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% $Id: YangBaxterGraph.tex,v 1.1.1.1 2004/10/04 17:25:13 nthiery Exp $
% File:		lib/EXPERIMENTAL/DOC/YangBaxter.tex
% Authors:	Nicolas M. Thiéry <nthiery@users.sourceforge.net>
% Purpose:	Documentation for the domain experimental::YangBaxter
% Created:	Tue Jul  6 10:38:35 2004
% License:	LGPL
% Project:	MuPAD-Combinat, http://mupad-combinat.sf.net
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\begin{Mhelppage}{YangBaxterGraph}{Nicolas M. Thiéry}
  [ImplState=beta,TestState=beta,DocState=beta]
%\Mproofread{user}{date}{comment}

\begin{MInfo}
  \eng{domain for Yang-Baxter graphs}
  \de{}
\end{MInfo}

\begin{MIntro}
  \eng{\Mthis{top, operators\Mopt{,
        SpectralParameters}=parameters\Mopt{, Values=values}}
    represents the Yang Baxter graph with top element \Mex{top},
    and operators \Mex{operators}.}

  \de{}
\end{MIntro}

\begin{MDetails}
  \eng{Let $t$ be some object, and $o_1,\dots, o_{n-1}$ be operators
    that can be repeatedly applied on $t$. Each sequence
    $i,j,k,\dots$, gives thus rise to an object that we write as $t
    o_io_jo_k\dots$ (action on the right, most of the time to model an
    action on positions). We further assume that the operators satisfy
    the braid relations $o_i \circ o_{i+1} \circ o_i = o_{i+1} \circ
    o_i \circ o_{i+1}$ and $o_io_j = o_j o_i$ for $j\geq i+2$, as well
    as quadratic relations such as $o_i^2=\lambda o_i + \mu \operatorname{id}$. In
    that case, the image $t^{o_io_jo_k\dots}$ can essentially be
    indexed by the permutation $\sigma=s_is_js_k\dots$, where $s_i$
    denotes the $i-th$ elementary transposition. We denote this image
    by $t^\sigma$.}
  
  \eng{The Yang-Baxter graph gives is the graph whose vertices are
    labelled by the images $t^\sigma$, and edges labelled by the
    operators.}
  
  \eng{In most input and output methods of this domain, permutations
    are represented by their lexicographically minimal reduced word
    (see \Mup{combinat::permutations::reducedWordLexMin}); some of the
    method do not require the input word to be reduced.}

  \eng{The spectral parameters $[x_1,\dots,x_n]$ allow to deform the
    operators. Namely, the image of $t^\sigma$ by the $i-th$ operator
    is given by $t^{\sigma s_i} = t^\sigma (o_i+\frac{1}{x_{\sigma(i)}
      - x_{\sigma(i+1)}})$. }
  
  \eng{The value parameter $w$ allow to restrict to a subgraph of the
    full permutahedron to take into symmetries account. Namely,
    $t^{\sigma, i}$ is only computed if $Y^\sigma$ is, and the
    transposition $s_i$ makes the word $w \sigma$ strictly smaller.}
\end{MDetails}

\begin{MExample}{ex-basic1}
  \eng{As first example, we construct the Yang Baxter graph with top
    element \Mex{[a,b,c]}, and the two elementary transpositions
    \Mex{s1} and \Mex{s2} as operators. The function
    \Mex{makeOperator} is used to build those operators:}
  \begin{Mexin}
makeOperator :=
proc(i)
    option escape;
begin
    w -> combinat::words::swap(w,i);
end:
s1 := makeOperator(1):
s2 := makeOperator(2):
YB := experimental::YangBaxterGraph([a,b,c], [s1,s2]):
  \end{Mexin}
  \begin{Mexout}
  \end{Mexout}

  \eng{\Mex{s1} and \Mex{s2} are just usual \mupad{} functions. For example:}
  \begin{Mexin}
s1([a, b, c])
  \end{Mexin}
  \begin{Mexout}
                           [b, a, c]
  \end{Mexout}
  They furthermore satisfy the braid relations:
  \begin{Mexin}
s1(s2(s1([a, b, c]))) = s2(s1(s2([a, b, c])))
  \end{Mexin}
  \begin{Mexout}
                     [c, b, a] = [c, b, a]
  \end{Mexout}
  
  \eng{Altogether, the vertices of the Yang-Baxter graph are simply
    all permutations of the list \Mex{[a,b,c]}:}
  \begin{Mexin}
YB::basis()
  \end{Mexin}
  \begin{Mexout}
[[a, b, c], [a, c, b], [b, a, c], [b, c, a], [c, a, b],

   [c, b, a]]
  \end{Mexout}

\end{MExample}

\begin{MBackground}
  \eng{Reference: lectures notes of Alain Lascoux,
    \url{http://phalanstere.univ-mlv.fr/~al/}.  FIXME.}
\end{MBackground}

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

\end{Mhelppage}

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