On the centers of cyclotomic quiver Hecke algebras

TL;DR

This paper proves conjectures about the centers of cyclotomic quiver Hecke algebras and affine Hecke algebras, showing their centers' surjectivity, stability under base change, and dimension related to partitions.

Contribution

It confirms that the centers of these algebras map surjectively onto their cyclotomic quotients and verifies related conjectures on their structure and dimensions.

Findings

Centers of cyclotomic quiver Hecke algebras surject onto their original centers.

Centers of affine Hecke algebras are stable under base change.

Dimension of the center equals the number of -partitions of n.

Abstract

Let $n\in\mathbb{N}$ and $K$ be any field. For any symmetric generalized Cartan matrix $A$, any $\beta$ in the positive root lattice with height $n$ and any integral dominant weight $\Lambda$, one can associate a quiver Hecke algebras $R_{\beta}(K)$ and its cyclotomic quotient $R_{\beta}^{\Lambda}(K)$ over $K$. It has been conjectured that the natural map from $R_{\beta}(K)$ to $R_{\beta}^{\Lambda}(K)$ maps the center of $R_{\beta}(K)$ surjectively onto the center of $R_{\beta}^{\Lambda}(K)$. A similar conjecture claims that the center of the affine Hecke algebra of type $A$ maps surjectively onto the center of its cyclotomic quotient---the cyclotomic Hecke algebra $H_n^{\Lambda}$ of type $G(\ell,1,n)$ over $K$. In this paper, we prove these two conjectures affirmatively. As a consequence, we show that the center of $H_n^{\Lambda}$ is stable under base change and it has dimension equal to the number of $\ell$-partitions of $n$. Finally, as a byproduct, we also verify a conjecture of Shan, Varagnolo and Vasserot on the grading structure of the center of $R_{\beta}^{\Lambda}(K)$.