Affine highest weight structures on module categories over quiver Hecke algebras

Abstract: We prove that the category of finitely generated graded modules over the quiver Hecke algebra of arbitrary type admits numerous stratifications in the sense of Kleshchev. A direct consequence is that the full subcategory corresponding to the quantum unipotent subgroup associated with any Weyl group element is an affine highest weight category. Our results significantly generalize earlier works by Kato, Brundan, Kleshchev, McNamara and Muth. The key ingredient is a realization of standard modules via determinantial modules. We utilize the technique of R-matrices to study these standard modules.