Positroid varieties and affine Stanley symmetric functions

The positroid stratification of the Grassmannian is obtained by intersecting the cyclic rotations of the Schubert stratification. This stratification was first studied by Postnikov in the context of the totally nonnegative part of the Grassmannian. I will begin by discussing some of the combinatorics involved in indexing this stratification, and describing its closure partial order. I will also talk about some of the remarkable geometric properties that positroid varieties have: they are normal, Cohen-Macaulay, have rational singularities, and are compatibly Frobenius split. Finally, I will discuss the class of a positroid variety in the cohomology of the Grassmannian. When identified with symmetric functions, one obtains certain affine Stanley symmetric functions.

This talk is based on joint work with Allen Knutson and David Speyer. If time permits, I may also discuss some recent related work, joint with XuHua He.