FUNCTION:  WeylDim - dimension of an irreducible rePresentation,    
CALLING SEQUENCE:  WeylDim(v,R);,                   WeylDim(v,R,q);
    ,PARAMETERS:  R = a (crystallographic) root system data Structure
             v = a linear combination of e1,e2,... representing a dominant
                 integral weight
             q = (optional) a variable or expression,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 any vector (not necessarily a dominant or integral weight),
  WeylDim(v,R) returns the Weyl product
                                Iprod(r,v+Rho(R))
                           prod -----------------,
                            r    Iprod(r,Rho(R))
  where r ranges over PosRoots(R). If v is a dominant integral weight,
  this is the dimension of the irreducible rePresentation of LieAlg(R)
  Indexed by v.
  If a third argument q is present, then the result is a "q-analogue" of
  the Weyl dimension formula; namely,
           -Iprod(CoRho(R),v)        1-q^Iprod(Co(r),v+Rho(R))
          q                    * prod -------------------------,
                                  r    1-q^Iprod(Co(r),Rho(R))  
  where Co(r) = 2*r/Iprod(r,r) denotes the co-root corresponding to r.
  This can be viewed as a formal substitution of q^CoRho(R) into the Weyl
  character of the rePresentation Indexed by v. As a formal series in
  q^(1/2), the coefficients represent weight multiplicities for the
  principal embedding of sl_2 in LieAlg(R).
  For a description of root system data Structures, see Coxeter[Structure].
EXAMPLES:   
  WeylDim(Weights(F4)[2],F4);    yields                273
  w:=Weights(C2); v:=expand(a*w[1]+b*w[2]);
  normal(WeylDim(v,C2));         yields  1/6*(1+a)*(1+b)*(2+a+b)*(3+2*a+b)
  WeylDim(2*e2+e1,B2,q);         yields   q^(-5)*(1-q^7)*(1-q^5)/(1-q)^2
    ,SEE ALSO:  CoRho, Rho, WeightCoords, Weights, Coxeter[Base],
           Coxeter[Iprod], Coxeter[PosRoots], Coxeter[Structure]):
