I claim there is currently a mathematical mistake in the way the
"is_real_root" method in the file root_lattice_realizations.py is
implemented. There is also a parallel mistake in "is_imaginary_root". Let
me illustrate with an example:
sage: R = RootSystem(["A",2,1]).weight_lattice()
sage: alpha = R.simple_roots()
sage: alpha[0].is_real_root()
True
sage: b = 2*alpha[0]
sage: b.is_real_root()
True
b.is_real_root() should return false because 2*alpha[0] is not a real root.
In particular, 2*alpha[0] is not a root.
Looking at the implementation of "is_real_root()", it is easy to see why
this isn't working properly.
def is_real_root(self):
r"""
Return ``True`` if ``self`` is a real root.
A root `\alpha` is real if it is `W` conjugate to a simple
root where `W` is the corresponding Weyl group.
EXAMPLES::
sage: Q = RootSystem(['B',2,1]).root_lattice()
sage: alpha = Q.simple_roots()
sage: alpha[0].is_real_root()
True
sage: elt = alpha[0] + alpha[1] + 2*alpha[2]
sage: elt.is_real_root()
False
"""
return self.norm_squared() > 0
self.norm_squared() is not the way to check whether an element of the root
space or weight_space is a real root.
The problem is that there is no check to see whether self is a root. In
affine types, the following should work:
return (self.norm_squared() > 0) and self.is_root()
where is_root() would be a method that checks whether self is a root. But
this isn't currently implemented, and I don't know what would be the most
intelligent way to implement this. In affine types, this is fairly straight
forward assuming you have a test for checking whether something is a root
in the finite root system, but I don't know for general Cartan types.
What do people think about this? I have not contributed to sage in the
past, but I am interested in implementing the fix.
Also, is there a better place for me to post this bug. I looked
here: http://wiki.sagemath.org/support/ReportingBugs, and it suggested I
post on sage-support since I don't have experience developing sage or using
trac.
Best,
Dinakar
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