Stable Centres of Iwahori-Hecke Algebras of type A

A celebrated result of Farahat and Higman constructs an algebra $\mathrm{F}\mathrm{H}$ 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 ${\mathrm{F}\mathrm{H}}_q$ “interpolating” the centres $Z(H_n(q))$. We prove that ${\mathrm{F}\mathrm{H}}_q$ is isomorphic to $\mathcal{R}[q,q^{-1}]{\otimes }_{\mathbb{Z}}\mathrm{\Lambda }$ (where $\mathcal{R}$ is the ring of integer-valued polynomials, and $\mathrm{\Lambda }$ 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.