Maximal Newton points and the quantum Bruhat graph

TL;DR

This paper reveals a deep connection between Newton points in affine Schubert cells and quantum cohomology, using the quantum Bruhat graph to describe maximal elements and their geometric implications.

Contribution

It provides a combinatorial formula for the maximum Newton point via paths in the quantum Bruhat graph, linking affine Schubert calculus and quantum cohomology.

Findings

Formula for the maximum Newton point in terms of quantum Bruhat graph paths

Paths encode saturated chains in the strong Bruhat order

Derived an inequality related to affine Deligne-Lusztig varieties

Abstract

We discuss a surprising relationship between the partially ordered set of Newton points associated to an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the unique maximum element in this poset in terms of paths in the quantum Bruhat graph, whose vertices are indexed by elements in the finite Weyl group. Key to establishing this connection is the fact that paths in the quantum Bruhat graph encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. One important geometric application of the present work is an inequality which provides a necessary condition for non-emptiness of certain affine Deligne-Lusztig varieties in the affine flag variety.