An equivariant Hochster's formula for Sn$\mathfrak {S}_n$?invariant monomial ideals

Let $R=\Bbbk[x_1,\dots,x_n]$ be a polynomial ring over field $\Bbbk$ and let $I\subset R$ monomial ideal preserved by the natural action of symmetric group $\mathfrak S_n$ on $R$. We give combinatorial method to determine S_n$-module structure $\mathrm{Tor}_i(I,\Bbbk)$. Our formula shows that $\mathrm{Tor}_i(I,\Bbbk)$ is built from induced representations tensor products Specht modules associated hook partitions, their multiplicities are determined topological Betti numbers certain simplicial complexes. This result can viewed as an S_n$-equivariant analogue Hochster's for ideals. apply our results extremal S_n$-invariant ideals, in particular recover formulas Castelnuovo--Mumford regularity projective dimension. also concrete recipe how change we increase number variables, characteristic zero (or $>n$) compute part terms $\mathrm{Tor}$ groups unsymmetrization $I$.