Stable centres of wreath products

Ryba, Christopher¹

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

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

Résumé

A result of Farahat and Higman shows that there is a “universal” algebra, $\mathrm{FH}$, interpolating the centres of symmetric group algebras, $Z\left(\mathbb{Z}{S}_n\right)$. We explain that this algebra is isomorphic to $ℛ\otimes \Lambda$, where $ℛ$ is the ring of integer-valued polynomials and $\Lambda$ 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 $\Gamma \wr S_n$ of a fixed finite group $\Gamma$. This involves constructing wreath-product versions ${ℛ}_{\Gamma }$ and $\Lambda \left({\Gamma }_{*}\right)$ of $ℛ$ and $\Lambda$, respectively, which are interesting in their own right (for example, both are Hopf algebras). We show that the universal algebra for wreath products, ${\mathrm{FH}}_{\Gamma }$, is isomorphic to ${ℛ}_{\Gamma }\otimes \Lambda \left({\Gamma }_{*}\right)$ and use this to compute the $p$-blocks of wreath products.

Détail

Reçu le : 2021-08-31
Révisé le : 2022-06-27
Accepté le : 2022-08-26
Publié le : 2023-05-03

DOI : 10.5802/alco.264

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

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits