On Soergel bimodules

Abstract: For 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 even when $V$ is not reflection faithful and they 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 less assumptions on $V$. In this paper, we give a "bimodule theoretic" definition of the category of Elias-Williamson and reprove the categorification theorem.