Some Restriction Coefficients for the Trivial and Sign Representations

TL;DR

This paper provides combinatorial interpretations for the multiplicities of trivial and sign representations in certain polynomial representations of general linear and symmetric groups, using character polynomials.

Contribution

It introduces new combinatorial interpretations for these multiplicities, specifically for two-column and hook partitions, via character polynomials.

Findings

Positive combinatorial interpretation for sign representation multiplicity

Positive combinatorial interpretation for trivial representation multiplicity

Method applicable to specific partition types

Abstract

We use character polynomials to obtain a positive combinatorial interpretation of the multiplicity of the sign representation in irreducible polynomial representations of $GL_n(\mathbb{C})$ indexed by two-column and hook partitions. Our method also yields a positive combinatorial interpretation for the multiplicity of the trivial representation of $S_n$ in an irreducible polynomial representation indexed by a hook partition.