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Cancellation-free version of the quantum $K$-theoretic divisor axiom for the flag manifold in the quasi-minuscule case

Ryo Kato, Daisuke Sagaki

Abstract

We prove a cancellation-free version of the quantum $K$-theoretic divisor axiom for the flag manifold in the quasi-minuscule case. Namely, we remove the cancellations from the quantum $K$-theoretic divisor axiom due to Lenart-Naito-Sagaki-Xu in the case where the fundametal weight corresponding to the divisor class is quasi-minuscule.

Cancellation-free version of the quantum $K$-theoretic divisor axiom for the flag manifold in the quasi-minuscule case

Abstract

We prove a cancellation-free version of the quantum -theoretic divisor axiom for the flag manifold in the quasi-minuscule case. Namely, we remove the cancellations from the quantum -theoretic divisor axiom due to Lenart-Naito-Sagaki-Xu in the case where the fundametal weight corresponding to the divisor class is quasi-minuscule.

Paper Structure

This paper contains 17 sections, 14 theorems, 100 equations.

Table of Contents

  1. Introduction.
  2. Acknowledgments.
  3. Preliminaries.
  4. Notation for root systems.
  5. The quantum Bruhat graph.
  6. Quantum Lakshmibai-Seshadri paths.
  7. $K$-theoretic Gromov-Witten invariants.
  8. Main theorem.
  9. Minuscule case.
  10. Quasi-minuscule case.
  11. Proof of Theorem \ref{['thm:main']}.
  12. Case 1.
  13. Case 2.
  14. Case 3.
  15. Subcase 3.1.
  16. ...and 2 more sections

Key Result

Theorem 2

Let $i \in I$ be such that $\varpi_{i}$ is quasi-minuscule. Let $w,x \in W$, and $d \in Q^{\vee,+}$. We have where $x_{\min}:=\min(xW_{J},\le_{w})$ and $\mathop{\mathrm{wt}}\nolimits(\eta) = \frac{1}{2}w\varpi_{i}+\frac{1}{2}x\varpi_{i} \in \varpi_{i} - Q^{+}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (33)

  • Definition 1
  • Theorem 2: $=$ Corollary \ref{['cor:main']}
  • Remark 3: Positivity
  • Definition 2.1
  • Proposition 2.2: LNSSS1
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5: LNSSS1
  • proof
  • ...and 23 more