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arXiv · 2109.09312

On Sch\"utzenberger modules of the cactus group

Jongmin Lim·Oded Yacobi
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Abstract

The cactus group acts on the set of standard Young tableau of a given shape by (partial) Sch\"utzenberger involutions. It is natural to extend this action to the corresponding Specht module by identifying standard Young tableau with the Kazhdan-Lusztig basis. We term these representations of the cactus group "Sch\"utzenberger modules", denoted $S^\lambda_{\mathsf{Sch}}$, and in this paper we investigate their decomposition into irreducible components. We prove that when $\lambda$ is a hook shape, the cactus group action on $S^\lambda_{\mathsf{Sch}}$ 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.

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Jongmin Lim, Oded Yacobi. 2021-09-20. On Sch\"utzenberger modules of the cactus group. https://arxiv.org/abs/2109.09312

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