A Special Class of Rank 10 and 11 Coxeter Groups

TL;DR

This paper classifies special classes of rank 10 and 11 Coxeter groups with specific incidence and exponent properties, extending previous work on subgroups of E(10) and exploring their embeddings into E(11).

Contribution

It provides a complete classification of rank 10 Coxeter groups with incidence index 3 and extends the classification to rank 11 with incidence index 4, identifying those embeddable into E(11).

Findings

21 rank 10 Coxeter groups identified, 7 previously known.

252 rank 11 Coxeter groups classified, 28 embeddable into E(11).

Coxeter graphs have incidence index 3 or 4, exponents 2 or 3.

Abstract

In the course of investigating regular subalgebras of E(10) related to cosmological solutions of 11-dimensional supergravity supporting an electric 4-form field, a class of rank 10 Coxeter subgroups of the Weyl group of E(10) was uncovered (hep-th/0606123). These Coxeter groups all share the property that their Coxeter graphs have incidence index 3, i.e. that each node is incident to three and only three single lines. Furthermore, the Coxeter exponents are either 2 or 3, but never infinity. We here go beyond subgroups of the Weyl group of E(10) and classify all rank 10 Coxeter groups with these properties. We find 21 distinct Coxeter groups of which 7 were already described in hep-th/0606123. Moreover, we extend the classification to the rank 11 case and we find 252 inequivalent rank 11 Coxeter groups with incidence index 4, of which at least 28 can be regularly embedded into E(11).