Pré-Publication, Document De Travail Année : 2022

Positivity of minuscule quantum K-theory

Anders S. Buch
  • Fonction : Auteur
Pierre-Emmanuel Chaput
Leonardo C. Mihalcea
  • Fonction : Auteur

Résumé

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.

Dates et versions

hal-04480428 , version 1 (27-02-2024)

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Anders S. Buch, Pierre-Emmanuel Chaput, Leonardo C. Mihalcea, Nicolas Perrin. Positivity of minuscule quantum K-theory. 2022. ⟨hal-04480428⟩
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