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Representability of permutation representations on coalgebras and the isomorphism problem

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
1660-5454
Costoya, Cristina
835f21d2-8417-43d4-8fda-79c32c6dbbd4
Méndez, David
082b86ce-ba78-4519-9713-ff1a38f65c96
Viruel, Antonio
e2cacd64-35e4-43de-9e5a-2df17c5775f8
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).

Record type: Article

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.

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1907.00626v1 - Accepted Manuscript
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Accepted/In Press date: 1 July 2019
e-pub ahead of print date: 20 August 2020
Additional Information: 13 pages
Keywords: Primary 20G05, Secondary 05E18, 16T15, math.CO, math.RT

Identifiers

Local EPrints ID: 441369
URI: http://eprints.soton.ac.uk/id/eprint/441369
ISSN: 1660-5454
PURE UUID: f8bd30f9-ec92-4d5a-92dc-4a59f52da41b
ORCID for David Méndez: ORCID iD orcid.org/0000-0003-4023-172X

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Date deposited: 10 Jun 2020 16:31
Last modified: 28 Apr 2022 04:53

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Contributors

Author: Cristina Costoya
Author: David Méndez ORCID iD
Author: Antonio Viruel

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