Mackaay, MarcoMazorchuk, VolodymyrMiemietz, VanessaTubbenhauer, Daniel2020-09-232020-09-232018-04-240016-2736http://hdl.handle.net/10400.1/14737The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum $\mathfrak{sl}_2$ representation category. It also establishes a precise relation between the simple transitive $2$-representations of both monoidal categories, which are indexed by bicolored $\mathsf{ADE}$ Dynkin diagrams. Using the quantum Satake correspondence between affine $\mathsf{A}_{2}$ Soergel bimodules and the semisimple quotient of the quantum $\mathfrak{sl}_3$ representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke algebras and Soergel bimodules. These have their own Kazhdan-Lusztig combinatorics, simple transitive $2$-representations corresponding to tricolored generalized $\mathsf{ADE}$ Dynkin diagrams.eng2-representation theoryQuantum groups and their fusion categoriesHecke algebrasSoergel bimodulesZigzag algebrasjournal article10.4064/fm566-3-2019