Maximal Newton Points and the Quantum Bruhat Graph

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