Maximal Newton Points and the Quantum Bruhat Graph

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.

Affine Deligne-Lusztig Varieties and Quantum Bruhat Graph

“…An explicit description of ν([b w ]) is given by Milićević in [21,Theorem 3.2] for elements w ∈ W that are suitably far from walls of any Weyl chamber. To quantify the later condition, let us define for any dominant coweight λ its depth as depth(λ) = min{ α, λ : α is a simple root}.…”

“…1.2.1. An uniform bound is given in [21,Corollary 3.3] in the quasi-simple case, which says that the description of ν([b w ]) provided in the article is valid whenever the following depth hypothesis is satisfied on λ:…”

Closure relations of Newton strata in Iwahori double cosets

“…The following example is a particularly where s i denote the simple reflections in S. In this example, the constant M for the regularity assumption on μ equals 74 and is thus satisfied. Then the pair (x, s) satisfies the requirements of Proposition 4.2 and the superregularity hypothesis of [13] and [14].…”

“…In order to find a pair (x, s) as in Proposition 4.2, we use the mathematics software system. SageMath Here, to compute generic Newton points and to check cordiality we use the results of [13,Thm. 3.2] and [14,Prop.…”

“…4.2], respectively. The results of [13] offer a very useful description of the generic Newton point for split groups G and any x ∈ W of the form x = vt μ w where α, μ > M for all simple roots α where M is a constant depending on G, v and w. Under the same assumption, [14] gives a criterion to check if x is cordial in terms of some paths in the quantum Bruhat graph. We include the SageMath code in the appendix.…”