Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras

Abstract

In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra [Formula: see text] associated to an arbitrary symmetrizable Cartan matrix [Formula: see text], where [Formula: see text] and [Formula: see text]. As applications, we obtain some necessary and sufficient conditions for the KLR idempotent [Formula: see text] (for any [Formula: see text]) to be nonzero in the cyclotomic quiver Hecke algebra [Formula: see text]. We prove several level reduction results which decompose [Formula: see text] into a sum of some products of [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] for each [Formula: see text]. Finally, we construct some explicit monomial bases for the subspaces [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is arbitrary and [Formula: see text] is a certain specific [Formula: see text]-tuple defined in (5.1).