双色Soergel演算与简单传递2-表示
Two-color Soergel calculus and simple transitive 2-representations
摘要：
本文在可分阶性假设下，完成了有限二面体型Soergel双模的单传递$2$ -表示的类$\mathrm{ADE}$分类.特别是，我们使用二部图和曲折代数的$\mathrm{ADE}$型给一个明确的建设的分级（非严格）版本的所有这些$2$ -表示。此外，我们给出了简单的组合标准时，两个这样的$2$ -表示是等价的，当他们的Grothendieck群产生同构表示。最后，我们还构造了一般二部图的一类无限二面角型的简单传递$2$ -表示。
英文摘要：
In this paper we complete the $\mathrm{ADE}$-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 $\mathrm{ADE}$ 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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