Some simplifications of the intersection condition of chiral form for polytopes

Abstract

To determine if a poset of type [Formula: see text] is a directly regular or chiral polytope, it is necessary to test whether or not its rotation group (as a quotient of the orientation-preserving subgroup of the Coxeter group [Formula: see text]) satisfies the so-called intersection condition of chiral form. However, due to the fact that many cases need to be checked, this process is often very tedious and takes much time. In this paper, under certain circumstances, we give some simplifications for checking the intersection condition, which leads to certain constructions for directly regular or chiral polytopes.