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

TL;DR

This paper constructs integral bases for the cocenter of cyclotomic Hecke algebras of type G(r,1,n), showing their dimensions are characteristic-independent, and verifies related conjectures and properties.

Contribution

It generalizes existing work to construct bases for cocenters and proves their dimension invariance across different characteristics.

Findings

Constructed integral bases for the cocenter of cyclotomic Hecke algebras.

Proved the dimensions of cocenters and centers are characteristic-independent.

Verified conjectures on polynomial coefficients and characteristic independence for related algebras.

Abstract

In this paper, we construct some integral bases for the cocenter of the cyclotomic Hecke algebra $\mathscr{H}_{n,K}$ 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 $\mathscr{H}_{n,K}$ 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.