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

Abstract

Let $R=\mathbb{k}[x_1,\dots ,x_n]$ be a polynomial ring over a field $\mathbb{k}$ and let $I\subset R$ be a monomial ideal preserved by the natural action of the symmetric group ${\mathfrak{S}}_n$ on $R$. We give a combinatorial method to determine the ${\mathfrak{S}}_n$-module structure of ${\mathrm{Tor}}_i(I,\mathbb{k})$. Our formula shows that ${\mathrm{Tor}}_i(I,\mathbb{k})$ 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 ${\mathfrak{S}}_n$-equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of ${\mathfrak{S}}_n$-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 $>n$) we compute the ${\mathfrak{S}}_n$-invariant part of ${\mathrm{Tor}}_i(I,\mathbb{k})$ in terms of $\mathrm{Tor}$ groups of the unsymmetrization of $I$.

31 pages

References in corpus (1)

• Elementary construction of minimal free resolutions of the Specht ideals of shapes $(n-2,2)$ and $(d,d,1)$

Cited by in corpus (2)

• Theorems of Carathéodory, Minkowski-Weyl, and Gordan up to symmetry
• Setting the scene for Betti characters