The syzygies of some thickenings of determinantal varieties

TL;DR

This paper computes the syzygies of certain GL-equivariant ideals related to minors of matrices, extending previous work and connecting algebraic and representation theoretic methods, with implications for Noetherianity in twisted algebras.

Contribution

It generalizes earlier results by explicitly calculating syzygies of ideals I_{a x b} for all a,b, using novel connections to superalgebra representation theory.

Findings

Computed syzygies for all a,b in I_{a x b} ideals

Established GL-equivariant structure of these syzygies

Connected algebraic results to superalgebra representation theory

Abstract

The vector space of m x n complex matrices (m >= n) admits a natural action of the group GL = GL_m x GL_n via row and column operations. For positive integers a,b, we consider the ideal I_{a x b} defined as the smallest GL-equivariant ideal containing the b-th powers of the a x a minors of the generic m x n matrix. We compute the syzygies of the ideals I_{a x b} for all a,b, together with their GL-equivariant structure, generalizing earlier results of Lascoux for the ideals of minors (b=1), and of Akin-Buchsbaum-Weyman for the powers of the ideals of maximal minors (a=n). Our methods rely on a nice connection between commutative algebra and the representation theory of the superalgebra gl(m|n), as well as on our previous calculation of Ext modules done in the context of describing local cohomology with determinantal support. Our results constitute an important ingredient in the proof by Nagpal-Sam-Snowden of the first non-trivial Noetherianity results for twisted commutative algebras which are not generated in degree one.