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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-Holder 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 Schutzenberger 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.
The crystals for a finite-dimensional complex reductive Lie algebra $mathfrak{g}$ encode the structure of its representations, yet can also reveal surprising new structure of their own. We study the cactus group $C_{mathfrak{g}}$, constructed using t
We construct a 2-representation categorifying the symmetric Howe representation of $mathfrak{gl}_m$ using a deformation of an algebra introduced by Webster. As a consequence, we obtain a categorical braid group action taking values in a homotopy category.
We study a presentation of Khovanov - Lauda - Rouquiers candidate $2$-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to comp
The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of $U_q(mathfrak{sl}_2)$ are considered. It is shown that these intermediate Casimirs are related by a co
We develop a theory of localization for braid group representations associated with objects in braided fusion categories and, more generally, to Yang-Baxter operators in monoidal categories. The essential problem is to determine when a family of brai