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 $R^{\hat{I}}›({\hat{I}}^2)$ associated to an {\it arbitrary} symmetrizable Cartan matrix $A=(a_{ij}{)}_{i,j}\in I$, where $\hat{I}›\in P^{+}$ and ${\hat{I}}^2\in Q_n^{+}$. As applications, we obtain some {\it necessary and sufficient conditions} for the KLR idempotent $e(\hat{I}½)$ (for any $\hat{I}½\in I^2$) to be nonzero in the cyclotomic quiver Hecke algebra $R^{\hat{I}}›({\hat{I}}^2)$. We prove several level reduction results which decomposes $\dim R^{\hat{I}}›({\hat{I}}^2)$ into a sum of some products of $\dim R^{\hat{I}{›}^i}({\hat{I}}_i^2)$ with $\hat{I}›={\sum }_i\hat{I}{›}^i$ and ${\hat{I}}^2={\sum }_i{\hat{I}}_i^2$, where $Λ^i\in P^+, β^i\in Q^+$ for each $i$. We construct some explicit monomial bases for the subspaces $e(\tilde{\hat{I}}½)R^{\hat{I}}›({\hat{I}}^2)e(\hat{I}¼)$ and $e(\tilde{\hat{I}}½)R^{\hat{I}}›({\hat{I}}^2)e(\hat{I}¼)$ of $R^{\hat{I}}›({\hat{I}}^2)$, where $\hat{I}¼\in I^2$ is {\it arbitrary} and $\tilde{\hat{I}}½\in I^2$ is a certain specific $n$-tuple (see Section 4).Finally, we use our graded dimension formulae to provide some examples which show that $R^{\hat{I}}›(n)$ is in general not graded free over its natural embedded subalgebra $R^{\hat{I}}›(m)$ with $m<n$.

To appear in Communications in Contemporary Mathematics