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Polyhedral realizations for B(∞) and extended Young diagrams, Young walls of type \rm A⁽¹⁾_n-1, \rm C⁽¹⁾_n-1, \rm A⁽²⁾_2n-2, \rm D⁽²⁾_n

2021/10/27 by Yuki Kanakubo, Kanakubo, Yuki
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2110.14140

openalex publication_date 2021/10/27 · openalex created_date 2021/11/08 · openalex updated_date 2026/07/28

Abstract

The crystal bases are quite useful combinatorial tools to study the representations of quantized universal enveloping algebras U_q(\mathfrakg). The polyhedral realization for B(∞) is a combinatorial description of the crystal base, which is defined as an image of embedding Ψ_ι:B(∞)\hookrightarrow ℤ^∞_ι, where ι is an infinite sequence of indices and ℤ^∞_ι is an infinite ℤ-lattice with a crystal structure associated with ι. It is a natural problem to find an explicit form of the polyhedral realization \rm Im(Ψ_ι). In this article, supposing that \mathfrakg is of affine type \rm A⁽¹⁾_n-1, \rm C⁽¹⁾_n-1, \rm A⁽²⁾_2n-2 or \rm D⁽²⁾_n and ι satisfies the condition of `adaptedness', we describe \rm Im(Ψ_ι) by using several combinatorial objects such as extended Young diagrams and Young walls.

Polyhedral realizations for B(∞) and extended Young diagrams, Young walls of type \rm A(1)n-1, \rm C(1)n-1, \rm A(2)2n-2, \rm D(2)n

Abstract

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Polyhedral realizations for B(∞) and extended Young diagrams, Young walls of type \rm A⁽¹⁾_n-1, \rm C⁽¹⁾_n-1, \rm A⁽²⁾_2n-2, \rm D⁽²⁾_n