Categorical actions of generalised braids for $n=3$

Khovanov and Thomas constructed a categorical action of the braid group $B{r}_n$ on the derived category $D(T^{\ast }F{l}_n)$ of coherent sheaves on the cotangent bundle of the variety $F{l}_n$ of the complete flags in ${\mathbb{C}}^n$. In this paper, we define the generalised braid category $GB{r}_3$, give a sufficient condition for the existence of a skein-triangulated $GB{r}_3$ categorical action, and construct a categorical action of $GB{r}_3$ on $D(T^{\ast }(F{l}_3(\bar{i}))$ that generalises Khovanov and Thomas categorical braid action on $D(T^{\ast }F{l}_3)$.