Hopf Modules and Representations of Finite Wreath Products
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Author(s)
Shelley-Abrahamson, Seth
Date Issued
June 2016
Journal
Algebras and Representation Theory
Publisher
Springer Netherlands
Citation
Shelley-Abrahamson, Seth. “Hopf Modules and Representations of Finite Wreath Products.” Algebras and Representation Theory 20, no. 1 (June 29, 2016): 123–145.
Version
Author's final manuscript
Abstract
For a finite group G and nonnegative integer n ≥ 0, one may consider the associated tower G≀S[subscript n]:=S[subscript n]⋉G[superscript n] of wreath product groups. Zelevinsky associated to such a tower the structure of a positive self-adjoint Hopf algebra (PSH-algebra) R(G) on the direct sum over integers n ≥ 0 of the Grothendieck groups K[subscript 0](Rep−G≀S[subscript n]). In this paper, we study the interaction via induction and restriction of the PSH-algebras R(G) and R(H) associated to finite groups H ⊂ G. A class of Hopf modules over PSH-algebras with a compatibility between the comultiplication and multiplication involving the Hopf k[superscript th]-power map arise naturally and are studied independently. We also give an explicit formula for the natural PSH-algebra morphisms R(H) → R(G) and R(G) → R(H) arising from induction and restriction. In an appendix, we consider a family of subgroups of wreath product groups analogous to the subgroups G(m, p, n) of the wreath product cyclotomic complex reflection groups G(m, 1, n).
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s10468-016-9633-4
