An explicit presentation of the centre of the restricted rational Cherednik algebra

TL;DR

This paper provides an explicit algebraic description of the center of the restricted rational Cherednik algebra for certain parameters, linking it to multipartitions and Calogero-Moser space smoothness.

Contribution

It offers the first explicit presentation of the centre of the restricted rational Cherednik algebra for specific parameters, connecting algebraic generators with multipartitions.

Findings

Explicit generators and relations for the centre are given.

The presentation applies when the Calogero-Moser space is smooth.

The centre's structure is derived from multipartitions of n.

Abstract

In this paper we give an explicit presentation of the centre of the restricted rational Cherednik algebra $\overline{H}_c(S_n\wr\mathbb{Z}/\ell\mathbb{Z})$ for a particular choice of parameter $c$. More precisely, we describe the centre of the indecomposable blocks of $\overline{H}_c(S_n\wr\mathbb{Z}/\ell\mathbb{Z})$ in terms of generators and relations. The parameters $c$ for which this presentation is valid are such that the Calogero-Moser space is smooth. Furthermore, we demonstrate how the explicit presentation of the centre of $\overline{H}_c(S_n\wr\mathbb{Z}/\ell\mathbb{Z})$ can be directly derived from the set of $\ell$-multipartitions of $n$.