Representability of permutation representations on coalgebras and the isomorphism problem
Representability of permutation representations on coalgebras and the isomorphism problem
Let G be a group and let ρ: G → Sym(V ) be a permutation representation of G on a set V . We prove that there is a faithful G-coalgebra C such that G arises as the image of the restriction of Aut(C) to G(C), the set of grouplike elements of C. Furthermore, we show that V can be regarded as a subset of G(C) invariant through the G-action, and that the composition of the inclusion G ֒→ Aut(C) with the restriction Aut(C) → Sym(V ) is precisely ρ. We use these results to prove that isomorphism classes of certain families of groups can be distinguished through the coalgebras on which they act faithfully.
Primary 20G05, Secondary 05E18, 16T15, math.CO, math.RT
Costoya, Cristina
835f21d2-8417-43d4-8fda-79c32c6dbbd4
Méndez, David
082b86ce-ba78-4519-9713-ff1a38f65c96
Viruel, Antonio
e2cacd64-35e4-43de-9e5a-2df17c5775f8
20 August 2020
Costoya, Cristina
835f21d2-8417-43d4-8fda-79c32c6dbbd4
Méndez, David
082b86ce-ba78-4519-9713-ff1a38f65c96
Viruel, Antonio
e2cacd64-35e4-43de-9e5a-2df17c5775f8
Costoya, Cristina, Méndez, David and Viruel, Antonio
(2020)
Representability of permutation representations on coalgebras and the isomorphism problem.
Mediterranean Journal of Mathematics, 17 (5), [157].
(doi:10.1007/s00009-020-01594-4).
Abstract
Let G be a group and let ρ: G → Sym(V ) be a permutation representation of G on a set V . We prove that there is a faithful G-coalgebra C such that G arises as the image of the restriction of Aut(C) to G(C), the set of grouplike elements of C. Furthermore, we show that V can be regarded as a subset of G(C) invariant through the G-action, and that the composition of the inclusion G ֒→ Aut(C) with the restriction Aut(C) → Sym(V ) is precisely ρ. We use these results to prove that isomorphism classes of certain families of groups can be distinguished through the coalgebras on which they act faithfully.
Text
1907.00626v1
- Accepted Manuscript
More information
Accepted/In Press date: 1 July 2019
e-pub ahead of print date: 20 August 2020
Published date: 20 August 2020
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© 2020, Springer Nature Switzerland AG.
Keywords:
Primary 20G05, Secondary 05E18, 16T15, math.CO, math.RT
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Local EPrints ID: 441369
URI: http://eprints.soton.ac.uk/id/eprint/441369
ISSN: 1660-5446
PURE UUID: f8bd30f9-ec92-4d5a-92dc-4a59f52da41b
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Date deposited: 10 Jun 2020 16:31
Last modified: 17 Mar 2024 05:28
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Author:
Cristina Costoya
Author:
David Méndez
Author:
Antonio Viruel
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