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 $R^\Lambda(\beta)$ associated to an {\it arbitrary} symmetrizable Cartan matrix $A=(a_{ij})_{i,j}\in I$, where $\Lambda\in P^+$ and $\beta\in Q_n^+$. As applications, we obtain some {\it necessary and sufficient conditions} for the KLR idempotent $e(\nu)$ (for any $\nu\in I^\beta$) to be nonzero in the cyclotomic quiver Hecke algebra $R^\Lambda(\beta)$. We prove several level reduction results which decomposes $\dim R^\Lambda(\beta)$ into a sum of some products of $\dim R^{\Lambda^i}(\beta_i)$ with $\Lambda=\sum_i\Lambda^i$ and $\beta=\sum_{i}\beta_i$, where $\Lambda^i\in P^+, \beta^i\in Q^+$ for each $i$. We construct some explicit monomial bases for the subspaces $e(\widetilde{\nu})R^\Lambda(\beta)e(\mu)$ and $e(\widetilde{\nu})R^\Lambda(\beta)e(\mu)$ of $R^\Lambda(\beta)$, where $\mu\in I^\beta$ is {\it arbitrary} and $\widetilde{\nu}\in I^\beta$ is a certain specific $n$-tuple (see Section 4).Finally, we use our graded dimension formulae to provide some examples which show that $R^\Lambda(n)$ is in general not graded free over its natural embedded subalgebra $R^\Lambda(m)$ with $m<n$.