**Data**

- Title: Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types
- Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer and Xiaoting Zhang
- Status: Preprint. Last update: Fri, 8 Jan 2021 10:29:03 UTC
- ArXiv link: https://arxiv.org/abs/1906.11468
- LaTex Beamer presentation: Slides1, Slides2a (joint with Marco Mackaay Slides2b and Slides2c), Slides3a and companion Slides3b, Slides4 (joint with Vanessa Miemietz Notes), Slides5

**Abstract**

**A few extra words**

First of all, the Classification Problem for $\mathcal{S}$ can be reduced to that of certain subquotients $\mathcal{S}_{\mathcal{H}}$. To do that, one uses the cell structure of $\mathcal{S}$, which is the categorical analog of the Kazhdan-Lusztig cell structure of Hecke algebras. By apex reduction and $\mathcal{H}$-reduction, it suffices to solve the classification problem for some $\mathcal{S}_{\mathcal{H}}$ instead of $\mathcal{S}$.

By the results in this paper, the benefits turn out to be even bigger. Based on Elias and Williamson's results, there exists a fusion bicategory $\mathcal{A}_{\mathcal{H}}$, which categorifies the corresponding asymptotic Hecke algebra. The main insight of this paper is that these asymptotic bicategories completely determine the (graded) simple transitive $2$-representations of $\mathcal{S}$. To be precise, our main result is the existence of a biequivalence of categories of simple transitive $2$-representations of $\mathcal{S}_{\mathcal{H}}$ and $\mathcal{A}_{\mathcal{H}}$.

Since $\mathcal{A}_{\mathcal{H}}$ is semisimple and well-understood in the great majority of cases, the upshot is that the classification boils down to certain computational data; which in almost all cases is enough to parameterize the simple transitive $2$-representations of $\mathcal{S}$. Here is an example how such computational data might look like.

This is the cell structure of Coxeter type $H_4$. In this case, which we barely understand, there would be still work to be done - even assuming our conjecture to be true. For example, we do not know what the asymptotic $2$-categories for the big cell are - we only know its numerical data (e.g. Fusion graph, Perron-Frobenius dimensions as illustrated). However, assuming our conjecture, types $H_3$ and $H_4$ are the only types which would remain open. That is, in all other types our conjecture would give a full classification of $2$-simples.