University of Minnesota Combinatorics Seminar
Friday, March 14, 2014
3:35pm in 570 Vincent Hall



Maximal Newton Polygons and Quantum Schubert Calculus

Elizabeth Beazley

Haverford


Abstract

We will discuss a surprising relationship between the quantum cohomology of the variety of complete flags and the partially ordered set of Newton polygons associated to an element in the affine symmetric group. One key to establishing this connection is the fact that paths in the quantum Bruhat graph, which is a weighted directed graph with vertices indexed by elements in the finite symmetric group, encode saturated chains in the strong Bruhat order on the affine symmetric group. This correspondence, which we will explain, is also fundamental in establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. In addition, using some geometry associated to the poset of Newton polygons, one obtains independent proofs for several combinatorial statements about paths in the quantum Bruhat graph and its symmetries, which were originally proved by Postnikov.