ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 06 Nov 2018 16:38:28 +0100Are subtypes of Cartan types implemented correctly?https://ask.sagemath.org/question/44202/are-subtypes-of-cartan-types-implemented-correctly/ Consider the following code
CartanType("B3").subtype([2, 3]).index_set()
CartanType("B3").subtype([1, 3]).index_set()
It gives
(2, 3)
(1, 2)
in Sage 8.4. Is it wrong to expect the index set of a subtype to be the subset used in defining the subtype? From my point of view this inconsistency leads to all kinds of unexpected behaviour in the relabellings for subtypes, but maybe it is intentional?pbelmansTue, 06 Nov 2018 16:38:28 +0100https://ask.sagemath.org/question/44202/WeylCharacterRing and coroots / Dynkin labelshttps://ask.sagemath.org/question/9383/weylcharacterring-and-coroots-dynkin-labels/This might be a stupid question coming from a poor physicist, but there we go:
I would like to work with representations of Lie algebras, having the weights expressed in terms of the times they contain each of the fundamental weights, what we call 'Dynkin labels'. It seems to correspond to the style='coroot' when declaring a WeylCharacterRing: if I want the representation whose highest weight is 3 times the 1st fundamental weight, I ask for
sage: WCR = WeylCharacterRing("A2",style='coroots')
sage: WCR(3,0,0,...,0)
That said, I get:
sage: A2 = WeylCharacterRing("A2",style='coroots')
sage: rep = A2(1,0)
sage: print rep.weight_multiplicities()
{(2/3, -1/3, -1/3): 1, (-1/3, 2/3, -1/3): 1, (-1/3, -1/3, 2/3): 1}
but I would like to get
sage: A2 = WeylCharacterRing("A2",style='coroots')
sage: rep = A2(1,0)
sage: print rep.weight_multiplicities()
{( 1, 0): 1, ( -1, 1): 1, ( 0, -1): 1}
which is the right answer in Dynkin labels.
It would be easy to go from one place to the other if I could get the fundamental weights in the same ambient space as the weights: in this case (2/3, -1/3, -1/3) and (1/3, 1/3, -2/3). But I find no way to get them in general.
From the 1st example:
sage: A2.fundamental_weights()
Finite family {1: (1, 0, 0), 2: (1, 1, 0)}
which is not the answer I would expect, and seems inconsistent to me, since the highest weight of the representation, (2/3, -1/3, -1/3), is precisely the 1st fundamental weight.
Cheers,
JesÃºsTorrado
JesustcWed, 03 Oct 2012 17:17:58 +0200https://ask.sagemath.org/question/9383/