Cuspidal systems for affine Khovanov–Lauda–Rouquier algebras

Abstract

A cuspidal system for an affine Khovanov–Lauda–Rouquier algebra $$R_\alpha $$ yields a theory of standard modules. This allows us to classify the irreducible modules over $$R_\alpha $$ up to the so-called imaginary modules. We describe minuscule imaginary modules, laying the groundwork for future study of imaginary Schur–Weyl duality. We introduce colored imaginary tensor spaces and reduce a classification of imaginary modules to one color. We study the characters of cuspidal modules. We show that under the Khovanov–Lauda–Rouquier categorification, cuspidal modules correspond to dual root vectors.