Multiplicity of a particular weight

asked 2012-12-28 02:04:35 -0600

emiliocba gravatar image

Given $V$ an irreducible representation of some compact group, I would like to have the multiplicity of some particular weight $\eta$, that is, $\dim (V(\eta))$.

For example,

D=WeylCharacterRing("D3")
D(2,1,0).weight_multiplicities()

gives

{(0, 1, 0): 4, (1, -1, 1): 2, (0, 2, -1): 1, (1, 2, 0): 1, (-1, 0, 2): 1, (0, -2, -1): 1, (0, -1, 0): 4, (0, 2, 1): 1, (1, -1, -1): 2, (0, 1, -2): 1, (-1, 2, 0): 1, (-2, 0, 1): 1, (-1, 1, -1): 2, (0, -1, 2): 1, (-1, -2, 0): 1, (-1, -1, -1): 2, (-2, 1, 0): 1, (1, 0, 0): 4, (-1, 0, -2): 1, (-2, -1, 0): 1, (1, 0, -2): 1, (0, 1, 2): 1, (0, 0, -1): 4, (0, -2, 1): 1, (2, 0, 1): 1, (1, 1, -1): 2, (0, 0, 1): 4, (1, 0, 2): 1, (2, 1, 0): 1, (2, -1, 0): 1, (-2, 0, -1): 1, (0, -1, -2): 1, (-1, 1, 1): 2, (-1, -1, 1): 2, (2, 0, -1): 1, (-1, 0, 0): 4, (1, 1, 1): 2, (1, -2, 0): 1}

the list of multiplicities of every weight of the representation $V$ (=the irreducible representation of $SO(6)$ of highest weight $(2,1,0)=2 e_1+ e_2$). I want only the multiplicity of some particular weight, for example FOR $\eta=(1,1,1)$, thus $2$.

Thanks in advance.-.

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Comments

Just a suggestion of how to go about it, there should be a better (and perhaps simpler) way of getting the multiplicity. If you define the following dictionary wm = dict([ (tuple(x.to_vector()),y) for x,y in D(2,1,0).weight_multiplicities().items()]) then you can get the multiplicity of (1,1,1) with wm[(1,1,1)]

fidbc gravatar imagefidbc ( 2012-12-28 05:40:23 -0600 )edit