Stable Centres I: Wreath Products
Christopher Ryba

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, , interpolating the centres of symmetric group algebras, . We explain that this algebra is isomorphic to , where is the ring of integer-valued polynomials and 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 of a fixed finite group . This involves constructing wreath-product versions and of and , respectively, which are interesting in their own right (for example, both are Hopf algebras). We show that the universal algebra for wreath products, , is…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Advanced Combinatorial Mathematics
