The University of Southampton
University of Southampton Institutional Repository

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
1660-5446
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.

Text
1907.00626v1 - Accepted Manuscript
Download (213kB)

More information

Accepted/In Press date: 1 July 2019
e-pub ahead of print date: 20 August 2020
Published date: 20 August 2020
Additional Information: Publisher Copyright: © 2020, Springer Nature Switzerland AG.
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: 1660-5446
PURE UUID: f8bd30f9-ec92-4d5a-92dc-4a59f52da41b
ORCID for David Méndez: ORCID iD orcid.org/0000-0003-4023-172X

Catalogue record

Date deposited: 10 Jun 2020 16:31
Last modified: 17 Mar 2024 05:28

Export record

Altmetrics

Contributors

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

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×