Specht modules decompose as alternating sums of restrictions of Schur modules

Schur modules give the irreducible polynomial representations of the general linear group GL(t). Viewing the symmetric group (sic)(t) as a subgroup of GL(t), we may restrict Schur modules to (sic)(t) and decompose the result into a direct sum of Specht modules, the irreducible representations of (sic)(t). We give an equivariant Mobius inversion formula that we use to invert this expansion in the representation ring for (sic)(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 signs alternating by degree into the Schur basis.