Syzygies of Determinantal Thickenings

TL;DR

This paper proves a conjecture relating the linear strands of minimal free resolutions of certain invariant ideals in polynomial rings to modules over the general linear Lie superalgebra, providing an algorithmic method for their classification.

Contribution

It confirms a conjecture by Raicu and Weyman about the classes of modules over rl(m|n) associated with rl-invariant ideals, and introduces an algorithmic approach for their computation.

Findings

Proved the conjecture on rl(m|n)-modules classes.

Provided an algorithm for classifying these modules.

Connected free resolutions to Lie superalgebra representations.

Abstract

Let $S = \mathbb{C}[x_{i,j}]$ be the ring of polynomial functions on the space of $m \times n$ matrices, and consider the action of the group $\mathbf{GL} = \mathbf{GL}_m \times \mathbf{GL}_n$ via row and column operations on the matrix entries. It is proven by Raicu and Weyman that for a $\mathbf{GL}$-invariant ideal $I \subseteq S$, the linear strands of its minimal free resolution translates via the BGG correspondence to modules over the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. When $I=I_{\lambda}$ is the ideal generated by the $\mathbf{GL}$-orbit of a highest weight vector of weight $\lambda$, they gave a conjectural description of the classes of these $\mathfrak{gl}(m|n)$-modules in the Grothendieck group. We prove their conjecture here. We also give a algorithmic description of how to get the classes of these $\mathfrak{gl}(m|n)$-modules for any $\mathbf{GL}$-invariant ideal $I \subseteq S$.