source: git/Singular/LIB/qmatrix.lib @ 1b2216

//
version="$Id$";
category="Noncommutative";
info="
LIBRARY: qmatrix.lib     Quantum matrices, quantum minors and symmetric groups
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:
quantMat(n, [p]);          generates the quantum matrix ring of order n;
qminor(u, v, nr);          calculate a quantum minor of a quantum matrix

SymGroup(n);                generates an intmat containing S(n), each row is an element of S(n)
LengthSymElement(v);        calculates the length of the element v of S(n)
LengthSym(M);               calculates the length of each element of M, being M a subset of S(n)
";

LIB "ncalg.lib";
LIB "nctools.lib";  // for rootofUnity
///////////////////////////////////////////////////////////////////////////////

proc SymGroup(int n)
"USAGE:   SymGroup(n); n an integer (positive)
RETURN:  intmat
PURPOSE: represent the symmetric group S(n) via integer vectors (permutations)
NOTE:    each row of the output integer matrix is an element of S(n)
SEE ALSO: LengthSym, LengthSymElement
EXAMPLE: example SymGroup; shows examples
"{
  if (n<=0)
  {
    "n must be positive";
    intmat M[1][1]=0;
  }
  else
  {
    if (n==1)
    {
      intmat M[1][1]=1;
    }
    else
    {
      def N=SymGroup(n-1); // N is the symmetric group S(n-1)
      int m=nrows(N); // The order of S(n-1)=(n-1)!
      intmat M[m*n][n]; // Matrix to save S(n), m*n=n*(n-1)!=n!=#S(n)
      int i,j,k;
      for (i=1; i<=m; i++)
      { // fixed an element i of S(n-1)
        for (j=n; j>0; j--)
        { // and fixed a position j to introduce an "n"
          for (k=1; k<j; k++)
          { // we want to copy the i-th element until position j-1
            M[n*(i-1)+(n-j+1),k]=N[i,k];
          }
          M[n*(i-1)+(n-j+1),j]=n; // we write the "n" in the position j
          for (k=j+1; k<=n; k++)
          {
            M[n*(i-1)+(n-j+1),k]=N[i,k-1]; // and we run until the end of copying
          }
        }
      }
    }
  }
  return (M);
}
example
{
  "EXAMPLE:";echo=2;
  //  "S(3)={(1,2,3),(1,3,2),(3,1,2),(2,1,3),(2,3,1),(3,2,1)}";
  SymGroup(3);
}

///////////////////////////////////////////////////////////////////////////////

// This procedure calculates the length of an element v of a symmetric group
// If size(v)=n, the group is S(n). The permutation is i->v[i].

proc LengthSymElement(intvec v)
"USAGE:  LengthSymElement(v); v intvec
RETURN:  int
PURPOSE: determine the length of the permutation given by v in some S(n)
ASSUME:  v represents an element of S(n); otherwise the output may have no sense
SEE ALSO: SymGroup, LengthSym
EXAMPLE: example LengthSymElement; shows examples
"{
  int n=size(v);
  int l=0;
  int i,j;
  for (i=1; i<n; i++)
  {
    for (j=i+1; j<=n; j++)
    {
      if (v[j]<v[i]) {l++;}
    }
  }
  return (l);
}
example
{
  "EXAMPLE:";echo=2;
  intvec v=1,3,4,2,8,9,6,5,7,10;
  LengthSymElement(v);
}

///////////////////////////////////////////////////////////////////////////////

proc LengthSym(intmat M)
"USAGE:   LengthSym(M); M an intmat
RETURN:  intvec
PURPOSE: determine a vector, where the i-th element is the length of the permutation of S(n) given by the i-th row of M
ASSUME: M represents a subset of S(n) (each row must be an element of S(n)); otherwise, the output may have no sense
SEE ALSO: SymGroup, LengthSymElement
EXAMPLE: example LengthSym; shows examples
"{
  int n=ncols(M); // this n is the n of S(n)
  int m=nrows(M); // m=num of element of S(n) in M, if M=S(n) m=n!
  intvec L=0;
  int i;
  for (i=1; i<=m; i++)
  {
    L=L,LengthSymElement(intvec(M[i,1..n]));
  }
  L=L[2..size(L)];
  return (L);
}
example
{
  "EXAMPLE:";echo=2;
  def M = SymGroup(3); M;
  LengthSym(M);
}

///////////////////////////////////////////////////////////////////////////////

proc quantMat(int n, list #)
"USAGE:   quantMat(n [, p]); n integer (n>1), p an optional integer
RETURN:  ring (of quantum matrices). If p is specified, the quantum parameter q
@*       will be specialized at the p-th root of unity
PURPOSE: compute the quantum matrix ring of order n
NOTE:    activate this ring with the \"setring\" command.
@*       The usual representation of the variables in this quantum
@*       algebra is not used because double indexes are not allowed
@*       in the variables. Instead the variables are listed by reading
@*       the rows of the usual matrix representation, that is, there
@*       will be n*n variables (one for each entry an n*N generic matrix),
@*       listed row-wise
SEE ALSO: qminor
EXAMPLE: example quantMat; shows examples
"{
  if (n>1)
  {
    int nv=n^2;
    intmat m[nv][nv];
    int i,j;
    for (i=1; i<=nv; i++)
    {
      m[i,nv+1-i]=1;
    }
    int chr = 0;
    if ( size(#) > 0 )
    {
      if ( typeof( #[1] ) == "int" )
      {
        chr = #[1];
      }
    }
    ring @rrr=(0,q),(y(1..nv)),Dp;
    minpoly = rootofUnity(chr);
    matrix C[nv][nv]=0;
    matrix D[nv][nv]=0;
    intvec idyi, idyj;
    for (i=1; i<nv; i++)
    {
      for (j=i+1; j<=nv; j++)
      {
        idyi=itoij(i,n);
        idyj=itoij(j,n);
        if (idyi[1]==idyj[1] || idyi[2]==idyj[2])
        {
          C[i,j]=1/q;
        }
        else
        {
          if (idyi[2]<idyj[2])
          {
            C[i,j]=1;
            D[i,j]=(1/q - q)*y(ijtoi(idyi[1],idyj[2],n))*y(ijtoi(idyj[1],idyi[2],n));
          }
          else
          {
            C[i,j]=1;
          }
        }
      }
    }
    def @@rrr=nc_algebra(C,D);
    return (@@rrr);
  }
  else
  {
    "ERROR: n must be greater than 1";
    return();
  }
}
example
{
  "EXAMPLE:"; echo=2;
  def r = quantMat(2); // generate O_q(M_2) at q generic
  setring r;   r;
  kill r;
  def r = quantMat(2,5); // generate O_q(M_2) at q^5=1
  setring r;   r;
}

///////////////////////////////////////////////////////////////////////////////

proc qminor(intvec I, intvec J, int nr)
"USAGE:        qminor(I,J,n); I,J intvec, n int
RETURN: poly, the quantum minor of a generic n*n quantum matrix
ASSUME: I is the ordered list of the rows to consider in the minor,
@*      J is the ordered list of the columns to consider in the minor,
@*      I and J must have the same number of elements,
@*      n is the order of the quantum matrix algebra you are working with (quantMat(n)).
@*      The base ring should be constructed using @code{quantMat}.
SEE ALSO: quantMat
EXAMPLE: example qminor; shows examples
"{
  poly d=0;
  poly f=0;
  int k=0;
  int n=size(I);
  if ( size(I)!=size(J) )
  {
    "#I must be equal to #J";
  }
  else
  {
    def Sn=SymGroup(n);
    def L=LengthSym(Sn);
    int m=size(L); // m is the order of S(n)
    int i,j;
    for (i=1; i<=m; i++)
    {
      f=(-q)^(L[i]);
      for (j=1; j<=n; j++)
      {
        k=ijtoi(I[j],J[Sn[i,j]],nr);
        f=f*y(k);
      }
      d=d+f;
    }
  }
  return (d);
}
example
{
  "EXAMPLE:";
  echo=2;
  def r = quantMat(3); // let r be a quantum matrix of order 3
  setring r;
  intvec u = 1,2;
  intvec v = 2,3;
  intvec w = 1,2,3;
  qminor(w,w,3);
  qminor(u,v,3);
  qminor(v,u,3);
  qminor(u,u,3);
}

///////////////////////////////////////////////////////////////////////////////

// For tecnical reasons we work with a list of variables {y(1)...y(n^2)}.
// In quantum matrices the usual is to work with a square matrix of variables {y(ij)}.
// The formulas are easier if we use matricial notation.
// The following two procedures change the index of a list to a matrix and viceversa in order
// to use matricial notation in the calculus but use list notation in input and output.
// n is the order of the quantum matrix algebra where we are working.

static proc itoij(int i, n)
{
  intvec ij=0,0;
  ij[1]=((i-1) div n)+1;
  ij[2]=((i-1) mod n)+1;
  return(ij);
}

static proc ijtoi(int i, j, n)
{
  return((j-1)+n*(i-1)+1);
}
///////////////////////////////////////////////////////////////////////////////