Specht modules decompose as alternating sums of restrictions of Schur modules

TL;DR

This paper develops a formula to decompose Schur modules into Specht modules when restricted to symmetric groups, revealing alternating sign patterns and introducing a new basis related to stable Kronecker coefficients.

Contribution

It provides an equivariant Möbius inversion formula for decomposing Schur modules into Specht modules and introduces a new symmetric function basis with stable Kronecker coefficients.

Findings

Coefficients in the decomposition alternate in sign by degree.

Explicit formulas involve plethysms.

New basis of symmetric functions with stable Kronecker coefficients.

Abstract

Schur modules give the irreducible polynomial representations of the general linear group $\mathrm{GL}_t$. Viewing the symmetric group $\mathfrak{S}_t$ as a subgroup of $\mathrm{GL}_t$, we may restrict Schur modules to $\mathfrak{S}_t$ and decompose the result into a direct sum of Specht modules, the irreducible representations of $\mathfrak{S}_t$. We give an equivariant M\"{o}bius inversion formula that we use to invert this expansion in the representation ring for $\mathfrak{S}_t$ for $t$ large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternate in sign by degree. In particular, this allows us to define a new basis of symmetric functions whose structure constants are stable Kronecker coefficients and which expand with alternating signs into the Schur basis.