2009 Algebraic Geometry Seminar - 02

The torus-equivariant K-theory of a (generalized) flag variety G/P is an algebra over a Laurent polynomial ring. This algebra has a natural basis consisting of structure sheaves of Schubert varieties. The structure constants for multiplication with respect to this basis are Laurent polynomials, and a fundamental problem is to determine them explicitly. Based on a wealth of evidence, Griffeth--Ram and Graham--Kumar conjectured that the coefficients of these polynomials are positive (with respect to a certain choice of generators for the polynomial ring). Their conjectures generalize theorems of Graham and Brion, treating equivariant cohomology and non-equivariant K-theory, respectively. In joint work with Stephen Griffeth and Ezra Miller, we prove these positivity conjectures. I will explain our methods, which combine earlier work of Brion with new equivariant transversality techniques.