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.
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.
We establish automorphisms with closed formulas on quasi-split $imath$quantum groups of symmetric Kac-Moody type associated to restricted Weyl groups. The proofs are carried out in the framework of $imath$Hall algebras and reflection functors, thanks
to the $imath$Hall algebra realization of $imath$quantum groups in our previous work. Several quantum binomial identities arising along the way are established.
We discuss some aspects of the deformed W-algebras W_{q,t}[g]. In particular, we derive an explicit formula for the Kac determinant, and discuss the center when t^2 is a primitive k-th root of unity. The relation of the structure of W_{q,t}[g] to the
representation ring of the quantum affine algebra U_q(hat g), as discovered recently by Frenkel and Reshetikhin, is further elucidated in some examples.