Schubert singularities and Levi subgroup actions on Schubert varieties

Abstract

In the first part of this thesis, we study the action of the Levi part, L_w, of the stabilizer Q_w in GLN(C) (for left multiplication) of a Schubert variety X(w) in the Grassmannian G_d,N . For the natural action of L_w on C[X(w)], the homogeneous coordinate ring of X(w) (for the Plücker embedding), we give a combinatorial description of the decomposition of C[X(w)] into irreducible L_w-modules; in fact, our description holds more generally for the action of the Levi part L of any parabolic group Q that is a subgroup of Q_w. Using this combinatorial description, we give a classification of all Schubert varieties X(w) in the Grassmannian G_d,N for which C[X(w)] has a decomposition into irreducible L_w-modules that is multiplicity free. This classification is then used to show that certain classes of Schubert varieties are spherical L_w-varieties. These classes include all smooth Schubert varieties, all determinantal Schubert varieties, as well as all Schubert varieties in G_2,N and G_3,N. Also, as an important consequence, we get interesting results related to the singular locus of X(w) and multiplicities at T-fixed points in X(w). In the second part, we construct free resolutions for a large class of closed affine subvarieties of the affine space of symmetric matrices. The class includes the determinental varieties in the space of symmetric matrices, whose free resolutions were studied by Józefiak-Pragacz-Weyman. We use techniques developed by Kummini-Lakshmibai-Sastry-Seshadri, and the geometry of Schubert varieties in the Lagrangian Grassmannian, to construct these resolutions. The approach is algebraic-group theoretic and suggests a framework for extending these results to determinental varieties in other matrix spaces.