The saturation conjecture (after A. Knutson and T. Tao)

In this exposition we give a simple and complete treatment of A. Knutson and T. Tao's recent proof (http://front.math.ucdavis.edu/math.RT/9807160) of the saturation conjecture, which asserts that the Littlewood-Richardson semigroup is saturated. The main tool is Knutson and Tao's hive model for Berenstein-Zelevinsky polytopes. In an appendix of W. Fulton it is shown that the hive model is equivalent to the original Littlewood-Richardson rule.Comment: Latex document, 12 pages, 24 figure

Specializations of Grothendieck polynomials

We prove a formula for double Schubert and Grothendieck polynomials specialized to two rearrangements of the same set of variables. Our formula generalizes the usual formulas for Schubert and Grothendieck polynomials in terms of RC-graphs, and it gives immediate proofs of many other important properties of these polynomials.Comment: 4 pages, 1 figur

Chern class formulas for quiver varieties

In this paper a formula is proved for the general degeneracy locus associated to an oriented quiver of type A_n. Given a finite sequence of vector bundles with maps between them, these loci are described by putting rank conditions on arbitrary composites of the maps. Our answer is a polynomial in Chern classes of the bundles involved, depending on the given rank conditions. It can be expressed as a linear combination of products of Schur polynomials in the differences of the bundles. The coefficients are interesting generalizations of Littlewood-Richardson numbers. These polynomials specialize to give new formulas for Schubert polynomials.Comment: 17 pages, 20 figures. The document is available as a .tar.gz file containing one LaTeX2e file and 20 (included) postscript files. Packages xypic and psfrag are used. Note that when viewed with xdvi, the text in figures looks bad, but it comes out right when printed. The paper is also available as one postscript file at http://www.math.uchicago.edu/~abuch/papers/quiver.ps.g

Projected Gromov-Witten varieties in cominuscule spaces

A projected Gromov-Witten variety is the union of all rational curves of fixed degree that meet two opposite Schubert varieties in a homogeneous space X = G/P. When X is cominuscule we prove that the map from a related Gromov-Witten variety is cohomologically trivial. This implies that all (3 point, genus zero) K-theoretic Gromov-Witten invariants of X are determined by the projected Gromov-Witten varieties, which extends an earlier result of Knutson, Lam, and Speyer. Our proof uses that any projected Gromov-Witten variety in a cominuscule space is also a projected Richardson variety.Comment: 13 page

Quantum K-theory of Grassmannians

We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the ordinary (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through 3 general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum K-theory ring of a Grassmannian, which determine the multiplication in this ring. Our formula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.Comment: 26 pages, 2 figures; comments welcom