[GAP Forum] simple Lie algebra questions

[R.N. Tsai]

I have two further question on simple Lie algebras :
 (1) How do you get all the weights of a representation; I found the function that
     provides the dominant weights, but I'd like to get an explicit list of all.
     The total number of such weights (including multiplicity) would be the dimension
     of the rep and their ordering is hopefully related to the ordering of the basis
     vectors of the rep.
 (2) "BilinearFormMat" returns a matrix that has the inner product of the roots <ai,aj>
     for example :
     L:=SimpleLieAlgebra("F",4,Rationals);
     R:=RootSystem(L);
     C:=CartanMatrix(R);
     K:=BilinearFormMat(R);
     PrintArray(C);
     [ [   2,   0,  -1,   0 ],
       [   0,   2,   0,  -1 ],
       [  -1,   0,   2,  -1 ],
       [   0,  -1,  -2,   2 ] ] 
     cartan matrix looks familar
     PrintArray(K);
    [ [   1/18,      0,  -1/36,      0 ],
      [      0,    1/9,      0,  -1/18 ],
      [  -1/36,      0,   1/18,  -1/18 ],
      [      0,  -1/18,  -1/18,    1/9 ] ]
     this doesn't look too familiar, but
    PrintArray(18*K);
   [ [     1,     0,  -1/2,     0 ],
     [     0,     2,     0,    -1 ],
     [  -1/2,     0,     1,    -1 ],
     [     0,    -1,    -1,     2 ] ]
   with the factor 18 this is a more familiar normalization of the root lengths
   (other algebras have different factors). There's probably a good reason for
   normalizing the root lengths this way; I'd like to know that if possible.
Thanks,
R.N.

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