Polyhedral realizations for crystal bases and Young walls of classical affine types

2024/03/02 by Yuki Kanakubo, Kanakubo, Yuki
Mathematics · #Finite Group Theory Research #Mathematics and Applications #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2403.01190

Abstract

For affine Lie algebra \mathfrakg of type A⁽¹⁾_n-1, B⁽¹⁾_n-1, C⁽¹⁾_n-1, D⁽¹⁾_n-1, A⁽²⁾_2n-2, A⁽²⁾_2n-3 or D⁽²⁾_n, let B(λ) and B(∞) be the crystal bases of integrable highest weight representation V(λ) and negative part U_q^-(\mathfrakg) of quantum group U_q(\mathfrakg). We consider the polyhedral realizations of crystal bases, which realize B(λ) and B(∞) as sets of integer points of some polytopes and cones in ℝ^∞. It is a natural problem to find explicit forms of the polytopes and cones. In this paper, we introduce pairs of truncated walls, which are defined as modifications of level 2-Young walls and describe inequalities defining the polytopes and cones in terms of level 1-proper Young walls and pairs of truncated walls. As an application, we also give combinatorial descriptions of ε_k^*-functions on B(∞) in terms of Young walls and truncated walls.

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