A bimodule description of the Hecke category

Abstract

AbstractFor a Coxeter system and a representation$V$of this Coxeter system, Soergel defined a category which is now called the category of Soergel bimodules and proved that this gives a categorification of the Hecke algebra when$V$is reflection faithful. Elias and Williamson defined another category when$V$is not reflection faithful and proved that this category is equivalent to the category of Soergel bimodules when$V$is reflection faithful. Moreover, they proved the categorification theorem for their category with fewer assumptions on$V$. In this paper, we give a bimodule description of the Elias–Williamson category and re-prove the categorification theorem.