FUNCTION: LieTools/HWVsInSgMonoSet
USAGE: 
SYNOPSIS: This is a routine to find the highest weight vectors 
  residing in the span of a set of monomials in S[g]
CAVEATS: requires a previously initialized global Lie algebra environment.
  (See LieTools[gSetup].)
EXAMPLE: 
> gSetup(A,3);
Setting up Gtype = A   rnk = 3
> DecompSg(3,A3);

  XX[3, 0, 3] + XX[1, 2, 1] + XX[2, 0, 2] + XX[0, 1, 2] + XX[2, 1, 0]

         + 2 XX[1, 0, 1] + XX[0, 0, 0]

> mset := fwgt2SgMonoSet([1,0,1],3);

  mset := {g[13] g[5] g[14], g[12] g[7] g[13], g[12] g[8] g[13],

        g[9] g[12] g[13], g[9] g[10] g[14], g[10] g[6] g[15],

        g[10] g[7] g[14], g[10] g[8] g[14], g[11] g[5] g[15],

        g[4] g[12] g[15], g[7] g[8] g[15], g[9] g[7] g[15],

        g[9] g[8] g[15], g[3] g[13] g[15], g[2] g[14] g[15],

                               2            2            2
        g[11] g[10] g[12], g[7]  g[15], g[8]  g[15], g[9]  g[15],

                  2
        g[1] g[15] }

> HWVsInSgMonoSet(mset);

                                                         2
  {2 g[3] g[13] g[15] + 3 g[9] g[10] g[14] + 2 g[1] g[15]

         + g[9] g[8] g[15] - 12 g[13] g[5] g[14] + 6 g[12] g[8] g[13]

         - 3 g[12] g[7] g[13] + 3 g[9] g[12] g[13]

         + 14 g[11] g[5] g[15] + g[7] g[8] g[15] + 2 g[9] g[7] g[15]

         + 2 g[10] g[6] g[15] - 3 g[10] g[7] g[14]

         - 6 g[10] g[8] g[14] + 2 g[2] g[14] g[15]

                                                             2
         + 2 g[4] g[12] g[15] - 12 g[11] g[10] g[12] + 4 g[8]  g[15],

                                   2
        16 g[11] g[10] g[12] + g[9]  g[15] + 16 g[13] g[5] g[14]

         - 8 g[12] g[8] g[13] - 4 g[9] g[10] g[14]

         + 4 g[12] g[7] g[13] - 4 g[9] g[12] g[13]

         - 16 g[11] g[5] g[15] - 2 g[9] g[7] g[15]

                                                         2
         + 4 g[10] g[7] g[14] + 8 g[10] g[8] g[14] + g[7]  g[15]

                 2
         - 4 g[8]  g[15]}

> nops(%);

                                  2


SEE ALSO: 
