  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-H\"older 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 Sch\"utzenberger 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.
