111 research outputs found
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 ). 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
Equivariant Giambelli and determinantal restriction formulas for the Grassmannian
The main result of the paper is a determinantal formula for the restriction
to a torus fixed point of the equivariant class of a Schubert subvariety in the
torus equivariant integral cohomology ring of the Grassmannian. As a corollary,
we obtain an equivariant version of the Giambelli formula.Comment: 16 pages, 3 figures, LaTex, uses epsfig and psfrag; for the revised
version: title changed; Proof of Theorem 3 changed; 3 references added and 1
deleted; other minor change
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