On the Center Conjecture for the Cyclotomic KLR Algebras

Abstract

Abstract The center conjecture for the cyclotomic KLR algebras $\mathscr{R}_{\beta }^{\Lambda }$ asserts that the center of $\mathscr{R}_{\beta }^{\Lambda }$ 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 $\overline{\iota }_{\beta }^{\Lambda ,i}$ from the cocenter of $\mathscr{R}_{\beta }^{\Lambda }$ to the cocenter of $\mathscr{R}_{\beta }^{\Lambda +\Lambda _{i}}$ for all $i\in I$ and $\Lambda \in P^{+}$. We prove that the map $\overline{\iota }_{\beta }^{\Lambda ,i}$ is given by multiplication with a center element $z(i,\beta )\in \mathscr{R}_{\beta }^{\Lambda +\Lambda _{i}}$ and we explicitly calculate the element $z(i,\beta )$ in terms of the KLR $x$ and $e(\nu )$ generators. We present explicit monomial bases for certain bi-weight spaces of the defining ideal of $\mathscr{R}_{\beta }^{\Lambda }$. 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 $\mathscr{R}_{\beta }^{\Lambda }$, prove the map $\overline{\iota }_{\beta }^{\Lambda ,i}$ is injective, and thus verify the center conjecture for these $\mathscr{R}_{\beta }^{\Lambda }$.