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∇(β) associated to an arbitrary symmetrizable Cartan matrix A = (aij)i,j I, where ∇ P+ and β Qn+. 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∇(β). We prove several level reduction results which decompose dim R∇(β) into a sum of some products of dim R∇i(βi) with ∇ =Σi∇i and β =Σiβi, where ∇i P+,βi Q+ for each i. Finally, we construct some explicit monomial bases for the subspaces e(ν) R∇(β)e(μ) and e(μ) R∇(β)e(ν) of R∇(β), where μ Iβ is arbitrary and ν Iβ is a certain specific n-tuple defined in (5.1).