A Special Class of Rank 10 and 11 Coxeter Groups

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. Out of the total number of rank 10 Coxeter groups, 2 give degenerate Cartan matrices, 9 are Lorentzian and finally 10 of them correspond to Cartan matrices with positive determinants but with signatures (--++++++++). We also 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)