Crystals of Lakshmibai-Seshadri paths and extremal weight modules over quantum hyperbolic Kac-Moody algebras of rank 2

2021/06/15 by Ryuta Hiasa, Hiasa, Ryuta
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2106.07918

openalex publication_date 2021/06/15 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

Let \mathfrakg be a hyperbolic Kac-Moody algebra of rank 2, and let λ be an arbitrary integral weight. We denote by \mathbbB(λ) the crystal of all Lakshmibai-Seshadri paths of shape λ. Let V(λ) be the extremal weight module of extremal weight λ generated by the (cyclic) extremal weight vector v_λ of weight λ, and let B(λ) be the crystal basis of V(λ) with u_λ∈ B(λ) the element corresponding to v_λ. We prove that the connected component B₀(λ) of B(λ) containing u_λ is isomorphic, as a crystal, to the connected component \mathbbB₀(λ) of \mathbbB(λ) containing the straight line π_λ. Furthermore, we prove that if λ satisfies a special condition, then the crystal basis B(λ) is isomorphic, as a crystal, to the crystal \mathbbB(λ). As an application of these results, we obtain an algorithm for computing the number of elements of weight μ in B(Λ₁-Λ₂), where Λ₁, Λ₂ are the fundamental weights, in the case that \mathfrakg is symmetric.

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