Simple transitive 2-representations of small quotients of Soergel bimodules

TL;DR

This paper classifies simple transitive 2-representations of certain quotients of Soergel bimodules across most finite Coxeter types, revealing new representations in specific dihedral cases.

Contribution

It provides a classification of simple transitive 2-representations for quotients of Soergel bimodules in all finite Coxeter types except three, highlighting new representations in dihedral types.

Findings

Most simple transitive 2-representations are cell 2-representations.

In dihedral types $I_2(2k)$ with $k extgreater 2$, there exist non-cell simple transitive 2-representations.

Abstract

In all finite Coxeter types but $I_2(12)$, $I_2(18)$ and $I_2(30)$, we classify simple transitive $2$-rep\-re\-sen\-ta\-ti\-ons for the quotient of the $2$-category of Soergel bimodules over the coinvariant algebra which is associated to the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive $2$-representations are exhausted by cell $2$-representations. However, in Coxeter types $I_2(2k)$, where $k\geq 3$, there exist simple transitive $2$-representations which are not equivalent to cell $2$-representations.