Stable Centres of Iwahori-Hecke Algebras of type A

TL;DR

This paper generalizes the construction of the center algebra from symmetric groups to type A Iwahori-Hecke algebras, providing explicit isomorphisms and character formulas.

Contribution

It extends Farahat-Higman's algebraic framework to Iwahori-Hecke algebras, constructing an interpolating algebra and proving a conjecture related to Jucys-Murphy elements.

Findings

Constructed algebra _q interpolating centers of Hecke algebras

Proved _q is isomorphic to a tensor product involving symmetric functions

Derived character formulas for Geck-Rouquier basis on Specht modules

Abstract

A celebrated result of Farahat and Higman constructs an algebra $\mathrm{FH}$ which "interpolates" the centres $Z(\mathbb{Z}S_n)$ of group algebras of the symmetric groups $S_n$. We extend these results from symmetric group algebras to type $A$ Iwahori-Hecke algebras, $H_n(q)$. In particular, we explain how to construct an algebra $\mathrm{FH}_q$ "interpolating" the centres $Z(H_n(q))$. We prove that $\mathrm{FH}_q$ is isomorphic to $\mathcal{R}[q,q^{-1}] \otimes_{\mathbb{Z}} \Lambda$ (where $\mathcal{R}$ is the ring of integer-valued polynomials, and $\Lambda$ is the ring of symmetric functions). The isomorphism can be described as "evaluation at Jucys-Murphy elements", leading to a proof of a conjecture of Francis and Wang. This yields character formulae for the Geck-Rouquier basis of $Z(H_n(q))$ when acting on Specht modules.