On characters of wreath products

TL;DR

This paper provides a concise combinatorial proof of a character identity relating hyperoctahedral and symmetric groups, and extends it to other wreath products, offering a new perspective beyond algebraic and Lie theoretic methods.

Contribution

It introduces a short combinatorial proof of a known character identity and generalizes it to a broader class of wreath products.

Findings

Combinatorial proof of the character identity for $B_n$ and $S_{2n}$

Extension of the identity to other wreath products

Simplification of the proof method compared to algebraic approaches

Abstract

A character identity which relates irreducible character values of the hyperoctahedral group $B_n$ to those of the symmetric group $S_{2n}$ was recently proved by L\"ubeck and Prasad. Their proof is algebraic and involves Lie theory. We present a short combinatorial proof of this identity, as well as a generalization to other wreath products.