A diagrammatic categorification of the $q$-Schur algebra

Abstract

In this paper we categorify the q-Schur algebra \mathbf{S}_q(n,d) as a quotient of Khovanov and Lauda’s diagrammatic 2-category \mathcal{U}(\mathfrak{sl}_n) (Khovanov and Lauda 2010). We also show that our 2-category contains Soergel’s (1992) monoidal category of bimodules of type A , which categorifies the Hecke algebra H_q(d) , as a full sub-2-category if d\leq n . For the latter result we use Elias and Khovanov’s diagrammatic presentation of Soergel’s monoidal category of type A ; see (Elias and Khovanov, 2009).