Given a crystallographic Coxeter group, it is easy to exhibit a lattice stabilized by it. We can
construct a basis for that lattice as follows. The basis vectors
of the lattice are multiples of the original
simple roots,
for some scalars
which we determine by applying the following
rules:
One easily verifies that for some integers
. Hence
is indeed stabilized. The
integers
are equal to
.
The rules are consistent as can be seen by starting from an arbitrary node, say , for which one takes
. One then proceeds to the next nodes in the (connected) Coxeter graph by applying the
above rules. If there is no closed circuit, there is no consistency problem since there is only
one way to proceed from
to any given node. If there are closed circuits, one must make
sure that one comes back to the same vector after one turn around any circuit. This can be
arranged if the number of steps where one multiplies or divides by
(respectively,
) is
even.
Our construction shows that the lattice is not unique. If there are only two different lengths for the
lattice vectors
, it is convenient to normalize the lengths so that the longest lattice vectors have length
squared equal to two. This choice simplifies the factors
.
The rank 10 hyperbolic Coxeter groups are all crystallographic. The lattices preserved by and
are unique up to an overall rescaling because the non-trivial
(
) are all equal to 3 and there is no choice in the ratios
, all equal to one (first rule above). The Coxeter group
preserves two (dual) lattices.
Since the vectors and
are proportional, they generate identical reflections. Even though they do
not necessarily have length squared equal to unity, the vectors
are more convenient to work with
because the lattice preserved by the Coxeter group is simply the lattice
of points with integer
coordinates in the basis
. For this reason, we shall call from now on “simple roots” the vectors
and, to follow common practice, will sometimes even rename them
. Thus, in the crystallographic case,
the (redefined) simple roots are appropriately normalized to the lattice structure. It turns out that it is with
this normalization that simple roots of Coxeter groups correspond to simple roots of Kac–Moody algebras
defined in the Section 4.6.3. A root is any point on the root lattice that is in the Coxeter orbit of
some (redefined) simple root. It is these roots that coincide with the (real) roots of Kac–Moody
algebras.
It is also useful to rescale the fundamental weights. The rescaled fundamental weights, of course
proportional to , are denoted
. The convenient normalization is such that
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