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Weight Rings
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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 :math:`GL(3)`, you may use the method
:meth:`mlist`, but another way is to coerce it into the
weight ring::

    sage: ad = A2(1,0,-1)
    sage: ad.mlist()
    [[(-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]]
    sage: a2(ad)
    a2(-1,0,1) + a2(-1,1,0) + a2(0,-1,1) + 2*a2(0,0,0) + a2(0,1,-1) + a2(1,-1,0) + a2(1,0,-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 :math:`\lambda` is a weight, or more generally,
an element of the ambient space, then :math:`e^\lambda` means the image of :math:`\lambda`
in the group algebra of the ambient space of the weight lattice :math:`\lambda`.
Since this group algebra is just the weight ring, we can interpret
:math:`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 = A2.coerce_to_sl(L.rho())
    sage: lhs = prod(a2(alpha/2)-a2(-alpha/2) for alpha in L.positive_roots()); lhs
    -a2(-1,0,1) + a2(-1,1,0) + a2(0,-1,1) - a2(0,1,-1) - a2(1,-1,0) + a2(1,0,-1)
    sage: rhs = sum((-1)^(w.length())*a2(w.action(rho)) for w in W); rhs
    -a2(-1,0,1) + a2(-1,1,0) + a2(0,-1,1) - a2(0,1,-1) - a2(1,-1,0) + a2(1,0,-1)
    sage: lhs == rhs
    True

Note that we have to be careful to use the right value of :math:`\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.