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.
Categories: math.RT math.RA