Trihedral Soergel bimodules
Volume 248 / 2020
Abstract
The 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.
