Stable centres of wreath products
Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 413-455.

A result of Farahat and Higman shows that there is a “universal” algebra, FH, interpolating the centres of symmetric group algebras, Z(S n ). We explain that this algebra is isomorphic to Λ, where 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 ΓS n of a fixed finite group Γ. This involves constructing wreath-product versions Γ and Λ(Γ * ) of 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 Γ Λ(Γ * ) and use this to compute the p-blocks of wreath products.

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DOI: 10.5802/alco.264
Classification: 20C30, 20E22, 16U70
Keywords: wreath products, Farahat–Higman algebra, Jucys–Murphy elements

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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