**Data**

- Title: $2$-representations of Soergel bimodules
- Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer and Xiaoting Zhang
- Status: Preprint. Last update: Thu, 27 Jun 2019 07:26:51 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**

Along the way we also show several results and provide examples which are interesting in their own right, e.g. we show that Duflo involutions have a Frobenius structure (in a certain quotient) and give an example of a left cell for which the underlying algebra of the cell $2$-representation is not symmetric.

**A few extra words**

We propose a general approach for attacking the classification problem of graded simple transitive $2$-representations of the $2$-category of Soergel bimodules for an arbitrary finite Coxeter group. Our approach is based on a connection between the $2$-category of Soergel bimodules and the associated asymptotic bicategory $\mathcal{A}$ which categorifies the asymptotic Hecke algebra, a (multi)fusion algebra. The asymptotic bicategory is no longer graded, but has the advantage of being semisimple, and even (multi)fusion. We formulate a very precise conjecture relating the $2$-representation theories of $\mathcal{S}$ and $\mathcal{A}$. What is of crucial importance is that $\mathcal{A}$ is explicitly known and rather simple in all but a handful of cases, and so is the classification of its simple transitive $2$-representations. Thus, our conjecture, if true, would reduce the classification of simple transitive $2$-representations of Soergel bimodules to a much easier problem. For example, in classical Weyl types the classification would boil down to computing the Schur multipliers of $(\mathbb{Z}/2\mathbb{Z})^k$, which is of course well-known. Another example, in dihedral types $\mathcal{A}$ is (related to) the semisimplified quotient of quantum $\mathfrak{sl}_2$-modules. In this case, our conjecture holds and yields the above mentioned ADE classification via the work of Kirillov-Ostrik.

In all other cases the classification would also boil down to certain computational data; which in almost all cases would be 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.