Stable Centres I: Wreath Products

TL;DR

This paper extends the Farahat-Higman algebra framework to wreath products, providing algebraic structures and character formulas that generalize symmetric group results and facilitate block computations.

Contribution

It introduces wreath-product analogues of key algebraic objects and establishes an isomorphism for the universal algebra of wreath products, generalizing previous symmetric group results.

Findings

Isomorphism between universal wreath product algebra and tensor product of new algebraic structures

Construction of Hopf algebra structures for these new algebras

Application to compute p-blocks of wreath products

Abstract

A result of Farahat and Higman shows that there is a ``universal'' algebra, $\mathrm{FH}$, interpolating the centres of symmetric group algebras, $Z(\mathbb{Z}S_n)$. We explain that this algebra is isomorphic to $\mathcal{R} \otimes \Lambda$, where $\mathcal{R}$ 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 $\mathcal{R}_\Gamma$ and $\Lambda(\Gamma_*)$ of $\mathcal{R}$ 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 $\mathcal{R}_\Gamma \otimes \Lambda(\Gamma_*)$ and use this to compute the $p$-blocks of wreath products.