3.4 Affine Coxeter groups

Affine Coxeter groups are by definition such that the bilinear form is positive semi-definite but not positive definite. The radical (defined as the subspace of vectors for which for all ) is then one-dimensional (in the irreducible case). Indeed, since is positive semi-definite, its radical coincides with the set of vectors such that as can easily be seen by going to a basis in which is diagonal (the eigenvalues of are non-negative). Furthermore, is at least one-dimensional since is not positive definite (one of the eigenvalues is zero). Let be a vector in . Let be the vector whose components are the absolute values of those of , . Because for (see definition of in Equation (3.28)), one has

and thus the vector belongs also to . All the components of are strictly positive, . Indeed, let be the set of indices for which and the set of indices for which . From () one gets, by taking in , that for all , , contrary to the assumption that the Coxeter system is irreducible ( is indecomposable). Hence, none of the components of any zero eigenvector can be zero. If were more than one-dimensional, one could easily construct a zero eigenvector of with at least one component equal to zero. Hence, the eigenspace of zero eigenvectors is one-dimensional.

Affine Coxeter groups can be identified with the groups generated by affine reflections in Euclidean space (i.e., reflections through hyperplanes that may not contain the origin, so that the group contains translations) and have also been completely classified [107]. The translation subgroup of an affine Coxeter group is an invariant subgroup and the quotient is finite; the affine Coxeter group is equal to the semi-direct product of its translation subgroup by . We list all the affine Coxeter groups in Table 2.

 Table 2: Affine Coxeter groups.
 Name Coxeter graph