Faithful permutation representations of toroidal regular maps
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Faithful permutation representations of toroidal regular maps Maria Elisa Fernandes1
· Claudio Alexandre Piedade1
Received: 6 April 2019 / Accepted: 18 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, we list all possible degrees of a faithful transitive permutation representation of the group of symmetries of a regular map of types {4, 4} and {3, 6}, and we give examples of graphs, called CPR graphs, representing some of these permutation representations. Keywords Regular polytopes · Regular toroidal maps · Permutation groups · CPR graphs Mathematics Subject Classification 52B11 · 05E18 · 20B25
1 Introduction The classification of highly symmetric objects, particularly regular maps and polytopes, is a problem that attracts both geometers and algebraists. The idea of using permutation representations to classify regular maps and polytopes is not new, but in 2008 the concept of a graph associated with a regular polytope, called a CPR graph, was introduced [15]. CPR graphs, that are faithful permutation representations, turned out to be a powerful tool in classification of abstract regular polytopes. The group of symmetries G of an abstract regular polytope of rank r is generated by r involutions ρ0 , . . . , ρr −1 . For each i ∈ {0, . . . , r − 1}, the i-faces correspond to the cosets of the group G i generated by all the generators of G except ρi [14]. The group G acts transitively on the set of i-faces and if, in addition, G i is core-free this action is faithful. Regular maps that have non-faithfull actions on the cells (vertices, edges or faces) or on darts are identified in [13].
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Maria Elisa Fernandes [email protected] Claudio Alexandre Piedade [email protected]
1
Department of Mathematics, University of Aveiro, Aveiro, Portugal
123
Journal of Algebraic Combinatorics
In this paper, we search for faithful transitive permutation representations of the group of symmetries of a toroidal regular map of types {4, 4} and {3, 6}, which includes the faithful actions on vertices, edges and faces of the map. CPR graphs give an efficient method to classify or construct abstract regular polytopes with a certain types of prescribed group of automorphisms. Several results were accomplished using these faithful permutation representations answering some conjectures arising from the atlas of abstract regular polytopes for small groups built in 2006 by Leemans and Vaulthier [12]. For instance, the symmetric group Sn (n ≥ 4) is the group of symmetries of a polytope of any rank between 3 and n −1 [5]. CPR graphs were used to construct examples of polytopes for all possible ranks, furthermore to prove that there are exactly two polytopes of rank r = n − 2 for Sn and to describe all polytopes of rank r = n − 3 for Sn [9]. For the alternating group An , it was possible to prove that the maximal rank is n−1 2 when n ≥ 12 [2] and to provide examples of polytopes with highest possible rank [7,8]. The concept of a hypertope, as a generalization of a polyto
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