Hypertopes with tetrahedral diagram
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.
scientific article; zbMATH DE number 6918055