Matrix Schubert varieties for the affine Grassmannian

Abstract

Schubert calculus has become an indispensable tool for enumerative geometry. It concerns the multiplication of Schubert classes in the cohomology of flag varieties, and is typically conducted using algebraic combinatorics by way of a polynomial ring presentation of the cohomology ring. The polynomials that represent the Schubert classes are called Schubert polynomials.

An ongoing project in Schubert calculus has been to provide geometric foundations for the combinatorics. An example is the recovery by Knutson and Miller of the Schubert polynomials for finite flag varieties as the equivariant cohomology classes of matrix Schubert varieties. The present thesis is the start of a project to recover Schubert polynomials for the Borel-Moore homology of the (special linear) affine Grassmannian by an analogous process. This requires finitizing an affine Schubert variety to produce a matrix affine Schubert variety. This involves a choice of ``window'', so one must then identify a class representative that is independent of this choice. Examples lead us to conjecture that this representative is a k-Schur function. Concluding the discussion is a preliminary investigation into the combinatorics of Gröbner degenerations of matrix affine Schubert varieties, which should lead to a combinatorial proof of the conjecture.