2018/03/02 by Satoshi Naito, Naito, Satoshi, Fumihiko Nomoto +3 · 1 citation
Mathematics · #14M15 #33D52 #81R10 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Primary 17B37 #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Secondary 14N15
paper · pdf · doi:10.48550/arxiv.1803.01727
openalex publication_date 2018/03/02 · openalex created_date 2018/03/29 · openalex updated_date 2026/07/28
Let λ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let QLS(λ) denote the quantum Lakshmibai-Seshadri (QLS) paths of shape λ. For an element w of a finite Weyl group W, the specializations at t = 0 and t = ∞ of the nonsymmetric Macdonald polynomial Ew λ(q, t) are explicitly described in terms of QLS paths of shape λ and the degree function defined on them. Also, for (level-zero) dominant integral weights λ, μ, we have an isomorphism Θ: QLS(λ+ μ) → QLS(λ) ⊗ QLS(μ) of crystals. In this paper, we study the behavior of the degree function under the isomorphism Θ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.