On the centers of cyclotomic quiver Hecke algebras
Let $n\in\mathbb{N}$ and $K$ be any field. For any symmetric generalized Cartan matrix $A$, any $\beta$ in the positive root lattice with height $n$ and any integral dominant weight $\Lambda$, one can associate a quiver Hecke algebras $R_{\beta}(K)$ and its cyclotomic quotient $R_{\beta}^{\Lambda}(K)$ over $K$. It has been conjectured that the natural map from $R_{\beta}(K)$ to $R_{\beta}^{\Lambda}(K)$ maps the center of $R_{\beta}(K)$ surjectively onto the center of $R_{\beta}^{\Lambda}(K)$. A similar conjecture claims that the center of the affine Hecke algebra of type $A$ maps surjectively onto the center of its cyclotomic quotient---the cyclotomic Hecke algebra $H_n^{\Lambda}$ of type $G(\ell,1,n)$ over $K$. In this paper, we prove these two conjectures affirmatively. As a consequence, we show that the center of $H_n^{\Lambda}$ is stable under base change and it has dimension equal to the number of $\ell$-partitions of $n$. Finally, as a byproduct, we also verify a conjecture of Shan, Varagnolo and Vasserot on the grading structure of the center of $R_{\beta}^{\Lambda}(K)$.