Syzygies of Determinantal Thickenings

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 $I \subseteq S$.