On the cocenter of the cyclotomic Hecke algebra of type $G(r,1,n)$

Abstract

In this paper, we construct some integral bases for the cocenter of the cyclotomic Hecke algebra ${\mathcal{H}}_{n,K}$ of type $G(r,1,n)$ by generalizing Geck and Pfeiffer's work on the cocenters of the Iwahori-Hecke algebras associated to finite Weyl groups. We show that the dimensions of both the cocenter and the center of the cyclotomic Hecke algebra ${\mathcal{H}}_{n,K}$ are independent of the characteristic of the ground field, its Hecke parameter and cyclotomic parameters. As applications, we verify Chavli-Pfeiffer's conjecture on the polynomial coefficient $g_{w,C}$ for the complex reflection group of type $G(r,1,n)$ and also show that both the cocenters and the centers of certain cyclotomic KLR algebras of affine type $A$ are independent of the characteristic of the ground field.

This paper is extracted from the previous preprint "Proof of the Center Conjectures for the cyclotomic Hecke and KLR algebras of type $A$", arXiv:2211.07069v6