Speaker: Allen Knutson, Cornell University
Title: Interval positroid varieties, Schubert calculus, and pipe dreams
Time: 2:30 PM, Monday, April 8, 2013
Place: Malott 206
Abstract:
Given a k x n matrix, we can measure the rank of each interval [i,j] of columns, and the achievable systems of ranks correspond to upper-triangular partial permutation matrices f, where rank([i,j]) = |[i,j]| - #{dots SW of (i,j)}. For each such f, define the interval positroid variety to be the closure of the set of such matrices. Passing to the row span, we get a subvariety of the Grassmannian. These subvarieties include Schubert varieties, opposite Schubert varieties, and their intersections. I'll define a "geometric shifting" operation for subvarieties of the Grassmannian, relate it to Erd"os-Ko-Rado shifting, and say what it does to interval positroid varieties. This will allow us to shift any interval positroid variety to a union of Schubert varieties, and compute e.g. the product of a Schubert polynomial S_pi by a Schur polynomial in the same number of variables. I'll show how to record such calculations using a new kind of pipe dream.