New structure on the quantum alcove model with applications to representation theory and Schubert calculus

TL;DR

This paper generalizes quantum Yang-Baxter moves within the quantum alcove model to arbitrary weights, ensuring invariance of key statistics and linking to representation theory and Schubert calculus.

Contribution

It extends quantum Yang-Baxter moves to all weights in the quantum alcove model, preserving important statistics and applications in representation theory.

Findings

Generated functions are independent of reduced alcove path choices.

Established identities for graded characters of Demazure modules.

Connected combinatorial models to representation-theoretic formulas.

Abstract

The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart-Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang-Baxter moves, which biject the objects of the model associated with two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang-Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula in the equivariant $K$-group of semi-infinite flag manifolds. The generalized quantum Yang-Baxter moves give rise to a "sijection" (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula. Other applications and some open problems involving "signed crystals" are discussed.