Crystal structure on Gelfand-Tsetlin-Zhelobenko patterns

Date
2020-01-01
Authors
Kingston, O'Neill
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Mathematics
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Abstract

We begin by presenting the crystal structure of finite-dimensional irreducible representations of the special linear Lie algebra in terms of Gelfand-Zeitlin patterns. We then define a crystal structure using the set of symplectic Zhelobenko patterns, parametrizing bases for finite-dimensional irreducible representations of sp4. This is obtained by a bijection with Kashiwara-Nakashima tableaux and the symplectic jeu de taquin of Sheats and Lecouvey. We offer some conjectures on the generalization of this structure to rank n as well as a bijection and crystal structure in certain special cases.

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Algebra, Combinatorics, Crystal basis, Gelfand-Tsetlin pattern, Representation theory, Zhelobenko pattern
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