A Nakayama result for the quantum K theory of homogeneous spaces

TL;DR

This paper proves that the relations in the quantum K theory of homogeneous spaces are generated by quantizations of classical relations, extending classical results to the quantum setting and illustrating with partial flag manifolds.

Contribution

It extends a classical Nakayama-type result to quantum K theory of homogeneous spaces, providing a new understanding of the structure of quantum K rings.

Findings

Relations in quantum K rings are generated by quantizations of classical relations.

The technique is illustrated on partial flag manifolds using quantum K Whitney relations.

The approach generalizes a known result from quantum cohomology to quantum K theory.

Abstract

We prove that the ideal of relations in the (equivariant) quantum K ring of a homogeneous space is generated by quantizations of each of the generators of the ideal in the classical (equivariant) K ring. This extends to quantum K theory a result of Siebert and Tian in quantum cohomology. We illustrate this technique in the case of the quantum K ring of partial flag manifolds, using a set of quantum K Whitney relations conjectured by the authors, and recently proved by Huq-Kuruvilla.