Free Resolutions of Some Schubert Singularities in the Lagrangian Grassmannian

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.