On Schützenberger modules of the cactus group

Abstract

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 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 Sch factors through Sn−1 and the resulting multiplicities are given by Kostka coefficients. Our proof relies on results of Berenstein and Kirillov and Chmutov, Glick, and Pylyavskyy.