New structure on the quantum alcove model with applications to representation theory and Schubert calculus
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 LenartLubovsky 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 YangBaxter 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 YangBaxter 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 semiinfinite flag manifolds. The generalized quantum YangBaxter 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 levelzero extremal weight modules over a quantum affine algebra, which can be thought of as a representationtheoretic analogue of the mentioned Chevalley formula. Other applications and some open problems involving "signed crystals" are discussed.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 arXiv:
 arXiv:2105.02546
 Bibcode:
 2021arXiv210502546K
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Quantum Algebra;
 Mathematics  Representation Theory;
 Primary 05E10;
 Secondary 14N15;
 14M15
 EPrint:
 49 pages, the title of this article has been changed