Free Resolutions of Some Schubert Singularities in the Lagrangian Grassmannian

TL;DR

This paper constructs free resolutions for specific symmetric matrix subvarieties, including symmetric determinantal varieties, using Schubert variety geometry, extending previous work with new algebraic and geometric techniques.

Contribution

It introduces a new approach to free resolutions of Schubert singularities in the Lagrangian Grassmannian, generalizing known results for symmetric determinantal varieties.

Findings

Constructed free resolutions for certain symmetric matrix subvarieties.

Extended techniques from Schubert variety geometry to new classes.

Provided explicit algebraic descriptions of these resolutions.

Abstract

In this paper we construct free resolutions of certain class of closed subvarieties of affine space of symmetric matrices (of a given size). Our class covers the symmetric determinantal varieties (i.e., determinantal varieties in the space of symmetric matrices), whose resolutions were first constructed by J\'ozefiak-Pragacz-Weyman. Our approach follows the techniques developed in Free Resolutions of Some Schubert Singularities by Kummini-Lakshmibai-Sastry-Seshadri, and uses the geometry of Schubert varieties.