Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras

In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra R^δ(β) associated to an arbitrary symmetrizable Cartan matrix A = (aij)i,j I, where δ ϵ P⁺ and β Q_n⁺. As applications, we obtain some necessary and sufficient conditions for the KLR idempotent e(ν) (for any ν Iβ) to be nonzero in the cyclotomic quiver Hecke algebra R^δ(β). We prove several level reduction results which decompose dim R^δ(β) into a sum of some products of dim R^δi(βi) with δ =Σ_i δ_i and β =Σiβi, where δi P+,βi Q+ for each i. Finally, we construct some explicit monomial bases for the subspaces e(ν)R^δ(β)e(μ) and e(μ)R^δ(β)e(ν) of R^δ(β), where μ Iβ is arbitrary and ν Iβ is a certain specific n-tuple defined in (5.1).