Hostname: page-component-cb9f654ff-plnhv Total loading time: 0.001 Render date: 2025-08-25T16:22:02.028Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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)

Information

Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 14
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bump, D. and Schilling, A., Crystal bases: representations and combinatorics, (World Scientific, New Jersey, 2017).CrossRefGoogle Scholar
Futorny, V., Molev, A. and Ovsienko, S., Gelfand-Tsetlin bases for representations of finite W-algebras and shifted Yangians, in Lie theory and its applications in physics, Proceedings of the VII International Workshop H.-D. Doebner and V.K. Dobrev, Editors), vol. VII, (Heron Press, Varna, Bulgaria, Sofia, 2008), 352–363. Preprint version arXiv:0711.0552v1 [math.RT].Google Scholar
Fulton, W., Young Tableaux: with applications to representation theory and geometry, London Mathematical Society Student Texts vol. 35, (Cambridge University Press, Cambridge, 1997).Google Scholar
Futorny, V. and Ovsienko, S., Galois orders in skew monoid rings, J. Algebra 324(4) (2010), 598–630.CrossRefGoogle Scholar
Gelfand, I. M. and Tsetlin, M. L., Finite-dimensional representations of the group of unimodular matrices, in Dokl. Akad. Nauk SSSR 71 (1950), 825--828 (Russian). English transl “Collected papers” (Gelfand, I. M., editor) vol. II (Springer-Verlag, Berlin, 1988), 653–656.Google Scholar
Gelfand, I. M. and Tsetlin, M. L., Finite-dimensional representations of groups of orthogonal matrices, in Dokl. Akad. Nauk SSSR 71 (1950), 1017--1020 (Russian). English transl. “Collected papers” (Gelfand, I. M., Editor), vol. II (Springer-Verlag, Berlin, 1988), 657–661.Google Scholar
Hong, J. and Kang, S.-J., Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, (American Mathematical Society, Providence, RI, 2002).CrossRefGoogle Scholar
Kashiwara, M., Crystalizing the q-analog of universal enveloping algebras, Comm. Math. Phys 133 (1990), 249–260.CrossRefGoogle Scholar
Kashiwara, M., On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516.CrossRefGoogle Scholar
Kashiwara, M. and Nakashima, T., Crystal graphs for representations of the q-analogue of classical lie algebras, J. Algebra 165(2) (1994), 295–345.CrossRefGoogle Scholar
Lusztig, G., Canonical bases arising from quantized enveloping algebras, ii, Prog Theor Phys Supp 102 (1990), 175–201.CrossRefGoogle Scholar
Molev, A., Yangians and classical Lie algebras, Mathematical Surveys and Monographs, vol. 143, (American Mathematical Society, Providence, RI, 2007).CrossRefGoogle Scholar
Sheats, J. T., A symplectic jeu de taquin bijection between the tableaux of King and of De Concini, Trans. Am. Math. Soc. 351(09) (1999), 3569–3608.CrossRefGoogle Scholar
Zelobenko, D. P., Compact Lie groups and their representations, Translations of Mathematical Monographs, vol. 40, (AMS, Providence RI, 1973).CrossRefGoogle Scholar
Watanabe, H. and Yamamura, K., Alcove paths and Gelfand-Tsetlin patterns, Ann Comb. 25(3) (2021), 645–676.CrossRefGoogle Scholar