------------ Weight Rings ------------ .. linkall You may wish to work directly with the weights of a representation. WeylCharacterRingElements 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 Weyl Character Ring, 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: A2 = WeylCharacterRing(['A',2]) sage: ad = A2(1,0,-1) sage: ad.weight_multiplicities() {(-1, 1, 0): 1, (0, 1, -1): 1, (1, 0, -1): 1, (0, 0, 0): 2, (-1, 0, 1): 1, (0, -1, 1): 1, (1, -1, 0): 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: .. MATH:: \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 :ref:SLvsGL. 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.