Stable Centres I: Wreath Products (2107.03752v1)

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 viaevaluation 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.