Multiplicity of a particular weight
 
  
 
  
 
 
 
 
 
 
    
        
        asked 12 years ago  
 
 emiliocba gravatar image  
 
 
 
 
 
Given V an irreducible representation of some compact group, I would like to have the multiplicity of some particular weight η, that is, dim(V(η)).
 
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)=2e1+e2). I want only the multiplicity of some particular weight, for example FOR η=(1,1,1), thus 2.
 
Thanks in advance.-.
 
Preview: (hide)
 
 
 
 
          
 
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 (
    
        12 years ago
    )
add a comment