source: git/Singular/LIB/qmatrix.lib @ cfc9c0

version="$Id: qmatrix.lib,v 1.9 2005-04-28 16:32:06 levandov Exp $";
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).

MAIN PROCEDURES:
quant(n, [p]);          generates the quantum matrix ring of order n; 
qminor(u, v, nr);       calculate a quantum minor of a quantum matrix

AUXILIARY PROCEDURES:
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";
///////////////////////////////////////////////////////////////////////////////

proc SymGroup(int n)
"USAGE:   SymGroup(n); n an integer, n>0
PURPOSE: present the symmetric group S(n) via integer vectors
RETURN:  intmat
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 an intvec representing an element of S(n)
PURPOSE: determine the length of v
RETURN:  int
NOTE:    if v doesn't represent an element of S(n), 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 representing a subset of S(n) (each row must be an element of S(n))
PURPOSE: determine a vector, which i-th element is the length of the i-th row of M
RETURN:  intvec 
NOTE:    If M is not a subset of S(n), the output may not have meaning
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 quant(int n, list #)
"USAGE:   quant(n [, p]); n integer (n>1), p an optional integer
PURPOSE: compute the quantum matrix ring of order n; 
RETURN:  ring (of quantum matrices). If p is specified, the quantum parameter q 
@*       will be specialized at p-th root of unity
NOTE:    You have to 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 reading
@*       the rows of the usual matrix representation.
SEE ALSO: qminor
EXAMPLE: example quant; 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;
          }
        }
      }
    }
    ncalgebra(C,D);
    return (@rrr);
  } 
  else 
  {
    "ERROR: n must be greater than 1";
    return();
  }
} 
example 
{
  "EXAMPLE:"; echo=2;
  def r=quant(2); // generate O_q(M_2) at q generic 
  setring r;   r;
  kill r;
  def r=quant(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,nr); where 
@*      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 and 
@*      nr is the order of the quantum matrix algebra you are working with (quant(nr)).
RETURN: poly, the quantum minor.
NOTE:    I and J must have the same number of elements.
SEE ALSO: quant
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: Let r be the quantum matrix of order 3."; echo=2;
  def r=quant(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);
}

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