Adjoint Representations of the Symmetric Group

TL;DR

This paper investigates how the adjoint representation of GL_n(C) decomposes when restricted to the symmetric group, identifying the irreducible components of symmetric and skew-symmetric matrix spaces as S_n-modules.

Contribution

It explicitly determines the irreducible constituents of the symmetric and skew-symmetric matrix spaces under the symmetric group's action, providing new insights into their module structure.

Findings

Identified irreducible constituents of symmetric matrices as S_n-modules.

Determined irreducible constituents of skew-symmetric matrices as S_n-modules.

Enhanced understanding of the representation theory of symmetric groups in matrix spaces.

Abstract

We study the restriction to the symmetric group, $\mc{S}_n$ of the adjoint representation of $\mt{GL}_n(\C)$. We determine the irreducible constituents of the space of symmetric as well as the space of skew-symmetric $n\times n$ matrices as $\mc{S}_n$-modules.