Representability of Permutation Representations on Coalgebras and the Isomorphism Problem
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Representability of Permutation Representations on Coalgebras and the Isomorphism Problem Cristina Costoya, David M´endez and Antonio Viruel Abstract. Let H be an arbitrary group and let ρ : H → Sym(V ) be any permutation representation of H on a set V . We prove that there is a faithful H-coalgebra C such that H 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 under the H-action and that the composition of the inclusion H → 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. Mathematics Subject Classification. Primary 20G05; Secondary 05E18, 16T15.
1. Introduction Given X an object in a category C, the study of its automorphism group, Aut(X), is a difficult task. In fact, even deciding which groups arise as the automorphism groups of objects in C is far from trivial, see [6,15]. Nonetheless, it is also rewarding, as it can be expected that distinguished objects have distinguished automorphism groups, which in turn may give valuable information regarding the object X. Not only that, but if we know the automorphism groups of enough objects in C, we can also draw conclusions regarding the category itself. One clear example of this comes from representation theory, as the automorphism groups of the objects of a category tell us a lot about which groups may act on which objects. Cristina Costoya and David M´endez are partially supported by Ministerio de Econom´ıa y Competitividad (Spain), grant MTM2016-79661-P (AEI/FEDER, UE, support included). David M´ endez is partially supported by Ministerio de Educaci´ on, Cultura y Deporte (Spain) grant FPU14/05137. David M´endez and Antonio Viruel are partially supported by Ministerio de Econom´ıa y Competitividad (Spain), grant MTM2016-78647-P (AEI/FEDER, UE, support included).
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The category of rings is one instance in which the possible automorphism groups of objects have been extensively studied. Indeed, there are many references in the literature regarding the realisability of groups as automorphisms of rings (see for example [5,7,16] regarding the associative case, and [4,12] on the non-associative one). However, very little is known about the problem of representing groups as automorphisms of coalgebras. Moreover, given that coalgebras are only truly dual of algebras in the finite-dimensional case, general results on automorphisms of coalgebras cannot be deduced from the preexisting literature on automorphism groups of rings. This article aims at providing the first result on the realisability of groups as automorphisms of coalgebras. Not only are we successful with regard to that task, we also prove results regarding the realisability of not necessarily faithful permutation representations of arbitrary groups. The key ingredi
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