Categorical actions of generalised braids for $n=3$

TL;DR

This paper extends the categorical braid group actions to a generalized braid category for n=3, providing new constructions and conditions for such actions on derived categories of flag varieties.

Contribution

It defines the generalized braid category GBr_3, establishes conditions for its categorical actions, and constructs a new action generalizing prior work by Khovanov and Thomas.

Findings

Defined the generalized braid category GBr_3

Provided sufficient conditions for skein-triangulated GBr_3 actions

Constructed a categorical action of GBr_3 on derived categories of flag cotangent bundles

Abstract

Khovanov and Thomas constructed a categorical action of the braid group $Br_n$ on the derived category $D(T^* Fl_n)$ of coherent sheaves on the cotangent bundle of the variety $Fl_n$ of the complete flags in $\mathbb{C}^n$. In this paper, we define the generalised braid category $GBr_3$, give a sufficient condition for the existence of a skein-triangulated $GBr_3$ categorical action, and construct a categorical action of $GBr_3$ on $D(T^*(Fl_3(\bar{i}))$ that generalises Khovanov and Thomas categorical braid action on $D(T^* Fl_3)$.