On Schützenberger modules of the cactus group

The cactus group acts on the set of standard Young tableau of a given shape by (partial) Schützenberger 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 "Schützenberger modules", denoted $S^λ_{\mathsf{Sch}}$, 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 $S^λ_{\mathsf{Sch}}$ 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.
v2: 15 pages, final version to appear in Algebraic Combinatorics