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Stable Centres of Iwahori-Hecke Algebras of Type A

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Abstract

A celebrated result of Farahat and Higman constructs an algebra FH which “interpolates” the centres \(Z(\mathbb {Z}S_{n})\) of group algebras of the symmetric groups Sn. We extend these results from symmetric group algebras to type A Iwahori-Hecke algebras, Hn(q). In particular, we explain how to construct an algebra FHq “interpolating” the centres Z(Hn(q)). We prove that FHq is isomorphic to \(\mathcal {R}[q,q^{-1}] \otimes _{\mathbb {Z}} {\Lambda }\) (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(Hn(q)) when acting on Specht modules.

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Acknowledgements

The author would like to thank Weiqiang Wang for helpful comments.

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  1. Department of Mathematics, University of California, Berkeley, Berkeley, CA, 94720, USA

    Christopher Ryba

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  1. Christopher Ryba
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Correspondence to Christopher Ryba.

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Presented by: Andrew Mathas

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Ryba, C. Stable Centres of Iwahori-Hecke Algebras of Type A. Algebr Represent Theor 26, 2343–2359 (2023). https://doi.org/10.1007/s10468-022-10184-9

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