Derived supersymmetries of determinantal varieties

TL;DR

This paper reveals that the linear strands of the Tor of determinantal varieties in symmetric and skew-symmetric matrices form irreducible representations of the periplectic Lie superalgebra, providing explicit realizations and simpler proofs.

Contribution

It extends the understanding of determinantal varieties by explicitly realizing their linear strands as representations of the periplectic Lie superalgebra, complementing prior work on general linear cases.

Findings

Linear strands are irreducible representations of the periplectic Lie superalgebra.

Provides explicit descriptions of these representations.

Offers a simpler proof of existing results.

Abstract

We show that the linear strands of the Tor of determinantal varieties in spaces of symmetric and skew-symmetric matrices are irreducible representations for the periplectic (strange) Lie superalgebra. The structure of these linear strands is explicitly known, so this gives an explicit realization of some representations of the periplectic Lie superalgebra. This complements results of Pragacz and Weyman, who showed an analogous statement for the generic determinantal varieties and the general linear Lie superalgebra. We also give a simpler proof of their result. Via Koszul duality, this is an odd analogue of the fact that the coordinate rings of these determinantal varieties are irreducible representations for a certain classical Lie algebra.