Specht modules decompose as alternating sums of restrictions of Schur modules

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 Möbius 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.

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