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On the center conjecture for the cyclotomic KLR algebras

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