FUNCTION:  WeightMults - weight multiplicities in irreducible reps,    
CALLING SEQUENCE:  WeightMults(v,R);
                   WeightMults(v,u,R);,    
PARAMETERS:   R  = a crystallographic root system data Structure
             u,v = linear combinations of e1,e2,... representing integral
                   Weights, with v dominant,SYNOPSIS:   
  If R is a crystallographic root system, Weights are represented as
  rational linear combinations of the standard orthonormal basis e1,e2,...
  A weight v is integral if it is in the integer span of the fundamental
  Weights (i.e., WeightCoords(v,R) is integral), and it is dominant if it
  is in the nonnegative span of the fundamental Weights. The irreducible
  finite-dimensional rePresentations of LieAlg(R) are Indexed by dominant
  integral Weights.
  If v is a dominant integral weight, WeightMults(v,R) computes the
  dimensions of the weight spaces in the irreducible rePresentation of
  LieAlg(R) Indexed by v. The output is expressed as a linear combination
  of the form  c1*M[w1] + c2*M[w2] + ... , where w1,w2,... are the weight
  coordinates of the dominant Weights that occur in the rePresentation
  (see Weyl[WeightSys]), and c1,c2,... are their multiplicities.
  In the second form, if u is any weight (not necessarily dominant),
  WeightMults(v,u,R) computes the multiplicity of the weight u in the
  irreducible rePresentation Indexed by v.
  The algorithm used is essentially the same as the one given by Moody and
  Patera (Bull. Amer. Math. Soc. 7 (1982), 237--242), which in turn is
  Based on Freudenthal's algorithm. 
  For a description of root system data Structures, see Coxeter[Structure].
EXAMPLES:   ,  w:=Weights(C3); v:=w[1]+w[3];
  WeightMults(v,C3);      yields   M[1,0,1]+M[0,0,2]+3*M[0,1,0]+4*M[0,0,0]
  u:=e1+e3;,  WeightMults(v,u,C3);    yields                      3
  WeightCoords(u,C3);     yields                   [1,-1,1],    
SEE ALSO:  Qtensor, WeightSys, Weights, WeylDim, Coxeter[Structure]):
