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\date{September, 12th 2012,\\
\textcolor{red}{S\'eminaire Lotharingien de Combinatoire}\\
Strobl (Austria)\\
(joint work with Matthieu Deneufch\^atel and Hoang Ngoc Minh)}

\title{Fuchsian type multipliers and shuffle algebras.}

\author{G.H.E.~Duchamp,\\
LIPN -- UMR 7030 CNRS,
Institut Galill\'ee --
Universit\'e Paris 13.}

\begin{document}

\maketitle

\begin{frame}{Summary}
\begin{enumerate}
	\item Introduction
	\item Strings of primitives
	\item Shuffle algebras
	\item Duality Series/Polynomials 
	\item Abstract setting 
	\item Sketch of the proof 
	\item Application to polylogs 
	\item Concluding remarks and perspectives
\begin{enumerate}
	\item Triangle condition applied to other entries and function fields
	\item Constants of (alien ?) derivation
	\item Better understanding of the combinatorics behind the differential Galois group
\end{enumerate}
\end{enumerate}
\end{frame}
%
\begin{frame}{Introduction}

\begin{itemize}
	\item Many discussions and interactions convinced us that a special care 
should be payed on the study of {\it coefficients} about polylogarithms :\\
indeed, {\it the only known proofs of the linear independance of polylogarithms and 
hyperlogarithms were through an argument of {\rm monodromy} (i.e. integration 
along a path with the addition of one, or several, loops containing singularities). 
A strong need was to examine whether one could prove independance without 
using {\rm monodromy} and, if so, if one could ``push'' further the set of coefficients.}     
	\item Today we will show an abstract theorem about differential algebra 
and some of its applications to polylogarithms.
	\item The road to polylogarithms is the following	
\begin{itemize}
		\item Riemann zeta values 
		\item One wants to produce a tractable $\Q$-algebra where these functions 
		live 
		\item Euler-Zagier sums 
		\item Polylogarithms (analytic function)
		\item Integral form of polylogarithms (today's concern)
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Strings of primitives 1/4}
We begin by a classical example, the generation of polylogarithms. \\
On the following domain, 
\begin{center}
\begin{tikzpicture}
\node (origin) at (0,0) [draw,circle,inner sep=1pt,fill=black!75,below] {};
\node () at (-.2,-.2) [below] {0};
\node () at (1.2,-.2) [below] {1};
\node (space) at (2,0) [below] {$\Omega$};
%\draw[step=.5] (-.1,-.1) grid (3.1,3.1);
\draw[->] (0,-1) node [below] {$\mathbb{C}-(]-\infty,0] \cup [1,+\infty[)$} -- (0,2);
\draw[thick, inner sep =1pt] (-2,0) -- (0,0);
\draw[dotted] (0,0) --  (1,0) node [draw,circle,inner sep=1pt,fill=black!75] {};
\draw[thick,->] (1,0) -- (4,0);
\end{tikzpicture}
\end{center}
let's play a game ...
\end{frame}

\begin{frame}{Strings of primitives 2/4}
Starting from $1$ (the constant function on $\Omega$), we read the words of 
$\{x_0,x_1\}^*$ and compute w.r.t. the following rule (using any ``primitive-making'' process).  
%\begin{center}
%\begin{tikzpicture}[level/.style={sibling distance = 50mm/#1}]
%\node {1} [grow'=up]
%	 {
%		child{node {$x_0$}
%		child{node {$x_0^2$}}
%		child{node {$x_1x_0$}}
%		}				
%		child{node {$x_1$}
%		child{node {$x_0x_1$}}
%		child{node {$x_1^2$}}
%		}
%	};
%\end{tikzpicture}
%\end{center}
%
%\hskip2cm
\begin{center}
\begin{tikzpicture}
\node (p1) at (-2,1) [] {};
\node (p2) at (2,1) [] {};
\draw[->] (0,0) -- node [below = 5pt] {$\int ?\frac{dz}{z}$} (p1);
\draw[->] (0,0) -- node [below = 5pt] {$\int ?\frac{dz}{1-z}$} (p2);
\end{tikzpicture}
\end{center}
We then get a collection of analytic functions $(S_w(z))_{w\in \{x_0,x_1\}^*}$ satisfying 
\begin{equation}\label{diffeq1}
	S_{x_0w}'(z)=\frac{1}{z} S_{w}(z)\ ;\ S_{x_1w}'(z)=\frac{1}{1-z} S_{w}(z)
\end{equation}
\end{frame}
%
\begin{frame}{Strings of primitives 3/4}
A classical method (Chen's iterated integrals) to obtain solutions of \mref{diffeq1} is to pick an element $z_0\in \Omega$ and compute the following iterated integrals   
\begin{equation}
   S_{x_{i_1}x_{i_2}\cdots x_{i_n}}(z)=\int_{z_0}^z d\omega_{i_1} \int_{z_0}^{y_1} \dots \int_{z_0}^{y_{n-1}} d\omega_{i_n}\ ;\  \omega_0=\frac{dz}{z}\ \omega_1=\frac{dz}{1-z}
\end{equation}
One can also initialize the process with a chain of well choosen primitives i.e.
\begin{equation}
   S_{x_{0}^n}(z):=\frac{log(z)^n}{n!}
\end{equation}
and integrate w.r.t. the other moves.\\
In these two cases, the solution is a morphism w.r.t. the shuffle product. 
\end{frame}

\begin{frame}{Strings of primitives 4/4}
We then get a map $w\mapsto S_w$ from $X^*=\{x_0,x_1\}^*$ to $\calC^{\omega}(\Omega,\C)$ (a ring) which we note as a sum $S=\sum_{w\in X^*}\scal{S}{w}w$ (a noncommutative series).\\ 
We recall that the algebra of noncommutative series is endowed with the convolution product (here also concatenation) 
\begin{equation}\label{convprod}
	ST:=\sum_{w\in X^*}\Big(\sum_{uv=w}\scal{S}{u}\scal{T}{v}\Big)w
\end{equation}
Now, with the extension of $\frac{d}{d z}$ to series, one has 
\begin{equation}\label{diffeq1}
	\frac{d}{d z}(S)=MS
\end{equation}
with $M=\frac{1}{z}.x_0+\frac{1}{1-z}.x_1$.\\ 
%
\red{This extension of the derivative to noncommutative series permits a closer study of solutions of equations of type \mref{diffeq1}.}\\
\blue{In particular, with certain initial conditions, solutions of \mref{diffeq1} are morphisms of the \red{shuffle algebra (this is the case of the two solutions given above)}.}
% 
\end{frame}
%
\begin{frame}{Shuffle algebras}
There is another product between series which is linked to the Hopf algebra structure of the noncommutative polynomials (viewed as the enveloping algebra of the free Lie algebra)
\begin{equation}
	(\ncp{k}{X},conc,1_{X^*},\red{\Delta_{\shuffle}},\epsilon,S)\ .
\end{equation}
The letters are primitive i.e. for all $x\in X$ 
\begin{equation}
\Delta_{\shuffle}(x)=x\otimes 1 + 1\otimes x\, .
\end{equation}
Then the shuffle product can be defined either by duality with $\Delta_\shuffle$ 
or by the following recursion
\begin{eqnarray}
                1 \shuffle w &=& w \shuffle 1  = w ~; \cr
                a.u \shuffle b.v & = & a.(u \shuffle b.v) + b (a.u \shuffle v) 
\end{eqnarray}
\end{frame}
%
\begin{frame}{Duality Series/Polynomials}
As $\ncp{k}{X}=k^{(X^*)}$ and $\ncs{k}{X}=k^{X^*}$, we get a natural pairing 
\begin{equation}
\ncs{k}{X}\otimes \ncp{k}{X}\ra k 	
\end{equation}
by $\scal{S}{P}=\sum_{w\in X^*}\scal{S}{w}\scal{P}{w}$.\\ 
Here is the meaning of the statement (see above) that 
certain series $S$ are morphisms for the shuffle product. We have 
\begin{equation}
\scal{S}{u\shuffle v}=\scal{S}{u}\scal{S}{v} 	
\end{equation}
\end{frame}
%
\begin{frame}{Abstract setting 1/2}
Let $(\A,d)$ be a $k$-commutative associative differential algebra with unit 
($ch(k)=0$ and $ker(d)=k$) and $\calC$ be a differential subfield of $\A$ (i.e. $d(\calC)\subset \calC$). 
We suppose that $S\in \ncs{\A}{X}$ is a solution of the differential equation 
\begin{equation}
	\bd(S)=MS\ ;\ \scal{S}{1}=1
\end{equation}
where the multiplier $M=\sum_{x\in X}u_x x\in \ncs{\calC}{X}$ is an homogeneous series 
(a polynomial in case $X$ is finite) of degree $1$.\\
\end{frame}
%
\begin{frame}{Abstract setting 2/2}
\begin{theo} 
The following condition are equivalent :\\
i) The family $(\scal{S}{w})_{w\in X^*}$ of coefficients of $S$ is free over $\calC$.\\
ii) The family of coefficients $(\scal{S}{y})_{y\in X\cup \{1_{X^*}\}}$ is free over $\calC$.\\
iii) The family $(u_x)_{x\in X}$ is such that, for $f\in \calC$ and $\al_x\in k$   
\begin{equation}\label{prim_ind}
d(f)=\sum_{x\in X} \al_x u_x \Longrightarrow (\forall x\in X)(\al_x=0)\ .
\end{equation}
\end{theo}
\begin{remark} (iii) can be rephrased as\\
iv) The family $(u_x)_{x\in X}$ is free over $k$ and 
\begin{equation}
	d(\calC)\cap span_k\Big((u_x)_{x\in X}\Big)=\{0\}\ .
\end{equation} 
\end{remark}
\end{frame}
%
\begin{frame}{Sketch of the proof 1/2}
\begin{itemize}
	\item (iii)$\Longrightarrow$(i) Suppose a linear dependence relation 
\begin{equation}\label{dep_rel}
\sum_{w\in X^*}\al_w \scal{S}{w}=0	
\end{equation}
 (sum with finite support). With $P=\sum_{w\in X^*}\al_w \, w=0$, \mref{dep_rel} can be rewritten 
\begin{equation}\label{dep_rel1}
\scal{S}{P}=0\ ;\ P\not=0\ .	
\end{equation}
Take the polynomial with the least degree (lenght of the leading monomial) such that \mref{dep_rel1} and least leading word among the non-trivial relations, normalize it (because $\calC$ is a field, one can divide by the leading coefficient) and write it
\begin{equation}
	P=w+\sum_{u<w}\scal{P}{u} u
\end{equation}
\end{itemize}
\end{frame}
%
\begin{frame}{Sketch of the proof 2/2}
Now 
\begin{eqnarray}
0&=&\scal{S}{P}'=\scal{S'}{P}+\scal{S}{P'}=\scal{MS}{P}+\scal{S}{P'}=\cr
&&\scal{S}{M^{\dagger}P}+\scal{S}{P'}=\scal{S}{M^{\dagger}P+P'}	
\end{eqnarray}
which implies that $P'=-M^{\dagger}P$ hence $\scal{P}{w}'=-\scal{M^{\dagger}P}{w}=-\sum_{x\in X}u_x\scal{P}{xw}$ 
From this and the fact that $ker(d)=k$, one obtains a contradiction (if we had $deg(P)\geq 1$). 
\end{frame}
%
\begin{frame}{Sketch of the proof 2/2}
\begin{itemize}
	\item (i)$\Longrightarrow$(ii) obvious
	\item (ii)$\Longrightarrow$(iii) An equality $d(f)=\sum_{x\in X} \al_x u_x$ for a function $f\in \calC$ can 
	be rewritten $d\Big(f-\sum_{x\in X} \al_x S_x\Big)=0$ and thus $f-\sum_{x\in X} \al_x S_x=c\in k$ which implies that 
$f=c$ and all the $\al_x$ are zero.   
\end{itemize}
\end{frame}
\begin{frame}{Application to polylogs} 
\begin{itemize}
	\item 
In order to get a field of function we have to withdraw the poles. We then consider fields of germs of functions. 
For simplicity, let's take the field of rational functions $\C(z)$, realized on $\Omega$ as functions defined on 
some $(\Omega-F)_{F\ finite}$. This field fulfils the conditions of the theorem w.r.t. $d=\frac{d}{dz}$ and then we have the independance of the polylogarithms w.r.t. the rational functions.
	\item
One can even go further (larger fields of functions, and/or other ``entries'' $u_x$) ... in particular    
	\item all other multipliers of ``Fuchsian type'' (by this we mean that all entries have distinct singularities of order one) yield the same conclusion.  
\end{itemize}

\end{frame}
%
\begin{frame}{Concluding remarks and perspectives}
\begin{enumerate}
	\item We did not touch the proof of the ``shuffle'' condition. It is an easy consequence 
	of the primitivity of the multiplier 
	($\Delta_\shuffle(M)=M\otimes 1+1\otimes M$)
	\item Constants of (alien ?) derivatives\\
The operators $x_0^{-1}$ and $x_1^{-1}$ act as (directional ?, alien ?) derivatives on 
$\A_\mathcal{C}=\mathcal{C}\otimes_\C \A_\C\ $. They coincide respectively with $z\frac{\sd}{\sd\, z}$  
and $(1-z)\frac{\sd}{\sd\, z}$ on $\A_\C$, but their constant field is much larger.\\
\centerline{What is the maximal constant field for these derivatives ?}
(presumably the field of germs of analytic functions which are inessential at $0$ and $1$).
	\item Better understanding of the combinatorics behind the differential Galois group. 
$$
S'=MS\ ;\ (SG)'=S'G=M(SG)
$$
\end{enumerate}
\end{frame}
\end{document}


