Data

  1. Title: $2$-representations of Soergel bimodules
  2. Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer and Xiaoting Zhang
  3. Status: Preprint. Last update: Thu, 27 Jun 2019 07:26:51 UTC
  4. ArXiv link: https://arxiv.org/abs/1906.11468
  5. LaTex Beamer presentation: Slides1, Slides2

Abstract

In this paper we study the graded $2$-representation theory of Soergel bimodules for a finite Coxeter group. We establish a precise connection between the graded $2$-representation theory of this non-semisimple $2$-category and the $2$-representation theory of the associated semisimple asymptotic bicategory. This allows us to formulate a conjectural classification of graded simple transitive $2$-representations of Soergel bimodules, which we prove under certain assumptions.
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

Classification problems are among the most important basic problems in mathematics. For example, classifying simple representations of Hecke algebras has played an important role in modern representation theory. The present paper is motivated by the problem of classifying graded simple transitive $2$-representations of the $2$-category $\mathcal{S}$ of Soergel bimodules associated to a finite Coxeter group, which one can see as a categorification of the classification problem for Hecke algebras. We do not give a complete answer, but we propose a precise conjecture for this classification and prove this conjecture under a technical assumption.
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.