Equivariant Matrix Factorizations of Symmetric Polynomials

The action of a finite group $G$ on a module over a graded polynomial ring extends to its free resolution, and this extension generalises to modules over hypersurface rings if the polynomial ideal is closed under the group action. We show how the equivariant structure transfers to matrix factorizations when the polynomial itself is $G$-invariant, and study the representations of these factorizations for the symmetric group. We show that the representations in $\mathfrak{S}_n$-equivariant factorizations from finite length modules have identical decompositions into irreducible representations, and provide explicit formulas for decompositions in factorizations of $f$ from the fat point modules $R/\mathfrak{m}^a$ when $0 < a < \deg f$.