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 $K$-theoretic divisor axiom for the flag manifold in the quasi-minuscule case under some assumptions. Namely, we remove the cancellations from the quantum $K$-theoretic divisor axiom due to Lenart-Naito-Sagaki-Xu in the case where the fundamental weight corresponding to the divisor class is quasi-minuscule.

In Version 2, there was a gap in the proof of the main theorem; equation (3.24) in Subcase 3.2 in the proof of Theorem 3.4 was incorrect. In this version, we proved Theorem 3.7 (= the main part of Theorem 3.4 in the old version) under the assumption that Conjecture 3.5 is true. We proved in Corollary 3.11 that the same formula as in Theorem 3.7 holds if the effective degree is sufficiently large