2207.06319Jul 13, 202213 pages

Stable Centres of Iwahori-Hecke Algebras of type A

Christopher Ryba

Representation Theory Rings and Algebras

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 $S_n$

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. 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}} Λ

(where

\mathcal{R}

is the ring of integer-valued polynomials, and

Λ

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.

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Stable Centres of Iwahori-Hecke Algebras of type A

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