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

Abstract

Let R = k[x 1 , . . . , x n ] be a polynomial ring over a field k and let I ⊂ R be a monomial ideal preserved by the natural action of the symmetric group S n on R. We give a combinatorial method to determine the S n -module structure of Tor i (I, k). Our formula shows that Tor i (I, 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 S n -equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of 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 S n -invariant part of Tor i (I, k) in terms of Tor groups of the unsymmetrization of I.

Theorems of Carathéodory, Minkowski-Weyl, and Gordan up to symmetry

“…One prominent theme in this theory is to find equivariant versions of known results related to ideals in Noetherian polynomial rings. Examples of successful extensions include equivariant Hilbert's basis theorem [2,4,5,9,18], equivariant Hilbert-Serre theorem [11,16,17], equivariant Buchberger algorithm [8], equivariant Hochster's formula [15]. See, e.g., also [7,12,13,14,19,20,21] for related results.…”

Finite group characters on free resolutions

“…In this case, the G-action extends to a minimal free resolution of M in an essentially unique way (see [Gal15,§1]). There has been significant interest in naturally occurring examples of minimal free resolutions with such group actions [ZGS14, ELSW18, GGW18, BDRH + 19, Gal20, BdAG + 20, Mur20, SY20, MR20,Rai21].…”