# 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, JesúsTorrado