Hypertopes with tetrahedral diagram
type: Item
instance of: scholarly article
title: Hypertopes with tetrahedral diagram
zbMATH Open document ID: 1402.52012
author: Maria Elisa Fernandes
author: Isabel Hubard
author: Dimitri Leemans
author: Domenico A. Catalano
published in: The Electronic Journal of Combinatorics
publication date: 2018-08-15T00:00:00Z
full work available at URL: v25i3p22
review text: Hypertopes, as introduced in an earlier paper by two of the authors, generalise the notion of abstract polytopes by not requiring a linear diagram. The concept of chirality can be extended to them, and in order to get a better understanding of chiral hypertopes, this paper provides examples of the latter by studying a family of rank \(4\) locally toroidal hypertopes -- more particularly, rank \(4\) hypertopes with rank 3 residues that are toroidal hypermaps of type \((3,3,3)\) -- having a group of rotational symmetries isomorphic to \(\mathsf{PSL}(2,q)\), where \(q=p\) or \(q=p^2\) for a prime \(p\) such that \(3\) is a divisor of \(q-1\), but, if \(q=p^2\), \(3\) is not a divisor of \(p-1\). The authors are able to construct these hypertopes as coset geometries (note that a coset geometry is not automatically a hypertope), by using \(\mathsf{PSL}(2,q)\) as a \(C^+\)-group whose four maximal parabolic subgroups (these will give the right rank 3 residues) are isomorphic to the alternating group \(A_4\) (occurs twice) or to \(E_q : C_3\), a cyclic extension of the elementary abelian \(p\)-group \(E_q\) or order \(q\) (also occurs twice). They also provide a geometric description. The thus obtained hypertopes turn out to be chiral if \(q=p\) and regular if \(q=p^2\). By construction, they have the complete graph \(K_4\) as their diagram. Interestingly, the chiral hypertopes admit both improper and proper correlations, i.e., there are correlations that interchange the two orbits and ones that preserve them, whereas this is impossible for chiral polytopes.
reviewed by: Anneleen De Schepper
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 52B15
zbMATH DE Number: 6918055
zbMATH Keywords: regularity
zbMATH Keywords: chirality
zbMATH Keywords: thin geometries
zbMATH Keywords: hypermaps
zbMATH Keywords: abstract polytopes
zbMATH Keywords: chiral polytopes
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6918055