Stable centres of wreath products

Ryba, Christopher

doi:10.5802/alco.264

Ryba, Christopher ¹

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

Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 413-455.

Abstract

A result of Farahat and Higman shows that there is a “universal” algebra, $FH$ , interpolating the centres of symmetric group algebras, $Z (ℤ S_{n})$ . We explain that this algebra is isomorphic to $ℛ \otimes Λ$ , where $ℛ$ is the ring of integer-valued polynomials and $Λ$ is the ring of symmetric functions. Moreover, the isomorphism is via “evaluation at Jucys–Murphy elements”, which leads to character formulae for symmetric groups. Then, we generalise this result to wreath products $Γ ≀ S_{n}$ of a fixed finite group $Γ$ . This involves constructing wreath-product versions $ℛ_{Γ}$ and $Λ (Γ_{*})$ of $ℛ$ and $Λ$ , respectively, which are interesting in their own right (for example, both are Hopf algebras). We show that the universal algebra for wreath products, ${FH}_{Γ}$ , is isomorphic to $ℛ_{Γ} \otimes Λ (Γ_{*})$ and use this to compute the $p$ -blocks of wreath products.

Received: 2021-08-31
Revised: 2022-06-27
Accepted: 2022-08-26
Published online: 2023-05-03

DOI: 10.5802/alco.264

Classification: 20C30, 20E22, 16U70
Keywords: wreath products, Farahat–Higman algebra, Jucys–Murphy elements

Author's affiliations:

Ryba, Christopher ¹

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Ryba, Christopher. Stable centres of wreath products. Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 413-455. doi : 10.5802/alco.264. https://alco.centre-mersenne.org/articles/10.5802/alco.264/

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