Stable Centres I: Wreath Products

Abstract

A result of Farahat and Higman shows that there is a "universal" algebra, FH, interpolating the centres of symmetric group algebras, Z(ZSn). 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 ≀ Sn 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.

The Farahat-Higman Algebra of Centralizers of Symmetric Group Algebras

“…The algebra G was shown to be isomorphic to the algebra of symmetric functions, and it has been studied in relation to enumeration problems for permutation factorisation, see for example [GJ94]. The algebras FH 0 and G have seen counterparts in the setting of the spin symmetric group algebras in [TW09], of wreath products of the symmetric groups with any finite group in [W03] and [Ryba21], and of the general linear group over finite fields in [WW19].…”

Stable Centres II: Finite Classical Groups

“…Wang [Wan04] defined an analogous version of FH for the wreath products G ≀ S n , where G is a fixed finite group, and used an associated graded version to study the cohomology of certain Hilbert schemes of points. We direct the reader to the prequel to the present paper, [Ryb21], for more details regarding the algebra FH.…”

“…Note that this paper is not the paper mentioned in [Ryb21] that will address Iwahori-Hecke algebras of type A and connections to GL n (F q ). That paper will appear in due course.…”