Syzygies of determinantal thickenings and representations of the general linear Lie superalgebra

TL;DR

This paper explores the algebraic structure of determinantal ideals under group actions, linking their resolutions to modules over the general linear Lie superalgebra, and provides conjectures and proofs for specific cases.

Contribution

It establishes a connection between minimal free resolutions of GL-invariant ideals and modules over the Lie superalgebra gl(m|n), offering new insights into their structure.

Findings

Linear strands of resolutions correspond to modules over gl(m|n)

Conjectural description of module classes in Grothendieck group

Proof of conjecture for the first resolution strand

Abstract

We let S denote the ring of polynomial functions on the space of m x n matrices, and consider the action of the group GL = GL_m x GL_n via row and column operations on the matrix entries. For a GL-invariant ideal I in S we show that the linear strands of its minimal free resolution translate via the BGG correspondence to modules over the general linear Lie superalgebra gl(m|n). When I is the ideal generated by the GL-orbit of a highest weight vector, we give a conjectural description of the classes of these gl(m|n)-modules in the Grothendieck group, and prove that our prediction is correct for the first strand of the minimal free resolution.