Here are some comments on #10744.
Very nice!
I do have some comments.
sage: A2=WeylCharacterRing("A2",style="coroots")
sage: a2=WeightRing(A2)
sage: rp=a2(A2(2,1))
sage: rp.plot()
Frequently one will want to call this for the character of
a representation. It would be good if one could:
A2(2,1).plot()
without having to create the weight ring. Of course some
times one might also want to be able to call it on an
element of the weight ring that is not W invariant (and
which could therefore not be coerced into the WeylCharacterRing).
I guess to accomplish this one could have to move some of the code
into an auxiliary method of the WeylCharacterRing that would be
called by plot methods of both the Weyl character ring and the
weight ring.
The plot that I like best is:
rp.plot(weight_lattice=false)
So maybe the weight lattice could be suppressed by default.
Here is a more serious criticism.
A grid is shown whose vertices are the weights.
But this grid is not invariant under the action of the Weyl group!
Although the grid that you've shown does pass through the weight
lattice, it is sort of unnatural for this reason. The grid
you've shown is one whose fundamental domain is the
parallelogram spanned by the fundamental weights, but since it is
not W invariant I think it is bad to use this grid.
A more natural and useful grid would be the one that passes through
the hyperplanes where the coroots take integer values. This is the
Stieffel diagram for the dual weight root system. Like the
grid you showed it is a grid whose vertices are the weights, but
unlike the one your code displays it is W-invariant.
It would also be good to be able to see the hyperplanes where the
roots take integer values. This is the Stieffel diagram. Then
you can see pictures of the fundamental alcoves.
You can see Stieffel diagrams on page 227 of Brocker and Tom
Dieck's book Representations of Compact Lie Groups. This happens to
be one that is displayed by Google Books. (Google search for
Stieffel Diagram.)
It would be good to be able to see these grids, but not by default,
since I think they make things too cluttered. Since I'm proposing
two optional methods showing the fundamental alcoves and dual
alcoves, maybe the name should be changed.
It seems to me (in the A2 case) that the root arrows are perfectly
placed, but the pink arrows corresponding to the fundamental
weights fall short of where they should. Especially the second
fundamental weight which points straight up. And maybe the pink
arrows are too thick.
It seems to me that the pictures are wider than tall. This is
particularly obvious for B2 and C2 since the lattices are
supposedly square.
I looked at the B2,C2 and G2 pictures of irreducible
characters. For G2 the weight multiplicities get large fast, and
I'm not sure whether some tuning of the default dot size might be
profitable, at least for G2. One idea would be that the dot size
could be proportional to the square root of the multiplicity.
(I'm not sure how well this would work, just an idea.)
Dan
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