ALGEBRAIC COMBINATORICS

no. 3
On Schützenberger modules of the cactus group
Lim, Jongmin1; Yacobi, Oded2
1 J. Lim: School of Mathematics and Statistics University of Sydney Australia
2 O. Yacobi: School of Mathematics and Statistics University of Sydney Australia
Algebraic Combinatorics, Volume 6 (2023) no. 3, pp. 773-788.

The cactus group acts on the set of standard Young tableaux of a given shape by (partial) Schützenberger involutions. It is natural to extend this action to the corresponding Specht module by identifying standard Young tableaux with the Kazhdan–Lusztig basis. We term these representations of the cactus group “Schützenberger modules”, denoted S Sch λ , and in this paper we investigate their decomposition into irreducible components. We prove that when λ is a hook shape, the cactus group action on S Sch λ factors through S n-1 and the resulting multiplicities are given by Kostka coefficients. Our proof relies on results of Berenstein and Kirillov and Chmutov, Glick, and Pylyavskyy.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.283
Classification: 05E10, 05E18, 20C30
Keywords: Kazhdan-Lusztig basis, crystal, cactus group, Young tableaux, Kotska numbers
Lim, Jongmin 1; Yacobi, Oded 2

1 J. Lim: School of Mathematics and Statistics University of Sydney Australia
2 O. Yacobi: School of Mathematics and Statistics University of Sydney Australia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Lim, Jongmin; Yacobi, Oded. On Schützenberger modules of the cactus group. Algebraic Combinatorics, Volume 6 (2023) no. 3, pp. 773-788. doi : 10.5802/alco.283. https://alco.centre-mersenne.org/articles/10.5802/alco.283/

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