source: git/Singular/LIB/gkdim.lib @ 341696

///////////////////////////////////////////////////////////////////////////////
version="$Id$";
category="Noncommutative";
info="
LIBRARY: gkdim.lib     Procedures for calculating the Gelfand-Kirillov dimension
AUTHORS: Lobillo, F.J.,     jlobillo@ugr.es
@*         Rabelo, C.,        crabelo@ugr.es

SUPPORT: 'Metodos algebraicos y efectivos en grupos cuanticos', BFM2001-3141, MCYT, Jose Gomez-Torrecillas (Main researcher).

PROCEDURES:
  GKdim(M);        Gelfand-Kirillov dimension computation of the factor-module, whose presentation is given by the matrix M.
";

///////////////////////////////////////////////////////////////////////////////////
static proc idGKdim(ideal I)
"USAGE:   idGKdim(I), I is a left ideal
RETURN:  int, the Gelfand-Kirillov dimension of the R/I
NOTE: uses the dim procedure, if the factor-module is zero, -1 is returned
"
{
  if (attrib(I,"isSB")<>1)
  {
    I=std(I);
  }

  int d    = dim(I);
  //  if (d==-1) {d++;} // The GK-dimension of a finite dimensional module is zero
  // levandov: but for consistency, GKdim(std(1)) == -1,
  //           mimicking the behaviour of dim() procedure.
  return (d);
}

///////////////////////////////////////////////////////////////////////////////
proc GKdim(list L)
"USAGE:   GKdim(L);   L is a left ideal/module/matrix
RETURN:  int
PURPOSE: compute the Gelfand-Kirillov dimension of the factor-module, whose presentation is given by L, e.g. R^r/L
NOTE:  if the factor-module is zero, -1 is returned
EXAMPLE: example GKdim; shows examples
"
{
  def M = L[1];
  int d = -1;
  if (typeof(M)=="ideal")
  {
    d=idGKdim(M);
  }
  else
  {
    if (typeof(M)=="matrix")
    {
      module N = module(M);
      kill M;
      module M = N;
    }
    if (typeof(M)=="module")
    {
      if (attrib(M,"isSB")<>1)
      {
        M=std(M);
      }
      int n = ncols(M); // Num of vectors defining M
      int m = nrows(M); // The rank of the free module where M is imbedded
      int i,j;
      for (j=1; j<=n; j++)
      {
        M[j] = leadmonom(M[j]); // Only consider the leader monomial of each vector
      }
      intmat v[1][m]; // v will be the dimension of each stable subset
      ideal I;
      for (i=1; i<=m; i++)
      {
        I=0;
        for (j=1; j<=n; j++)
        { // Extract each row like an ideal .LV: ????
          I=I, M[i,j];
        }
        v[1,i] = idGKdim(I);
        if (v[1,i]>d) { d = v[1,i]; }
      }
    }
    else
    {
      ERROR("The input must be an ideal, a module or a matrix.");
    }
  }
  return (d);
}
example
{
  "EXAMPLE:";echo=2;
  ring R = 0,(x,y,z),Dp;
  matrix C[3][3]=0,1,1,0,0,-1,0,0,0;
  matrix D[3][3]=0,0,0,0,0,x;
  def r = nc_algebra(C,D); setring r;
  r;
  ideal I=x;
  GKdim(I);
  ideal J=x2,y;
  GKdim(J);
  module M=[x2,y,1],[x,y2,0];
  GKdim(M);
  ideal A = x,y,z;
  GKdim(A);
  ideal B = 1;
  GKdim(B);
  GKdim(ideal(0)) == nvars(basering);  // should be true, i.e., evaluated to 1
}
///////////////////////////////////////////////////////////////////////////////
proc gkdim(list L)
{
  return(GKdim(L));
}
///////////////////////////////////////////////////////////////////////////////