Categorical braid group actions and cactus groups

Abstract

Let g be a semisimple simply-laced Lie algebra of finite type. Let C be an abelian categorical representation of the quantum group Uq(g) categorifying an integrable representation V. The Artin braid group B of g acts on Db(C) by Rickard complexes, providing a triangulated equivalenceΘw0:Db(Cμ)→Db(Cw0(μ)) where μ is a weight of V, and Θw0 is a positive lift of the longest element of the Weyl group.We prove that this equivalence is t-exact up to shift when V is isotypic, generalising a fundamental result of Chuang and Rouquier in the case g=sl2. For general V, we prove that Θw0 is a perverse equivalence with respect to a Jordan-Hölder filtration of C.Using these results we construct, from the action of B on V, an action of the cactus group on the crystal of V. This recovers the cactus group action on V defined via generalised Schützenberger involutions, and provides a new connection between categorical representation theory and crystal bases. We also use these results to give new proofs of theorems of Berenstein-Zelevinsky, Rhoades, and Stembridge regarding the action of symmetric group on the Kazhdan-Lusztig basis of its Specht modules.