ALGEBRAIC COMBINATORICS

no. 2
Stable centres of wreath products
Ryba, Christopher1
1 Department of Mathematics, University of California, Berkeley, CA 94720, USA
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(ℤS n ). We explain that this algebra is isomorphic to ℛ⊗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 ⊗Λ(Γ * ) and use this to compute the p-blocks of wreath products.

Received:
Revised:
Accepted:
Published online:
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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