Resolutions of standard modules over KLR algebras of type $A$

Abstract

Khovanov-Lauda-Rouquier algebras $R_{\theta }$ of finite Lie type are affine quasihereditary with standard modules $\Delta (\pi )$ labeled by Kostant partitions $\pi$ of $\theta$. In type $A$, we construct explicit projective resolutions of standard modules $\Delta (\pi )$.