Ouch, the Cartan matrix was broken for A,2n,2 (because the conversion
into BC n left 2n/2 as a rational number which broke the graph code!).
sage: C=CartanType(['A',4,2])
sage: C.cartan_matrix()
[ 2 -2 0]
[-1 2 -1]
[ 0 -1 2]
sage: C.cartan_matrix().rank()
3
sage: C.dynkin_diagram()
O=<=O=<=O
0 1 2
BC2~
This is fixed now!
There remains to double check the translation factors for type A_2n^2;
For A_{2n}^{(2)},
c_\alpha = 1,1,2 for (alpha,alpha)=1,2,4 respectively.
This seems to be opposite of what is there right now
sage: C=CartanType(['A',4,2])
sage: C.dynkin_diagram()
O=<=O=<=O
0 1 2
BC2~
sage: R = RootSystem(C).weight_lattice()
sage: c = R.cartan_type().translation_factors()
sage: c
Finite family {0: 2, 1: 1, 2: 1}
But then, I think you have dualized everything, since I would also expect
c_alpha = 1 for type C_3^{(1)}
and you get
sage: C=CartanType(['C',3,1])
sage: R = RootSystem(C).weight_lattice()
sage: c = R.cartan_type().translation_factors()
sage: c
Finite family {0: 1, 1: 2, 2: 2, 3: 1}
If you have dualized, then it seems to be correct.
Cheers,
Anne
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