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Syzygies of Determinantal Thickenings and Representations of the General Linear Lie Superalgebra

$icon-email$ Claudiu Raicu , Jerzy Weyman

Abstract

We let $S = C [x_{i, j}]$ denote the ring of polynomial functions on the space of $m \times n$ matrices and consider the action of the group $GL = {GL}_{m} \times {GL}_{n}$ via row and column operations on the matrix entries. For a $GL$ -invariant ideal $I \subseteq S$ , 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 $I = I_{λ}$ is the ideal generated by the $GL$ -orbit of a highest weight vector of weight $λ$ , 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.

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Syzygies of Determinantal Thickenings and Representations of the General Linear Lie Superalgebra

Abstract