Testing chirality on hypertopes

TL;DR

The paper addresses the problem of efficiently certifying chirality for coset geometries arising from $C^+$-groups by deriving group-theoretical criteria on maximal parabolic subgroups that characterize when the associated incidence system is a chiral hypertope, avoiding the costly construction of incidence graphs. It presents a main theorem establishing equivalence between chirality and a collection of conditions on truncations, intersections of parabolic subgroups, and the absence of an automorphism inverting all generators, along with a residual connectedness guarantee. The contributions include a practical, scalable criterion for chirality, a formal bridge from $C^+$-group data to chiral hypertopes, and Magma code that automates the verification for large groups. This advances the theoretical and computational study of chiral hypertopes, enabling analysis of larger coset geometries and facilitating proofs of structural results.

Abstract

In this paper we give group-theoretical conditions on the maximal parabolic subgroups of a coset geometry for it to be a chiral hypertope, bypassing the need to construct the incidence graph of the coset geometry to determine whether or not it is a chiral hypertope. This result permits to study much larger coset geometries with computers and gives hope on proving theoretical results about coset geometries that are chiral hypertopes.

Theorems & Definitions (15)

• Theorem 1.1
• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.6
• Corollary 2.7
• Theorem 2.8
• Theorem 3.1
• Theorem 3.2