question about weyl_dimension

[Martin R]

I am confused about the following result:

sage: ct = CartanType(["D", 5])
sage: L = RootSystem(ct).weight_lattice()
sage: La = L.fundamental_weights();
sage: [L.weyl_dimension(wt) for wt in La]
[10, 45, 120, 16, 16]
sage: [crystals.HighestWeight(wt).q_dimension().subs(q=1) for wt in La]
[10, 45, 120, 210, 126]

Am I doing something wrong or is this a bug?  I would expect the former dimensions, but I am not versed enough in the theory to be sure that I did not make a mistake myself.  The discrepancy seems to occur only with type D.

All the best,

Martin

[dwb...@gmail.com]

It looks like Sage is constructing the wrong crystals for the two spin weights. D5 does have

two irreducible representations of dimensions 210 and 126 but their highest weights are not

the spin weights s1=(1/2,1/2,1/2,1/2,1/2) and s2=(1/2,1/2,1/2,1/2,-1/2) which are the two

fundamental weights. Instead the highest weights of the representations with these degrees

210 and 126 are s1+s2 and 2*s1.

This appears to me to be a bug in the crystal code.

Daniel Bump

[Martin R]

Thank you, that explains it to me!

Currently, the code that translates dominant weights to crystals is (line 174 of highest_weight_crystals.py)

        sh = sum([[i]*c for i,c in dominant_weight], [])
        sh = Partition(reversed(sh))
        return CrystalOfTableaux(cartan_type, shape=sh.conjugate())

So I guess there might also be a less visible problem for type B, and in the case of the spin weights we should rather be returning CrystalOfSpins, CrystalOfSpinsPlus, CrystalOfSpinsMinus.

So, maybe we should use dominant.to_ambient?

Martin

[dwb...@gmail.com]

You can construct the two spin crystals using crystals.SpinsMinus("D5") and crystals.SpinsPlus("D5").

If you want other spin weights, you can tensor these crystals with an orthogonal crystal. But

the method you tried should either work or fail gracefully instead of returning a wrong result.

Daniel Bump

[Martin R]

This is now https://trac.sagemath.org/ticket/31391

[Martin R]

I wonder whether there is already a function that transform a weight in `C.weight_lattice_realization` (which is the ambient space of the root system) into the corresponding tableau.

Otherwise, this probably has to be done case by case.