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In the previous paper, we defined a new category which categorifies the Hecke algebra. This is a generalization of the theory of Soergel bimodules. To prove theorems, the existences of certain homomorphisms between Bott–Samelson bimodules are assumed. In this paper, we prove this assumption. We only assume the vanishing of certain two-colored quantum binomial coefficients.

In this paper, we complete the ADE-like classification 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.

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.

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A HOMOMORPHISM BETWEEN BOTT–SAMELSON BIMODULES

6 - An Introduction to Diagrammatic Soergel Bimodules

Summary

Two-color Soergel Calculus and Simple Transitive 2-representations

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3 results

A HOMOMORPHISM BETWEEN BOTT–SAMELSON BIMODULES

6 - An Introduction to Diagrammatic Soergel Bimodules

Summary

Two-color Soergel Calculus and Simple Transitive 2-representations