Christian,
I've got a lot of thoughts swirling in my head about this stuff.
I agree that a lot of the general methods involving Weyl groups
are special cases of those for reflection groups and should be moved there.
There needs to be some mild reorganization of the class
CartanType_abstract and associated factories.
The problem is that in a current CartanType, the name
of a root system (like ['A',2]) is mandatory
and the GCM (generalized Cartan matrix) is derived from it,
while it should be that the GCM is mandatory (say for an unnamed
Kac-Moody root system that can be specified only by a GCM)
but it can be optionally specify it indirectly using a standard
root system name like ['A',2] or a Dynkin diagram, that is,
an index set I and for a pair i<j of elements of I,
an optional directed edge with label (a_{ij},a_{ji}).
Two integer labels are needed even for affine type.
Note that the GCM and the Dynkin diagram are the dense and sparse
versions of the same data and are easily obtained from each other.
> - construct a root system as the orbit of the simple roots under the
> reflection group as a matrix group.
Here you mean real roots. For practical reasons this is problematic for
infinite root systems. One could make a lazy version but I can't see a
significant benefit. However see below for the membership test in the Weyl
group case.
> - is there a way to obtain the imaginary roots as well (my knowledge here
> is very limited, but I will be happy to learn more!)?
The following algorithm decides whether a general element mu
of the nonnegative span Q^+ of the simple roots,
is a positive root and whether it is real or imaginary. This is for
Kac-Moody Weyl groups. See Kac's book "Infinite Dimensional Lie Algebras",
Theorem 5.4 for the imaginary root case.
Repeatly apply simple reflections to mu in height-decreasing directions i
until the first occurrence of one of the following conditions.
1. The result is a simple root. In this case mu is a real positive root.
2. The result is not in Q^+. Then mu is not a root.
3. The result has no height-decreasing directions, in which case
mu is a positive imaginary root if the support
of the result is connected and is not a root otherwise.
Here the support of an element of Q is the
subgraph of the Dynkin diagram induced by the set of nodes i
such that the coefficient of alpha_i in the element is nonzero.
--Mark
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