**Data**

- Title: Two-color Soergel calculus and simple transitive $2$-representations
- Authors: Marco Mackaay and Daniel Tubbenhauer
- Status: Canad. J. Math. 71 (2019), no. 6, 1523-1566. Last update: Mon, 23 Apr 2018 14:56:01 GMT
- ArXiv link: https://arxiv.org/abs/1609.00962
- ArXiv version = 0.99 published version
- LaTex Beamer presentation: Slides1, Slides2, Slides3, Slides4, Slides5, Slides6, Slides7, Slides8

**Abstract**

Moreover, we give simple combinatorial criteria for when two such $2$-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations.

Finally, our construction also gives a large class of simple transitive $2$-representations in infinite dihedral type for general bipartite graphs.

**A few extra words**

Basically the same question arises in $2$-representation theory, where the actions of algebras on vector spaces are replaced by functorial actions of $2$-categories on certain $2$-categories. Examples are $2$-representations of the $2$-categories which categorify representations of quantum groups, due to (Chuang-)Rouquier and Khovanov-Lauda, and $2$-representations of the $2$-category of Soergel bimodules, which categorify representations of Hecke algebras.

An appropriate categorical analogue of the simple representations of finite-dimensional algebras are the so-called simple transitive $2$-representations (of finitary $2$-categories). The problem of their classification is very hard in general and not well understood.

In this paper, we construct all (graded) simple transitive $2$-representations (of the small quotient) of the Soergel bimodules of dihedral type (including the infinite dihedral type) by using the so-called two-color Soergel calculus.

Our construction is completely explicit and emphasizes the usage of bipartite graphs which we use to cook up algebras and the $2$-categories of their projective endofunctors, on which the two-color Soergel calculus acts.

The main results are then that we can show that two simple transitive $2$-representations are equivalent if and only if the corresponding bipartite graphs are isomorphic (as bipartite graphs), and all such $2$-representations arise in this way.

Finally, we also determine when simple transitive $2$-representations decategorify to isomorphic representations of the corresponding Hecke algebra, using a purely graph-theoretic property.

Hereby we obtain examples of inequivalent simple transitive $2$-representations of the same $2$-category which decategorify to isomorphic representations, e.g.: The to the bipartite graphs $G$ and $G^{\prime}$ associated $2$-representations are inequivalent, but categorify the same module of the associated dihedral group.