Categorical braid group actions and cactus groups

Abstract

Let $\mathfrak{g}$ be a semisimple simply-laced Lie algebra of finite type. Let $\mathcal{C}$ be an abelian categorical representation of the quantum group $U_q(\mathfrak{g})$ categorifying an integrable representation $V$. The Artin braid group $B$ of $\mathfrak{g}$ acts on $D^b(\mathcal{C})$ by Rickard complexes, providing a triangulated equivalence ${\Theta }_{w_0}:D^b({\mathcal{C}}_{\mu })\to D^b({\mathcal{C}}_{w_0(\mu )})$, where $\mu$ is a weight of $V$ and ${\Theta }_{w_0}$ 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 $\mathfrak{g}={\mathfrak{sl}}_2$. For general $V$, we prove that ${\Theta }_{w_0}$ is a perverse equivalence with respect to a Jordan-Hölder filtration of $\mathcal{C}$. 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.

v3: 26 pages, fixed a mistake in Theorem 3.6. v2: 26 pages, revision based on referee comments. v1: 23 pages