Positivity of minuscule quantum K-theory

TL;DR

This paper establishes sign patterns and interval properties for quantum K-theory of minuscule flag varieties, providing new geometric and combinatorial insights and a uniform proof of the quantum-to-classical correspondence.

Contribution

It introduces new geometric descriptions, computes cohomology groups, and proves a uniform quantum-to-classical theorem for minuscule flag varieties.

Findings

Signs of structure constants alternate with codimension

Powers of q form an integer interval in products

Quantum invariants relate to classical intersection numbers

Abstract

We prove that the Schubert structure constants of the quantum $K$-theory ring of any minuscule flag variety or quadric hypersurface have signs that alternate with codimension. We also prove that the powers of the deformation parameter $q$ that occur in the product of two Schubert classes in the quantum cohomology or quantum $K$-theory ring of a cominuscule flag variety form an integer interval. Our proofs are based on several new results, including an explicit description of the most general non-empty intersection of two Schubert varieties in an arbitrary flag manifold, and a computation of the cohomology groups of any negative line bundle restricted to a Richardson variety in a cominuscule flag variety. We also give a type-uniform proof of the quantum-to-classical theorem, which asserts that the (3-point, genus 0) Gromov-Witten invariants of any cominuscule flag variety are classical triple-intersection numbers on an associated flag variety. Finally, we prove several new results about the geometry and combinatorics related to this theorem.