%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% $Id: RQuasiSymmetricFunctions.tex,v 1.1.1.1 2004/10/04 17:25:13 nthiery Exp $
% File:		lib/EXPERIMENTAL/DOC/RQuasiSymmetricFunctions.tex
% Authors:	Nicolas M. Thiéry <nthiery@users.sourceforge.net>
% Purpose:	Documentation for the domain experimental::RQuasiSymmetricFunctions.mu
% Created:	Mon Apr 19 18:12:16 2004
% License:	LGPL
% Project:	MuPAD-Combinat, http://mupad-combinat.sf.net
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{Mhelppage}{RQuasiSymmetricFunctions}{Nicolas M. Thiéry}
  [ImplState=beta,TestState=beta,DocState=alpha]
%\Mproofread{user}{date}{comment}

\begin{MInfo}
  \eng{The domain of R-quasi symmetric functions}
  \de{}
\end{MInfo}

\begin{MIntro}
  \eng{\Mthis{} represents The domain of R-quasi symmetric functions.}
  \de{}
\end{MIntro}

\begin{MChanges}
  \version{3.1}
  \eng[New domain]{\Mthis{} is a new domain} %
\end{MChanges}

For 2-QSym on 3 variables is not Cohen-Macaulay for the chain product:
\begin{Mexin}
r:=2: n:=3:
M:=experimental::RQuasiSymmetricFunctions(r,n):
for d from 0 to 15 do
    for p in combinat::partitions::list(d, MaxPart=n) do
         prog::test(M::checkSymDimension(p),M::dimension(p));
         dim := M::SymDimension(p);
         if dim<>0 then
             print(p, dim)
         end_if;
    end_for;
end_for:
\end{Mexin}
\begin{Mexout}
                             [], 1

                         [2, 2, 1], 1

                         [3, 2, 1], 1

                         [3, 3, 1], 2

                         [3, 3, 2], 2

                       [3, 2, 2, 1], -1
\end{Mexout}

Indeed, we have the following relation with the chain product:
\begin{displaymath}
  m321 * e2 = m431 = m32 * e3
\end{displaymath}
Reciprocally, in degree (3,3,2) there is just one decomposable
element, namely \Mex{e3^2*e2}. So, we need to add two generators among
m332, m323, and m233.

One can get a nice picture of this by drawing the lattice under 332
and 3221. To check the dimension:

\begin{Mexin}
r:=2: n:=3: part := [3,2,2,1]:
M:=experimental::RQuasiSymmetricFunctions(r,n):
for d from 0 to _plus(op(part)) do
    for p in combinat::submultisets(part,d) do
         print(p, M::dimension(p),M::SymDimension(p));
    end_for
end_for:
\end{Mexin}
\begin{Mexout}
                           [], 1, 1

                           [3], 1, 0

                           [2], 1, 0

                           [1], 1, 0

                         [3, 2], 1, 0

                         [3, 1], 1, 0

                         [2, 2], 1, 0

                         [2, 1], 1, 0

                        [3, 2, 2], 1, 0

                        [3, 2, 1], 2, 1

                        [2, 2, 1], 2, 1

                      [3, 2, 2, 1], 2, -1
\end{Mexout}

This gives an example in 3 variables of non Cohen-Macaulay
"equivalence relation style" ring for the chain product.

%\begin{question}
  Is 2-QSym on 3 variables Cohen-Macaulay for the usual product?
%\end{question}
Probably. At least, the relation above does not exists in the
polynomials: here we check that there is no relation at degree 8 in
the Sym-module generated by m0, m231 and m23:

\begin{Mexin}
P:=examples::MultivariatePolynomials([x,y,z]):
S:=examples::SymmetricFunctions():
m0   := P::fromExpr(1):
m321 := P::fromExpr(x^3*y^2*z + x^3*y*z^2 + x*y^3*z^2):
m32  := P::fromExpr(x^3*y^2   + x^3  *z^2 +   y^3*z^2):
e[1] := P::fromExpr(S::e([1])(P::variables)):
e[2] := P::fromExpr(S::e([2])(P::variables)):
e[3] := P::fromExpr(S::e([3])(P::variables)):
basis := [_mult(x, e[i] $ i in p)
          $ p in combinat::partitions::list(8-degree(x), MaxPart=3)
          $ x in [m0,m321,m32] ]:
mat := _concat(P::sparseVector(x) $ x in basis):
prog::test(linalg::rank(mat), nops(basis)):
\end{Mexin}
\end{Mhelppage}

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