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\newtheorem{defn}{Definition}
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\def\ia{\rightarrow\hspace{-40pt}+\ }
\def\ca{\nearrow\hspace{-32pt}\searrow}
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\def\SGS{\bigoplus_{n\geq0}{\mathbb Q}[\SG_n]}
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\begin{poster}

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\begin{center}
\titreorange{\textbf{\huge Free Quasi-Symmetric Functions, Product
Actions and Quantum Field Theory of Partitions\\}}

\vspace*{1cm}

\textsl{\LARGE G. H.E. Duchamp$^1$ J.-G. Luque$^2$, K. A. Penson$^3$ and  C. Tollu$^1$}

\vspace*{1cm}

\begin{tabular}{ccc}

\begin{minipage}[b]{0.3\textwidth} \large %\center

      $^1$ Institut Galil\'ee, LIPN\\
      Universit\'e Paris 13\\
      F- 93430 Villetaneuse, France\\
      e-mail: \textsf{\{ghed,ct\}@lipn-univ.paris13.fr}
    \end{minipage} &


    \begin{minipage}[b]{0.3\textwidth} \large %\center

      $^2$ Institut G.~Monge  UMR-CNRS 8049\\
      Univ. Marne la Vall{\'e}e\\
      F-77454 Marne la Vall{\'e}e, Cedex 2, France \\
      e-mail: \textsf{luque@univ-mlv.fr}
    \end{minipage} &



    \begin{minipage}[b]{0.3\textwidth} \large %\center

      $^3$ LPTCM CNRS UMR 7600\\
      Univ. Marne Pierre et Marie Curie,\\
       F 75252 Paris Cedex 05, France. \\
      e-mail: \textsf{penson@lptl.jussieu.fr}
    \end{minipage}
  \end{tabular}
\end{center}

\vspace*{3cm}

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\begin{multicols}{2}

\Large \titreic{Introduction} In a relatively recent paper,
Bender, Brody and Meister introduce a special Field Theory
described by\\
\pbox{0.9\columnwidth}{}{linewidth=2mm,framearc=0.1,linecolor=blue,fillstyle=gradient,gradangle=0,gradbegin=white,gradend=white,gradmidpoint=1.0,framesep=1em}{
\begin{equation}
G(z)=\Big(e^{(\sum_{n\geq 1}L_n
\frac{z^n}{n!}\frac{\partial}{\partial x})}\Big)
\Big(e^{(\sum_{m\geq 1}V_m \frac{x^m}{m!})}\Big)
\Big|_{x=0}\nonumber
\end{equation}
}\\ \\ in order to prove that any sequence of numbers $\{a_n\}$
can be generated by a suitable set of rules applied to some type
of Feynman diagrams . These diagrams actually are bicoloured
multigraphs with no isolated vertex.\\

\titrebleu{Actions of a direct product of permutation groups}
\sstitre{Direct product actions}
Two pairs $(G_1,X_1)$ and $(G_2,X_2)$,  each $G_i$ is
a permutation group acting on $X_i$.\\
 {\it \color{red} Intransitive action}
of $G_1\times G_2$ on $X_1\sqcup
 X_2$ :
\begin{equation}
(\sigma_1,\sigma_2)x=\left\{\begin{array}{ll}\sigma_1x&\mbox{ if
}x\in X_1\\ \sigma_2x&\mbox{ if }x\in X_2
\end{array}\right. .\nonumber
\end{equation}
 $(G_1,X_1)\ia(G_2,X_2):=(G_1\times G_2,X_1\sqcup
 X_2)$.\\
  {\it \color{red} Cartesian action} of
$G_1\times G_2$ on $X_1\times X_2$:
\begin{equation} (\sigma_1,\sigma_2)(x_1,x_2)=(\sigma_1
x_1,\sigma_2 x_2).\nonumber
\end{equation}
$(G_1,X_1)\ca(G_2,X_2):=(G_1\times G_2,X_1\times X_2)$.\\ \\
\sstitre{Explicit realization}
 Denote
 \begin{enumerate}
 \item[$\bullet\ \ $] by $\circ_N$ the natural action of $\SG_n$ on $\{0,\dots,n-1\}$,
\item[$\bullet\ \ $]  by $\circ_I$ the intransitive action of $\SG_n\times \SG_m$ on
$\{0,\cdots,n+m-1\}$
\item[$\bullet\ \ $] by $\circ_C$ the cartesian action of $\SG_n\times \SG_m$ on $\{0,\dots,nm-1\}$.
\end{enumerate}
  More precisely,
\begin{equation}
(\sigma_1,\sigma_2)\circ_Ii=\left\{\begin{array}{ll}
\sigma_1\circ_Ni&\mbox{if }0\leq i\leq n-1\\
\sigma_2\circ_N(i-n)+n&\mbox{if }n\leq i\leq
n+m-1\end{array}\right. \nonumber.
\end{equation}
and
\begin{equation}
(\sigma_1,\sigma_2)\circ_C(j+nk)=(\sigma_1\circ_Nj)+n(\sigma_2\circ_Nk)\nonumber
\end{equation}
for $0\leq i\leq n+m-1$, $0\leq j\leq n-1$ and $0\leq k\leq m-1$.\\
Let the map
$\ia:\SG_n\times\SG_m\rightarrow \SG_{n+m}$ defined by
\begin{equation}
\sigma_1\ia\sigma_2=\sigma_1\sigma_2[n]\nonumber
\end{equation}
\pbox{0.9\columnwidth}{}{linewidth=2mm,framearc=0.1,linecolor=lblue,fillstyle=gradient,gradangle=0,gradbegin=white,gradend=white,gradmidpoint=1.0,framesep=1em}{
 $\sigma_1={\color{blue}1320}\in\SG_4$,  $\sigma_2={\color{red}534120}\in\SG_6$.
\begin{equation}
\sigma_1\ia\sigma_2={\color{blue}1320}{\color{red}978564}\nonumber,\,
\sigma_2\ia\sigma_1={\color{red}534120}{\color{blue}7986}\nonumber
\end{equation}
}
\newpage
{\bf Proposition}\\
$(\sigma_1\ia\sigma_2)\circ_Ni=(\sigma_1,\sigma_2)\circ_Ii .$\\ \\
Let the map $\ca:\SG_n\times\SG_m\rightarrow \SG_{nm}$ defined by
\begin{equation}\label{defcar}
\sigma_1\ca\sigma_2=\prod_{i,j}c_i\ca c'_j\nonumber
\end{equation}
where $\sigma_1=c_1\cdots c_{k}$ and $\sigma_2=c'_1\cdots c'_{k'}$
are
 the decompositions of $\sigma_1$ and $\sigma_2$ in a product
of cycles
and \begin{equation}\label{cycleintr} c\ca
c'=\prod_{s=0}^{l\wedge l'-1}(\phi(s,0),\phi(s+1,1)\cdots,
\phi(s+l\vee l'-1,l\vee l'-1)),\nonumber
\end{equation}
($\wedge:=$  gcd,  $\vee:=$  lcm, $c=(i_0,\cdots, i_{l-1})$,
$c'=(j_0,\cdots,j_{l'-1})$ are two cycles and
$\phi(k,k')=i_{k\mbox{ mod }l}+nj_{k'\mbox{ mod }l'}$.)\\
 The cartesian action is
compatible with the natural action.\\
{\bf Proposition}\\
$(\sigma_1\ca\sigma_2)\circ_Ni=(\sigma_1,\sigma_2)\circ_Ci\ .$\\ \\
\pbox{0.9\columnwidth}{}{linewidth=2mm,framearc=0.1,linecolor=lblue,fillstyle=gradient,gradangle=0,gradbegin=white,gradend=white,gradmidpoint=1.0,framesep=1em}{ $c_1=(0,2,3,1)$, $c_2=(7,6,5,4,3,2)$.\\
 $c_1\ca c_2:$\\
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\put(13.8,7){{\tiny$\nearrow$}} \put(23.8,7){{\tiny$\nearrow$}}
% \put(9.5,16){{$\downarrow$}}\put(19.6,16){{$\downarrow$}}
% \put(29.5,16){{$\downarrow$}}
%  \put(16,9.5){{$\leftarrow$}}

 %croix
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  %\put(4.5,17){{$\downarrow$}}\put(14.6,17){{$\downarrow$}}
 % \put(24.5,17){{$\downarrow$}}
  %\put(16,14.5){{$\leftarrow$}} \put(16,4.6){{$\leftarrow$}}

   %croix
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\end{picture}}\nonumber\\
$\color{blue}(28,26,23,17,12,10,31,25,20,18,15,9)$\\
$\color{red}(30,27,21,16,14,11,29,24,22,19,13,8).$
\end{center}}\\ \\
\sstitre{Algebraic structure} {\bf Proposition}{\it\ \ \
Associativity}\\ Let $\sigma_1\in \SG_n$, $\sigma_2\in\SG_m$ and
$\sigma_3\in\SG_p$ be $3$ permutations
\begin{enumerate}
\item
$\sigma_1\ia(\sigma_2\ia\sigma_3)=(\sigma_1\ia\sigma_2)\ia\sigma_3$
\item $\sigma_1\ca(\sigma_2\ca\sigma_3)=(\sigma_1\ca\sigma_2)\ca\sigma_3$
\end{enumerate}
{\bf Proposition}{\it\ \  Semi-distributivity}\\
 $\sigma_1\in \SG_n$, $\sigma_2\in\SG_m$ and $\sigma_3\in\SG_p$
\[\sigma_1\ca(\sigma_2\ia\sigma_3)=(\sigma_1\ca\sigma_2)\ia(\sigma_1\ca\sigma_3) \]

\newpage

\end{multicols}
\end{poster}

\begin{poster}
\Large
\begin{multicols}{2}

\titrebleu{Cycle index algebra}


\sstitre{Cartesian product in $Sym$} Let $c_j(\sigma)$ be the
number of cycles of $\sigma$ of length $j$
\begin{equation}
\Zy(\sigma)=\prod_{j=0}^{\infty}\psi_j^{c_j(\sigma)}
\end{equation}
where  $\psi_i$ is a power sum symmetric function.\\ One defines
\begin{equation}
\prod_{1\leq i\leq \infty} \psi_i^{\alpha_i}\star \prod_{1\leq
j\leq \infty}\psi_j^{\beta_j}= \prod_{1\leq i,j\leq \infty}
\psi_{i\vee j}^{\alpha_i\beta_j (i\wedge j)}\nonumber
\end{equation}
{\bf Proposition} i)  $\Zy: \SGS\mapsto Sym$ is morphism sending
the two laws $\ia\;\ \ca$ respectively to $\times;\ \star$.\\ More
precisely, for $\sigma,\ \tau\in \sqcup_{n\geq 0}\SG_n$ one has
\begin{equation}\label{2assmorph}
\Zy(\sigma\ia \tau)=\Zy(\sigma)\Zy(\tau)\ ;\ \Zy(\sigma\ca
\tau)=\Zy(\sigma)\star\Zy(\tau)
\end{equation}
ii) $\star$ is commutative and distributive over $\times$.\\ \\

{\bf Proposition}
 $P:=\{\prod_{i=1}^\infty \psi_i^{\al_i}\}_{(\al_i)_{i\geq 1}\in \N^{(\N^*)}}$
is closed by $\times$ and $\star$. \\
 More precisely: $(P,\times,\star)$ is isomorphic to a subsemiring of the semiring
$\N[(\N^{(\p)},sup)]$ ( $\p:=$  the set of prime numbers).
\\ \\

\sstitre{Cycle index polynomial} {\it\color{red}  Poly\`a cycle
index polynomial} :
\begin{equation}
Z(G)=\Zy\left({1\over|G|}\sum_{\sigma\in G}\sigma\right).\nonumber
\end{equation}
\pbox{0.9\columnwidth}{}{linewidth=2mm,framearc=0.1,linecolor=lblue,fillstyle=gradient,gradangle=0,gradbegin=white,gradend=white,gradmidpoint=1.0,framesep=1em}{
\begin{enumerate}
\item Symmetric group $\SG_n$ : $Z(\SG_n)=h_n$.
\item Alternating group $A_n$ :
$Z(A_n)=h_n+e_n$.
\end{enumerate}
} As $\Zy$ is a morphism of $2$-associative algebra, one recovers
the classical relations
\begin{equation}
Z(G_1\ia G_2)=Z(G_1)Z(G_2),\, Z(G_1\ca G_2)=Z(G_1)\star Z(G_2)\nonumber
\end{equation}
\pbox{0.9\columnwidth}{}{linewidth=2mm,framearc=0.1,linecolor=lblue,fillstyle=gradient,gradangle=0,gradbegin=white,gradend=white,gradmidpoint=1.0,framesep=1em}{
\begin{enumerate}
\item Intransitive product of
 $\SG_n$ and $\SG_m$:
$$Z(\SG_n\ia\SG_m)=h_nh_m.$$
\item Cartesian product of  $\SG_n$ and $\SG_m$ :
$$Z(\SG_n\ca\SG_m)=h_n\star h_m=\sum_{|\lambda|=n,\atop
|\rho|=m}\frac1{z_\lambda
z_\rho}\prod_{i,j}\psi_{\lambda_i\vee\rho_j}^{\lambda_i\wedge\rho_j},$$
where $z_\lambda=\prod i^{n_i}n_i!$ if $n_i$ is the number of
parts of $\lambda$ equal to $i$.

\end{enumerate}
} \vspace{1cm} \sstitre{Enumeration of a type of Feynman diagrams
related to quantum field theory of partitions} The {\it
\color{red} type} $t(f)$ of $f:X\rightarrow
L=\{l_0<l_1\cdots<l_k\cdots\}$ is the vector $(i_0,\dots,
i_p,\dots)$ where $i_k$ is the number of elements of $X$ whose
image by $f$ is $l_k$.\\ The {\it \color{red}shape} $s(f)$ of $f$
is the partition obtained by sorting in the decreasing order
$t(f)$ and erasing the zeroes.\\ The number $d^s_\lambda(G,L)$ of
$G$-classes on $L^X$ with the shape $\lambda$ is the coefficient
of $m_\lambda$ in the expansion of $Z(G)$ in the basis of monomial
symmetric functions:
\begin{equation}
Z(G)=\sum_\lambda d^s_\lambda(G,L)m_\lambda .\nonumber
\end{equation}
Let us define the generating series of the type of our Feynman
diagrams\\
\pbox{0.9\columnwidth}{}{linewidth=2mm,framearc=0.1,linecolor=blue,fillstyle=gradient,gradangle=0,gradbegin=white,gradend=white,gradmidpoint=1.0,framesep=1em}{
\begin{equation}
F(n,m):=\sum_{I=(i_0,\dots,i_p,\dots)}f^t_I(n,m)\prod_{k=0}^\infty
y_k^{i_k},\nonumber
\end{equation}}
where $f^t_I(n,m)$ denotes the number of Feynman diagrams of type
$I$.\newpage {\bf \color{red} Theorem} One has the following
decomposition of the cycle index polynomial.\\
\begin{equation}
Z(\SG_n\ca\SG_m)=\sum_{(1,1)\leq_{lex} (k,p)\leq_{lex}
(n,m)}F(k,p)y_0^{nm+kp-np-mk}+y_0^{nm}.\nonumber
\end{equation}
\\ \\
\pbox{0.9\columnwidth}{}{linewidth=2mm,framearc=0.1,linecolor=lblue,fillstyle=gradient,gradangle=0,gradbegin=white,gradend=white,gradmidpoint=1.0,framesep=1em}{
\[\begin{array}{rcl}
F(2,3)&=&Z(\SG_3\ca\SG_2)-F(1,3)y_0^3-F(2,2)y_0^2\\
&&-F(2,1)y_0^4-F(1,2)y_0^4-F(1,1)y^5_0-y_0^6\\
&=&y_2^6+y_2^5y_1+3y_2^4y_1+3y_2^4y_1y_0+2y_2^4y_0^2\\
&&+3y_2^3y_1^3+6y_2^3y_1^2y_0+5y_2^3y_1y_0^2
+y_2^3y_0^3+3y_2^2y_1^4\\
&&+3y_2^2y_1^3y_0+8y_2^2y_1^2y_0^2+3y_2^2y_1y_0^3+y_2y_1^5+3y_2y_1^4y_0\\
&&+5y_2y_1^3y_0^2+3y_2y_1^2y_0^3+y_1^6+y_1^5y_0+y_1^3y_0^3+2y_1^2y_0^4.
\end{array}\]
For example, there is $8$    $(2,2,2)$- Feynman diagrams:
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}
}
\vspace{4cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\titrebleu{Non commutative realizations}
\label{sec:princ}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\sstitre{Free quasi-symmetric cycle index algebra}

Recall that the algebra $\FQSym$ is defined by one of its bases, indexed to $\SG$ and defined as follows
\begin{equation}
\F_\sigma=\sum_{Std(w)=\sigma^{-1}}w\in  \ncs{\Z}{A}\nonumber
\end{equation}
One has
\begin{equation}
F^\sigma F^\tau=F^{\sigma{\rightarrow\hspace{-25pt}+}\
\tau}.\nonumber
\end{equation}
This induces naturally a morphism of algebra
\begin{eqnarray}
\underline\Zy:\left(\bigoplus{\mathbb
Q}[\SG_n],\ia,+\right)&\rightarrow&(\FQSym,.,+)\nonumber\\
\sigma&\rightarrow&\F^\sigma.\nonumber
\end{eqnarray}
One defines the product $\star$ on $\FQSym$ by $F^\sigma\star
F^\tau:=F^{\sigma{\nearrow\hspace{-18pt}\searrow} \tau}$. By this
way, $\underline\Zy$ becomes a morphism of $2$-associative
algebra. Furthermore, $\star$ is associative, distributive over
the sum and semi-distributive over the shifted concatenation .
\sstitre{Free quasi-symmetric Poly\`a cycle index polynomial}
\begin{equation}
\underline Z(G):=\underline\Zy\left({1\over|G|}\sum_{\sigma\in
G}\sigma\right).\nonumber
\end{equation}
{\bf Proposition}\\
Let $G_1, G_2\in \SG_{sg}$ be two permutation groups, one has
\begin{enumerate}
\item $\underline Z(G_1\ia G_2)=\underline Z(G_1)\underline
Z(G_2)$.
\item $\underline Z(G_1\ca G_2)=\underline Z(G_1)\star\underline
Z(G_2)$.
\end{enumerate}
\begin{eqnarray}
\SG_{sg}&\displaystyle\mathop{\longrightarrow}^{\underline
Z}&\FQSym\nonumber\\ {\small Z}\downarrow&\displaystyle\mathop{\
}^z\swarrow&\uparrow{\small \underline\Zy}\nonumber\\
Sym&\displaystyle\mathop{\longleftarrow}_\Zy&\bigoplus{\mathbb
Q}[\SG_n]\nonumber
\end{eqnarray}
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\begin{enumerate}
\item Free quasi-symmetric cycle index of $\SG_n$ :
$${\bf H}_n:=\underline Z(\SG_n)=\frac1{n!}\sum_{\sigma\in
\SG_n}F^\sigma.$$ Analogous of $z({\bf
H_n})=Z(\SG_n)=h_n$.
\item  Free cycle index polynomial of the
alternative groups: $$ {\bf E}_n:=\underline Z(A_n)-\underline
Z(\SG_n).$$  analogous of elementary symmetric functions :
$$z({\bf E_n})=Z(A_n)-Z(\SG_n)=e_n.$$
\end{enumerate}}
%si biblio
%\bibliographystyle{unsrt}
%\bibliography{bib}

\end{multicols}
\end{poster}

\end{document}

