# 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.

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You can get the coroot vector components from the inner product with the simple coroots, of course:

sage: B5 = WeylCharacterRing('B5',style='coroots')
sage: Rep = 2*B5(1,0,0,0,0) + B5(0,1,2,3,4)
sage: Rep.degree()    # dimension
3777283147
sage: for highest_weight, multiplicity in Rep:
....:     coroots = [ highest_weight.inner_product(coroot)
....:                 for coroot in list(B5.simple_coroots()) ]
....:     print coroots, highest_weight, multiplicity
....:
[1, 0, 0, 0, 0] (1, 0, 0, 0, 0) 2
[0, 1, 2, 3, 4] (8, 8, 7, 5, 2) 1


Note that I used a more complicated group where simple roots do not coincide with the simple coroot vectors as in SU(4).

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I answer myself, just in case someone finds this and have the same problem:

My bad: the solution to the "strange" fundamental weights is explained in http://www.sagemath.org/doc/thematic_tutorials/lie/weyl_character_ring.html#sl-versus-gl.

Following the example:

sage: [fw.coerce_to_sl() for fw in A2.fundamental_weights()]
[(2/3, -1/3, -1/3), (1/3, 1/3, -2/3)]


Anyway, I am considering proposing to add a style='coroots' option to weight_multiplicities() to get the output in 'Dynkin labels'.

Cheers!

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