Hopf Modules and Representations of Finite Wreath Products
Author(s)
Shelley-Abrahamson, Seth
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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).
Date issued
2016-06Department
Massachusetts Institute of Technology. Department of MathematicsJournal
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
ISSN
1386-923X
1572-9079