Abstract

We construct an injective weight-preserving map (called the forgetful map) from the set of all admissible subsets in the quantum alcove model associated to an arbitrary weight. The image of this forgetful map can be explicitly described by introducing the notion of “interpolated quantum Lakshmibai-Seshadri (QLS for short) paths”, which can be thought of as a generalization of quantum Lakshmibai-Seshadri paths. As an application, we reformulate, in terms of interpolated QLS paths, an identity of Chevalley type for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra, which is a representation-theoretic analog of the Chevalley formula for the torus-equivariant K-group of a semi-infinite flag manifold.

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 models associated to 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 for 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.

Abstract

We provide identities of inverse Chevalley type for the graded characters of level-zero Demazure submodules of extremal weight modules over a quantum affine algebra of type C. These identities express the product $$e^{\mu } \text {gch} ~V_{x}^{-}(\lambda )$$ of the (one-dimensional) character $$e^{\mu }$$, where $$\mu $$ is a (not necessarily dominant) minuscule weight, with the graded character gch$$V_{x}^{-}(\lambda )$$ of the level-zero Demazure submodule $$V_{x}^{-}(\lambda )$$ over the quantum affine algebra $$U_{\textsf{q } }(\mathfrak {g}_{\textrm{af } })$$ as an explicit finite linear combination of the graded characters of level-zero Demazure submodules. These identities immediately imply the corresponding inverse Chevalley formulas for the torus-equivariant K-group of the semi-infinite flag manifold $$\textbf{Q}_{G}$$ associated to a connected, simply-connected and simple algebraic group G of type C. Also, we derive cancellation-free identities from the identities above of inverse Chevalley type in the case that $$\mu $$ is a standard basis element $${\varepsilon }_{k}$$ in the weight lattice P of G.