irreducible characters
This work addresses theoretical problems in representation theory for mathematicians, with incremental contributions building on known concepts.
The paper tackles the problem of computing expectations of absolute values and tensor powers of irreducible characters in finite groups, presenting a simple formula in terms of character ratios and discussing asymptotic properties and relations to dimension growth in tensor powers.
Let $ Ï$ 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 $(|Ï|/Ï(1))^{t} $ in terms of character ratios $ (|Ï(g)|/Ï(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