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.
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