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