Hopf Modules and Representations of Finite Wreath Products
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...
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| Format: | Article |
| Language: | English |
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Springer Netherlands
2017
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| Online Access: | http://hdl.handle.net/1721.1/107447 https://orcid.org/0000-0002-9807-1805 |
| _version_ | 1826205699002073088 |
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| author | Shelley-Abrahamson, Seth |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Shelley-Abrahamson, Seth |
| author_sort | Shelley-Abrahamson, Seth |
| collection | MIT |
| description | 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). |
| first_indexed | 2024-09-23T13:17:31Z |
| format | Article |
| id | mit-1721.1/107447 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:17:31Z |
| publishDate | 2017 |
| publisher | Springer Netherlands |
| record_format | dspace |
| spelling | mit-1721.1/1074472022-09-28T13:12:36Z Hopf Modules and Representations of Finite Wreath Products Shelley-Abrahamson, Seth Massachusetts Institute of Technology. Department of Mathematics Shelley-Abrahamson, Seth 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). 2017-03-16T20:20:36Z 2017-04-11T21:29:35Z 2016-06 2015-08 2017-02-08T04:30:10Z Article http://purl.org/eprint/type/JournalArticle 1386-923X 1572-9079 http://hdl.handle.net/1721.1/107447 Shelley-Abrahamson, Seth. “Hopf Modules and Representations of Finite Wreath Products.” Algebras and Representation Theory 20, no. 1 (June 29, 2016): 123–145. https://orcid.org/0000-0002-9807-1805 en http://dx.doi.org/10.1007/s10468-016-9633-4 Algebras and Representation Theory Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Science+Business Media Dordrecht application/pdf Springer Netherlands Springer Netherlands |
| spellingShingle | Shelley-Abrahamson, Seth Hopf Modules and Representations of Finite Wreath Products |
| title | Hopf Modules and Representations of Finite Wreath Products |
| title_full | Hopf Modules and Representations of Finite Wreath Products |
| title_fullStr | Hopf Modules and Representations of Finite Wreath Products |
| title_full_unstemmed | Hopf Modules and Representations of Finite Wreath Products |
| title_short | Hopf Modules and Representations of Finite Wreath Products |
| title_sort | hopf modules and representations of finite wreath products |
| url | http://hdl.handle.net/1721.1/107447 https://orcid.org/0000-0002-9807-1805 |
| work_keys_str_mv | AT shelleyabrahamsonseth hopfmodulesandrepresentationsoffinitewreathproducts |