A character relationship between symmetric group and hyperoctahedral group

TL;DR

This paper explores the connection between the character theories of symmetric groups and hyperoctahedral groups, aiming to deepen understanding of their algebraic relationships and implications for reductive groups with automorphisms.

Contribution

It establishes a relationship between the character theory of symmetric groups and hyperoctahedral groups, contributing to the broader understanding of automorphism-fixed subgroup character theories.

Findings

Character theory of $S_{2n}$ and $S_{2n+1}$ relates to that of $B_n$

Provides insights into automorphism-fixed subgroup character theories

Enhances understanding of algebraic structures in symmetric and hyperoctahedral groups

Abstract

We relate character theory of the symmetric groups $S_{2n}$ and $S_{2n+1}$ with that of the hyperoctahedral group $B_n = ({\mathbb Z}/2)^n \rtimes S_n$, as part of the expectation that the character theory of reductive groups with diagram automorphism and their Weyl groups, is related to the character theory of the fixed subgroup of the diagram automorphism.