FUNCTION: LieTools/HLWVsInSgMonoSet
USAGE: HLWVsInSgMonoSet(mset);
SYNOPSIS: This is a routine to find the lowest harmonic weight vectors 
  residing in the span of a set mset of monomials in S[g] of fixed degree
  and weight.
CAVEATS: requires a previously initialized global Lie algebra environment.
  (See LieTools[gSetup].)
EXAMPLE:
> gSetup(G,2);
  Setting up Gtype = G   rnk = 2
> DecompSg(3,G2);

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

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

                                    2                           2
  mset := {g[4] g[6] g[7], g[3] g[8] , g[4] g[6] g[8], g[5] g[6] ,

                                  2
        g[1] g[6] g[12], g[3] g[7] , g[3] g[7] g[8], g[3] g[6] g[9],

        g[2] g[7] g[9], g[2] g[8] g[9], g[4] g[2] g[12],

        g[1] g[9] g[10], g[5] g[3] g[10], g[4] g[3] g[11],

        g[5] g[2] g[11], g[1] g[7] g[11], g[1] g[8] g[11],

                                              2            2
        g[3] g[2] g[13], g[3] g[1] g[14], g[3]  g[12], g[4]  g[10]}

> HLWVsInSgMonoSet(mset);

                                 2                                2
  {3 g[4] g[6] g[8] - 3 g[5] g[6]  - 3 g[1] g[6] g[12] + g[3] g[7]

         + 3 g[3] g[7] g[8] + g[3] g[6] g[9] - 3 g[2] g[7] g[9]

         - 9 g[2] g[8] g[9] - 3 g[4] g[2] g[12] + 9 g[1] g[9] g[10]

         - 9 g[5] g[3] g[10] + g[4] g[3] g[11] - 9 g[5] g[2] g[11]

                                   2               2
         + 3 g[1] g[7] g[11] + g[3]  g[12] + 3 g[4]  g[10]}

SEE ALSO: LieTools[HWVsInSgMonoSet], LieTools[HHWVsInSgWgtSet],
          LieTools[HLWVsinSgWgtSet], LieTools[fwgt2SgMonoSet]
