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

Cristian Lenart, Satoshi Naito, Daisuke Sagaki

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.

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

Abstract

We give a Chevalley formula for an arbitrary weight for the torus-equivariant -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 -theory of an (ordinary) flag manifold ; this has been a longstanding conjecture about the multiplicative structure of . In type , we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum -theory ; we also obtain very explicit information about the coefficients in the respective Chevalley formula.

Paper Structure

This paper contains 22 sections, 43 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Background on the quantum Bruhat graph and its parabolic version
  3. Root systems
  4. The quantum Bruhat graph
  5. Additional results
  6. Background on the combinatorial models
  7. Quantum LS paths
  8. The quantum alcove model
  9. Galleries
  10. Additional shellability results
  11. Chevalley formulas for semi-infinite flag manifolds
  12. Chevalley formula for dominant weights
  13. Chevalley formula for anti-dominant weights
  14. Chevalley formula for arbitrary weights
  15. Proof of Theorem \ref{['genchev']}
  16. ...and 7 more sections

Key Result

Proposition 3

Let $v,\,w \in W^{J}$. If ${\bf p}$ and ${\bf q}$ are shortest directed paths in $\mathop{\rm QB}\nolimits(W^{J})$ from $v$ to $w$, then $\mathop{\mathrm{wt}}\nolimits^{J}({\bf p}) \equiv \mathop{\mathrm{wt}}\nolimits^{J}({\bf q})$ modulo $Q_{J}^{\vee}$. In particular, if $J=\emptyset$, then $\matho

Theorems & Definitions (90)

  • Definition 1
  • Remark 2: see lnsumk1
  • Proposition 3
  • Theorem 4: BFP
  • Lemma 5: lnsumk2
  • Definition 6
  • Lemma 7: lnsumk2
  • Lemma 8: lnsumk2
  • Proposition 9: lnsumk1
  • Proposition 10: nospcf
  • ...and 80 more