Categorical braid group actions and cactus groups

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