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Gelfand–Tsetlin crystals

Part of: Lie algebras and Lie superalgebras Algebraic combinatorics

Published online by Cambridge University Press:  22 May 2025

Jonas T. Hartwig Open the ORCID record for Jonas T. Hartwig [Opens in a new window]  and
O’Neill Kingston
Jonas T. Hartwig*
Affiliation:
Department of Mathematics, Iowa State University, Ames, IA, USA
O’Neill Kingston
Affiliation:
Department of Science and Mathematics, Clarke University, Dubuque, IA, USA
*
Corresponding author: Jonas T. Hartwig; Email: jth@iastate.edu
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Abstract

We give a crystal structure on the set of Gelfand–Tsetlin patterns (GTPs), which parametrize bases for finite-dimensional irreducible representations of the general linear Lie algebra. The crystal data are given in closed form and are expressed using tropical polynomial functions of the entries of the patterns. We prove that with this crystal structure, the natural bijection between GTPs and semistandard Young tableaux is a crystal isomorphism.

Keywords

crystal graphGelfand-Tsetlin patterntropical mathematics

MSC classification

Primary: 05E10: Combinatorial aspects of representation theory
Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations 17B10: Representations, algebraic theory (weights)
Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 14
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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