#15325: Highly confusing behavior of RootSystem related objects (lattices and
ambient space)
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Reporter: davidamadore | Owner:
Type: defect | Status: new
Priority: minor | Milestone: sage-5.13
Component: combinatorics | Keywords: root-system
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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I'm not sure to what extent the following behavior is deliberate or simply
unavoidable, so maybe this bug report should be reclassified as bad
documentation. Also, I'm not sure whether I should be filing several
different bug reports (but I do believe the issues I raise are very
closely related).
My main issue is that for mathematicians who are used to thinking of the
weight lattice of a root system living inside of the root lattice, itself
living inside some common euclidean ambient space (which, I think, is
standard practice), Sage's behavior is mind-boggingly confusing, if not
outright insane.
Exhibit A (shows that elements of the root lattice and its ambient space,
while comparing for equality, behave very differently when it comes to
scalar product):
{{{
sage: rt = RootSystem(['E',8])
sage: lat = rt.root_lattice()
sage: spc = rt.ambient_space()
sage: lat.simple_root(1) == spc.simple_root(1)
True
sage: lat.simple_coroot(2) == spc.simple_coroot(2)
True
sage: lat.simple_root(1).scalar(lat.simple_coroot(2))
0
sage: spc.simple_root(1).scalar(spc.simple_coroot(2))
0
sage: spc.simple_root(1).scalar(lat.simple_coroot(2))
-1/2
sage: lat.simple_root(1).scalar(spc.simple_coroot(2))
---------------------------------------------------------------------------
TypeError Traceback (most recent call
last)
<ipython-input-9-61492d33e12b> in <module>()
----> 1 lat.simple_root(Integer(1)).scalar(spc.simple_coroot(Integer(2)))
/work/sage-5.12/local/lib/python2.7/site-
packages/sage/combinat/root_system/root_space.pyc in scalar(self,
lambdacheck)
245 # Find some better test
246 if not (lambdacheck in self.parent().coroot_lattice() or
lambdacheck in self.parent().coroot_space()):
--> 247 raise TypeError, "%s is not in a coroot
lattice/space"%(lambdacheck)
248 zero = self.parent().base_ring().zero()
249 cartan_matrix = self.parent().dynkin_diagram()
TypeError: (1, 1, 0, 0, 0, 0, 0, 0) is not in a coroot lattice/space
}}}
Exhibit B (shows that Sage does not consider the root lattice as living
inside the weight lattice):
{{{
sage: rt = RootSystem(['E',8])
sage: wlat = rt.weight_lattice()
sage: lat = rt.root_lattice()
sage: wlat.simple_root(1) == lat.simple_root(1)
True
sage: lat.simple_root(1) in lat
True
sage: wlat.simple_root(1) in lat
False
}}}
Exhibit C (shows how confusing the embedding of the weight lattice in the
ambient space is):
{{{
sage: rt = RootSystem(['A',2])
sage: wlat = rt.weight_lattice()
sage: lat = rt.root_lattice()
sage: spc = rt.ambient_space()
sage: lat.simple_root(2) == wlat.simple_root(2)
True
sage: spc.simple_root(2)
(0, 1, -1)
sage: spc(lat.simple_root(2))
(0, 1, -1)
sage: spc(wlat.simple_root(2))
(1, 2, 0)
}}}
I'm not sure how all of this should be fixed, but certainly when x==y
returns True then x and y should have a very similar behavior, and in
particular x in y.parent() should also return True.
--
Ticket URL: <http://trac.sagemath.org/ticket/15325>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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