On Soergel bimodules

Noriyuki Abe

Published 2019-01-08Version 1

For a Coxeter system and its representation $V$, 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 which 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.