On the center conjecture for the cyclotomic KLR algebras

Abstract

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 $R_{\beta }^{\Lambda }$.