  There are two longstanding conjectures on the centers of the cyclotomic Hecke
algebra $\mathscr{H}_{n,K}^\Lambda$ of type $G(r,1,n)$ which assert that: 1)
the dimension of the center $Z(\mathscr{H}_{n,K}^\Lambda)$ is independent of
the characteristic of the ground field, its Hecke parameter and cyclotomic
parameters; 2) the center $Z(\mathscr{H}_{n,K}^\Lambda)$ of
$\mathscr{H}_{n,K}^\Lambda$ is the set of symmetric polynomial in its
Jucys-Murphy operators. In this paper we prove these two conjectures
affirmatively. At the same time we show that the center conjecture holds for
the cyclotomic KLR algebras $\mathscr{R}_{\alpha,K}^\Lambda$ associated to the
cyclic quiver $A_{e-1}^{(1)}$ (for $e>1$) and the linear quiver $A_{\infty}$
(for $e=0$) when the ground field $K$ has characteristic $p$ which satisfies
either $p=0$, or $p>0=e$, or $p=e>1$, or $p>0$, $e>1$ and $p$ is coprime to
$e$. As applications, we show that the cohomology of the Nakajima quiver
variety $\mathfrak{M}(\Lambda,\alpha)$ with coefficient in $K$ is isomorphic to
the center of $\mathscr{R}_{\alpha,K}^\Lambda$ in the affine type $A$ case when
${\rm{char}} K=0$; we also verify Chavli-Pfeiffer's conjecture on the
polynomial coefficient $g_{w,C}$ for the complex reflection group of type
$G(r,1,n)$.
