An equivariant Hochster's formula for S_n-invariant monomial ideals

Let (Formula presented.) be a polynomial ring over a field (Formula presented.) and let (Formula presented.) be a monomial ideal preserved by the natural action of the symmetric group (Formula presented.) on (Formula presented.). We give a combinatorial method to determine the (Formula presented.) -module structure of (Formula presented.). Our formula shows that (Formula presented.) is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an (Formula presented.) -equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of (Formula presented.) -invariant monomial ideals, and in particular recover formulas for their Castelnuovo–Mumford regularity and projective dimension. We also give a concrete recipe for how the Betti numbers change as we increase the number of variables, and in characteristic zero (or (Formula presented.)) we compute the (Formula presented.) -invariant part of (Formula presented.) in terms of (Formula presented.) groups of the unsymmetrization of (Formula presented.).