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

Socle pairings on tautological rings

We study some aspects of the $\lambda_g$ pairing on the tautological ring of $M_g^c$, the moduli space of genus $g$ stable curves of compact type. We consider pairing kappa classes with pure boundary strata, all tautological classes supported on the boundary, or the full tautological ring. We prove that the rank of this restricted pairing is equal in the first two cases and has an explicit formula in terms of partitions, while in the last case the rank increases by precisely the rank of the $\lambda_g\lambda_{g - 1}$ pairing on the tautological ring of $M_g$.Comment: 18 pages, 1 figure; v3: journal version; v2: minor revisions to sections 1.1 and 4.1, results unchange