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 . 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.
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