Representations of Khovanov-Lauda-Rouquier algebras of affine Lie type

Abstract

We study representations of Khovanov-Lauda-Rouquier (KLR) algebras of affine Lie type. Associated to every convex preorder on the set of positive roots is a system of cuspidal modules for the KLR algebra. For a balanced order, we study imaginary semicuspidal modules by means of `imaginary Schur-Weyl duality'. We then generalize this theory from balanced to arbitrary convex preorders for affine ADE types. Under the assumption that the characteristic of the ground field is greater than some explicit bound, we prove that KLR algebras are properly stratified. We introduce affine zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above. Finally, working in finite or affine affine type A, we show that skew Specht modules may be defined over the KLR algebra, and real cuspidal modules associated to a balanced convex preorder are skew Specht modules for certain explicit hook shapes.