Two-color Soergel calculus and simple transitive 2-representations
In this paper we complete the ADE-like
of simple transitive $2$-representations
of Soergel bimodules
in finite dihedral type, under the assumption of gradeability.
In particular, we use bipartite
graphs and zigzag algebras of ADE type to give an explicit
construction of a graded (non-strict)
version of all these $2$-representations.
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.
$2$-representation theory, categorification, Soergel bimodule, Kazhdan--Lusztig theory, Hecke algebras for dihedral groups, zigzag algebra
20C08 - Hecke algebras and their representations
17B10 - Representations, algebraic theory (weights)
18D05 - Double categories, $2$-categories, bicategories and generalizations
18D10 - Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]
20F55 - Reflection and Coxeter groups [See also 22E40, 51F15]