Richardson Varieties and Equivariant K-Theory

We generalize Standard Monomial Theory (SMT) to intersections of Schubert varieties and opposite Schubert varieties; such varieties are called Richardson varieties. The aim of this article is to get closer to a geometric interpretation of the standard monomial theory. Our methods show that in order to develop a SMT for a certain class of subvarieties in G/B (which includes G/B), it suffices to have the following three ingredients, a basis for the space of sections of an effective line bundle on G/B, compatibility of such a basis with the varieties in the class, certain quadratic relations in the monomials in the basis elements. An important tool will be the construction of nice filtrations of the vanishing ideal of the boundary of the varieties above. This provides a direct connection to the equivariant K-theory, where the combinatorially defined notion of standardness gets a geometric interpretation.Comment: 38 page

Singularities of Affine Schubert Varieties

This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type $\mathrm{A}^{(1)}_\ell$). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also determine explicitly the tangent spaces at singular points. For a general affine Schubert variety, we give partial results on the singular locus.Comment: 13 figure