Weight Rings

You may wish to work directly with the weights of a representation.

Weyl character ring elements are represented internally by a dictionary of their weights with multiplicities. However these are subject to a constraint: the coefficients must be invariant under the action of the Weyl group.

The WeightRing is also a ring whose elements are represented internally by a dictionary of their weights with multiplicities, but it is not subject to this constraint of Weyl group invariance. The weights are allowed to be fractional, that is, elements of the ambient space. In other words, the weight ring is the group algebra over the ambient space of the weight lattice.

To create a WeightRing first construct the WeylCharacterRing, then create the WeightRing as follows:

sage: A2 = WeylCharacterRing(['A',2])
sage: a2 = WeightRing(A2)

You may coerce elements of the WeylCharacterRing into the weight ring. For example, if you want to see the weights of the adjoint representation of \(GL(3)\), you may use the method mlist, but another way is to coerce it into the weight ring:

sage: from pprint import pprint
sage: A2 = WeylCharacterRing(['A',2])
sage: ad = A2(1,0,-1)
sage: pprint(ad.weight_multiplicities())
{(0, 0, 0): 2, (-1, 1, 0): 1, (-1, 0, 1): 1, (1, -1, 0): 1,
 (1, 0, -1): 1, (0, -1, 1): 1, (0, 1, -1): 1}

This command produces a dictionary of the weights that appear in the representation, together with their multiplicities. But another way of getting the same information, with an aim of working with it, is to coerce it into the weight ring:

sage: a2 = WeightRing(A2)
sage: a2(ad)
2*a2(0,0,0) + a2(-1,1,0) + a2(-1,0,1) + a2(1,-1,0) + a2(1,0,-1) + a2(0,-1,1) + a2(0,1,-1)

For example, the Weyl denominator formula is usually written this way:

\[\prod_{\alpha\in\Phi^+}\left(e^{\alpha/2}-e^{-\alpha/2}\right) = \sum_{w\in W} (-1)^{l(w)}e^{w(\rho)}.\]

The notation is as follows. Here if \(\lambda\) is a weight, or more generally, an element of the ambient space, then \(e^\lambda\) means the image of \(\lambda\) in the group algebra of the ambient space of the weight lattice \(\lambda\). Since this group algebra is just the weight ring, we can interpret \(e^\lambda\) as its image in the weight ring.

Let us confirm the Weyl denominator formula for A2:

sage: A2 = WeylCharacterRing("A2")
sage: a2 = WeightRing(A2)
sage: L = A2.space()
sage: W = L.weyl_group()
sage: rho = L.rho().coerce_to_sl()
sage: lhs = prod(a2(alpha/2)-a2(-alpha/2) for alpha in L.positive_roots()); lhs
a2(-1,1,0) - a2(-1,0,1) - a2(1,-1,0) + a2(1,0,-1) + a2(0,-1,1) - a2(0,1,-1)
sage: rhs = sum((-1)^(w.length())*a2(w.action(rho)) for w in W); rhs
a2(-1,1,0) - a2(-1,0,1) - a2(1,-1,0) + a2(1,0,-1) + a2(0,-1,1) - a2(0,1,-1)
sage: lhs == rhs
True

Note that we have to be careful to use the right value of \(\rho\). The reason for this is explained in SL versus GL.

We have seen that elements of the WeylCharacterRing can be coerced into the WeightRing. Elements of the WeightRing can be coerced into the WeylCharacterRing provided they are invariant under the Weyl group.