A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

TL;DR

This paper develops a Chevalley formula for semi-infinite flag manifolds using the quantum alcove model, proving a longstanding conjecture for quantum K-theory of flag manifolds and relating quantum Grothendieck polynomials to Schubert classes.

Contribution

It introduces a general Chevalley formula for semi-infinite flag manifolds and confirms conjectures about quantum K-theory and quantum Grothendieck polynomials in type A.

Findings

Chevalley formula for arbitrary weights in semi-infinite flag manifolds.

Proof of the Chevalley formula for anti-dominant fundamental weights in quantum K-theory of flag manifolds.

Quantum Grothendieck polynomials represent Schubert classes in quantum K-theory in type A.

Abstract

We give a Chevalley formula for an arbitrary weight for the torus-equivariant $K$-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum $K$-theory $QK_{T}(G/B)$ of an (ordinary) flag manifold $G/B$; this has been a longstanding conjecture about the multiplicative structure of $QK_{T}(G/B)$. In type $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum $K$-theory $QK(SL_{n}/B)$; we also obtain very explicit information about the coefficients in the respective Chevalley formula.