#18034: symmetric_form weirdness for affine roots
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Reporter: bump | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.6
Component: PLEASE CHANGE | Keywords:
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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Affine weight lattices have a method called symmetric_form which
implements the symmetric bilinear form on the weight lattice (or weight
space) used extensively in representation theory. See Kac, *Infinite-
dimensional Lie algebras* Chapter 2. It is an inner product, that is, a
pairing of the space with itself. Here are some examples where it doesn't
work right unless you coerce the root into the weight lattice.
{{{
sage: RS = RootSystem(['A',2,1])
sage: P = RS.weight_lattice(extended=true)
sage: Q = RS.root_lattice()
sage: alpha = Q.simple_roots()
sage: alphacheck = Q.simple_coroots()
sage: omega = P.fundamental_weights()
sage: [alpha[1].symmetric_form(omega[i]) for i in [0,1,2]]
[-1, 2, -1]
sage: [alpha[1].symmetric_form(alphacheck[i]) for i in [0,1,2]]
[-1, 2, -1]
sage: [P(alpha[1]).symmetric_form(alphacheck[i]) for i in [0,1,2]]
[0, 1, 0]
sage: [P(alpha[1]).symmetric_form(omega[i]) for i in [0,1,2]]
[0, 1, 0]
}}}
The first three answers are (in my opinion) wrong for the following
reasons.
The last answer is correct. The first one is therefore wrong since
the symmetric form should not depend on whether alpha is regarded as
an element of Q or of P. The second two pairings should be undefined since
alphacheck cannot be coerced into P.
--
Ticket URL: <http://trac.sagemath.org/ticket/18034>
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