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 $d\le n$. 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).