On the center conjecture for the cyclotomic KLR algebras

The center conjecture for the cyclotomic KLR algebras $R_\beta^\Lambda$ asserts that the center of $R_\beta^\Lambda$ consists of symmetric elements in its KLR $x$ and $e(\nu)$ generators. In this paper we show that this conjecture is equivalent to the injectivity of some natural map $\bar{\iota}_\beta^{\Lambda,i}$ from the cocenter of $R_\beta^\Lambda$ to the cocenter of $R_\beta^{\Lambda+\Lambda_i}$ for all $i\in I$ and $\Lambda\in P^+$. We prove that the map $\bar{\iota}_\beta^{\Lambda,i}$ is given by multiplication with a center element $z(i,\beta)\in R_\beta^{\Lambda+\Lambda_i}$ and we explicitly calculate the element $z(i,\beta)$ in terms of the KLR $x$ and $e(\nu)$ generators. We present an explicit monomial basis for certain bi-weight spaces of the defining ideal of $R_\beta^\Lambda$ and of $R_\beta^\Lambda$. For $\beta=\sum_{j=1}^n\alpha_{i_j}$ with $\alpha_{i_1},\cdots, \alpha_{i_n}$ pairwise distinct, we construct an explicit monomial basis of $R_\beta^\Lambda$, prove the map $\bar{\iota}_\beta^{\Lambda,i}$ is injective and thus verify the center conjecture for these $R_\beta^\Lambda$.