---
abstract: 'In all finite Coxeter types but $I_2(12)$, $I_2(18)$ and $I_2(30)$, we classify simple transitive $2$-representations 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.'
author:
- |
    Tobias Kildetoft, Marco Mackaay,\
    Volodymyr Mazorchuk and Jakob Zimmermann
title: |
    Simple transitive $2$-representations\
    of small quotients of Soergel bimodules
---

Introduction and description of the results {#s1}
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Classical representation theory makes a significant emphasis on problems and techniques related to the classification of various classes of representations. The recent “upgrade” of classical representation theory, known as [*$2$-representation theory*]{}, has its abstract origins in [@BFK; @CR; @KL; @Ro]. The first classification result in $2$-representation theory was obtained in [@CR Subsection 5.4.2] where certain “minimal” $2$-representations of the Chuang-Rouquier $2$-analogue of $U(\mathfrak{sl}_2)$ were classified.

The series [@MM1; @MM2; @MM3; @MM4; @MM5; @MM6] of papers initiated a systematic study of the 