%\documentclass[12pt]{amsart}
\documentclass[submission]{FPSAC2019}

\articlenumber{73}
\addbibresource{73.bib}

%\usepackage[dvips]{graphicx,mathtools}
%\usepackage{amsmath}
%\usepackage{color}
%\usepackage{amsfonts}
%\usepackage{amssymb}%
%\setcounter{MaxMatrixCols}{30}
%\providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
%\providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
%\providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
%\providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
%\providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
%\providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
%\providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
%\textheight 220mm
%\textwidth 170mm
%\hoffset -20mm
%\voffset -15mm
%\font\goth=eufm10
%\font\sevenbf=cmbx7
%\font\bb=msbm10

\newtheorem{theorem}{Theorem}[section]
\newtheorem{acknowledgement}[theorem]{Acknowledgment}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Note}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{examples}[theorem]{Examples}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{remarks}[theorem]{Remarks}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\def\w{{\rm w}}
\def\Tab{{\rm Tab}}
\def \N{\mathbb{N}}
\def \Q{\mathbb{Q}}
\def \H{\mathbb{H}}
\def \Z{\mathbb{Z}}
\def \R{\mathbb{R}}
\def \T{\mathbb{T}}
\def \I{\mathbb{I}}
\def \E{\mathbb{E}}
\def \PP{\mathbb{P}}
\def \d{\delta}
\def \e{\varepsilon}
\def \g{\gamma}
\def \a{\alpha}
\def \b{\beta}
\def \l{\lambda}
\def \o{\omega}
\def \G{\Gamma}
\def \D{\Delta}
\def \O{\Omega}
\def \r{\rho}
\def \s{\sigma}
\def \t{\theta}
\def \cA{\mathcal A}
\def \cB{\mathcal B}
\def \cH{\mathcal H}
\def \cN{\mathcal N}
\def \C{\mathcal C}
\def \cD{\mathcal D}
\def \cE{\mathcal E}
\def \cP{\mathcal P}
\def \cT{\mathcal T}
\def \bA{\overline{A}}
\def \bB{\overline{B}}
\def \bD{\overline{D}}
\def \bC{\overline{C}}
\def \bX{\overline{X}}
\def \bG{\overline{G}}
\def \bP{\overline{P}}
\def \bH{\overline{H}}
\def \bg{\overline{g}}
\def \bp{\overline{p}}
\def \bd{\overline{d}}
\def \bo{{\bf o}}
\def \bx{{\bf x}}
\def \by{{\bf y}}
\def \bz{{\bf z}}
\def \bu{{\bf u}}
\def \bv{{\bf v}}
\def \bbo{\bar{\bf o}}
\def \bbx{\bar{\bf x}}
\def \bby{\bar{\bf y}}
\def \bbz{\bar{\bf z}}
\def \bbu{\bar{\bf u}}
\def \bbv{\bar{\bf v}}
\def \bbA{\bar{\bf A}}
\def \bbB{\bar{\bf B}}
\def \fin{$\Box$}
\def \ffin{\Box}
\def\ds{\displaystyle}
\def \pg{\left (}
\def \pd{\right )}
\def \Db{\overline{D}}
\def \fdem{$\Box$}
\def \L{{\cal{L}}}
\def \A{{\cal{A}}}
\def \B{{\cal{B}}}
\def \P{{\cal{P}}}
\def \sp{(n+{1 \over 2})}
\def \mb{\overline{M}}
\def \eqalign#1{\null\,
\vcenter{\openup\jot\ialign{\strut\hfil$\displaystyle{##}$&
$\displaystyle{{}##}$\hfil \crcr#1\crcr}}\,}
\def \dD{\partial D}
\def \XA{ (\Omega,{\cal A},P)}
\def\r{\rightarrow}
\def\st {X_{(1)},\cdots,X_{(n)}}
\setcounter{tocdepth}{2}

\newcommand{\revise}[1]{\textcolor{red}{#1}}
\newcommand{\casetwoother}[3]{\left\{ \begin{array}{ll} #1 &\mbox{if $#2$} \\[1mm] #3 &\mbox{otherwise}\,. \end{array} \right.}
\newcommand{\casethree}[6]{\left\{ \begin{array}{ll} #1 &\mbox{if $#2$} \\[1mm] #3 &\mbox{if $#4$} \\[1mm]#5 &\mbox{if $#6$}\,. \end{array} \right.}


\newlength{\cellsize}
\cellsize=2.5ex

% \tableau{} sets up an alignment with zero spacing, boxing each entry
% with \tableaucell{}.

\newcommand\tableau[1]{
\vcenter{
\let\\=\cr
\baselineskip=-16000pt
\lineskiplimit=16000pt
\lineskip=0pt
\halign{&\tableaucell{##}\cr#1\crcr}}}

% \tableaucell{} generates a square box of side length \cellsize.  If
% its argument is non-void, it is typeset in math mode, centered in the
% box, and a frame is drawn.

\newcommand{\tableaucell}[1]{{%
\def \arg{#1}\def \void{}%


\ifx \void \arg
\vbox to \cellsize{\vfil \hrule width \cellsize height 0pt}%
\else
\unitlength=\cellsize
\begin{picture}(1,1)
\put(0,0){\makebox(1,1){$#1$}}
\put(0,0){\line(1,0){1}}
\put(0,1){\line(1,0){1}}
\put(0,0){\line(0,1){1}}
\put(1,0){\line(0,1){1}}
\end{picture}%
\fi}}

%\addbibresource{fpsac.bib}

\usepackage{lipsum}

%% define your title in the usual way
\title[Alcove random walks and $k$-Schur functions]{Alcove random walks and $k$-Schur functions}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use
%\thanks{} for contact details

\author{C\'edric Lecouvey\thanks{\href{mailto:cedric.lecouvey@lmpt.univ-tours.fr}{
			cedric.lecouvey@lmpt.univ-tours.fr}}\addressmark{1}
	\and Pierre Tarrago\thanks{\href{mailto:pierre.tarrago@upmc.fr}{pierre.tarrago@upmc.fr}}\addressmark{2}
}
%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Facult\'e des Sciences et Techniques, Universit\'e de Tours, 37000 Tours, France\\ \addressmark{2}Sorbonne Universit\'e, 4 place de Jussieu, 75005 Paris}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have
%%one yet
%% please don't use custom commands in your abstract / resume, as these
%%will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out
%%your citation as something like (author year)
\abstract{
	We use $k$-Schur functions to get the minimal boundary of the $k$-bounded
partition poset. This permits to describe the central random walks on affine
Grassmannian elements of type $A$ and yields a rational expression for their
drift. We also recover Rietsch's parametrization of totally nonnegative
unitriangular Toeplitz matrices without using quantum cohomology of flag
varieties. All the homeomorphisms we define can moreover be made explicit by
using the combinatorics of $k$-Schur functions and elementary computations
based on Perron-Frobenius theorem.
}

%% put your French abstract here, or comment this out if you don't have one
\resumetitle{R\'{e}sum\'{e}}
\resume{
	Nous obtenons la fronti\`ere minimale du graphe des partitions $k$-born\'ees en utilisant l'alg\`ebre des fonctions de $k$-Schur. Nous en d\'eduisons une description des mesures de Gibbs pour les marches sur les Grassmaniennes affines de type $A$, ainsi qu'une expression rationnelle de la d\'erive d'une marche al\'eatoire suivant une telle mesure. Cela donne une preuve alternative et combinatoire de la param\'etrisation des matrices de Toeplitz unitriangulaires totalement positives obtenue par Rietsch. Tous les hom\'eomorphismes sont de plus explicites et d\'ecoulent de la combinatoire des fonctions de $k$-Schur et de manipulations du th\'eor\`eme de Perron-Frobenius.
}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{$k$-bounded partitions, $k$-Schur functions, affine Grassmannian, central probability measure	
}

%% you can include your bibliography however you want, but using an
%external .bib file is STRONGLY RECOMMENDED and will make the editor's life
%much easier
%% regardless of how you do it, please use numerical citations, ie. [xx,
%yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs
%in a more flexible way than bibtex
%% but you can use bibtex if you want
% \usepackage[backend=bibtex]{biblatex}
% \addbibresource{sample.bib}
%% note the \printbibliography command at the end of the file which goes
%with these biblatex commands




\begin{document}
	
\maketitle

\section{Introduction}

A real function on the Young graph is harmonic when its value on any Young diagram
$\lambda$ is equal to the sum of its values on the Young diagrams obtained by
adding one box to $\lambda$. The set of extremal nonnegative such functions
(i.e. those that cannot be written as a convex combination) is called the
minimal boundary of the Young graph. It is homeomorphic to the Thoma simplex.
Kerov and Vershik proved that the extremal nonnegative harmonic functions give
the asymptotic characters of the symmetric group. Kerov-Vershik approach of these
harmonic functions yields both a simple parametrization of the set of infinite
totally nonnegative unitriangular Toeplitz matrices (see \cite{Ker}) and a
characterization of the morphisms from the algebra $\Lambda$ of symmetric
functions to $\mathbb{R}$ which are nonnegative on the Schur functions. These
results were generalized in \cite{LLP2} and \cite{LT}.\ A crucial observation
here is the connection between the Pieri rule on Schur functions and the
structure of the Young graph (which is then said multiplicative in
Kerov-Vershik terminology).

In \cite{Ri}, Rietsch obtained a parametrization for the variety $T_{\geq0}$
of finite unitriangular $(k+1)\times(k+1)$ totally nonnegative Toeplitz
matrices by $\mathbb{R}_{\geq0}^{k}$ from the quantum cohomology of partial
flag varieties. More precisely, such a matrix is proved to be completely
determined by the datum of its $k$ initial minors obtained by considering its
southwest corners. On the combinatorial side, there is also an interesting
$k$-analogue $\mathcal{B}_{k}$ of the Young lattice of partitions whose
vertices are the $k$-bounded partitions (i.e. those with no parts greater than
$k$). Its oriented graph structure is isomorphic to to the Hasse poset on the
affine Grassmannian permutations of type $A$ which are minimal length coset
representatives in $\widetilde{W}/W$, where $\widetilde{W}$ is the affine type
$A_{k}^{(1)}$ group and $W$ the symmetric group of type $A_{k}$. The graph
$\mathcal{B}_{k}$ is also multiplicative but we have then to replace the
ordinary Schur functions by the $k$-Schur functions (see \cite{LLMSSZ} and the
references therein) and the algebra $\Lambda$ by $\Lambda_{(k)}=\mathbb{R}%
[h_{1},\ldots,h_{k}]$.\ The $k$-Schur functions were introduced by Lascoux,
Lapointe and Morse \cite{LLM} as a basis of $\Lambda_{(k)}$. It was
established by Lam \cite{LamLR} that their corresponding constant structures
(called $k$-Littlewood-Richardson coefficients) are nonnegative.\ This was
done by interpreting $\Lambda_{(k)}$ in terms of the homology ring of the
affine Grassmannian which, by works of Lam and Shimozono, can be conveniently
identified with the quantum cohomology ring of partial flag varieties studied
by Rietsch. By merging these two geometric approaches one can theoretically
deduce that the set of morphisms from $\Lambda_{(k)}$ to $\mathbb{R}$,
nonnegative on the $k$-Schur functions, is also parametrized by
$\mathbb{R}_{\geq0}^{k}$.

In this note, we shall use another approach to avoid sophisticated geometric
notions and make our construction as effective as possible. Our starting point
is the combinatorics of $k$-Schur functions. The latter yield an
explicit parametrization of the morphisms $\varphi$ which are nonnegative on the
$k$-Schur functions, or equivalently of all the minimal $t$-harmonic functions
with $t\geq0$ on $\mathcal{B}_{k}$. Both notions are related by the simple
equality $t=\varphi(s_{(1)})$. Each such morphism is in fact completely
determined by its values $\vec{r}=(r_{1},\ldots,r_{k})\in\mathbb{R}_{\geq
0}^{k}$ on the Schur functions indexed by the rectangle partitions
$R_{a}=(k-a+1)^{a}.$ We get a bi-continuous (homeomorphism) parametrization
which is moreover effective in the sense one can compute from $\vec{r}$ the
values of $\varphi$ on any $k$-Schur function from the Perron-Frobenius vector
of a matrix $\Phi$ encoding the multiplication by $s_{(1)}$ in $\Lambda_{(k)}%
$. Also, applying the primitive element theorem in the field of fractions of $\Lambda_{(k)}$ permits to prove that for any fixed
$t\geq0$ each $\varphi(s_{\lambda}^{(k)})$ is a rational functions on
$\mathbb{R}_{\geq0}^{k}$. It becomes then quite easy to derive Rietsch's
parametrization.\ So, the only place where geometry is needed in this paper is
in Lam's proof of the nonnegativity of the $k$-Schur coefficients. As far as
we are aware a complete combinatorial $k$-Littlewood-Richardson rule is not
yet available (see nevertheless \cite{MS}).

Random walks on reduced alcove paths have been considered by Lam in
\cite{Lam2}.\ They are random walks on a particular tessellation of
$\mathbb{R}^{k}$ by alcoves supported by hyperplanes, where each hyperplane
can be crossed only once.\ The random walks considered in this paper are
central and thus differ from those of \cite{Lam2}. Two trajectories with the
same ends will have the same probability. We characterize all the possible
laws of these alcove random walks and also get a simple algebraic expression
of their drift as a rational function on $\mathbb{R}_{\geq0}^{k}$. Our results
are more precisely summarized in the following Theorem.

\begin{figure}
\begin{center}
\scalebox{0.8}{\includegraphics[
height=7.0424cm,
width=7.0973cm
]
{marches_alcoves.pdf}}
\caption{A reduced alcove walk on Grassmannian elements for $k=2$.}
\end{center}
\vspace{-4ex}
\end{figure}

\begin{theorem}\label{main}
\ 

\begin{enumerate}
\item To each $\vec{r}\in\mathbb{R}_{\geq0}^{k}$ corresponds a unique morphism
$\varphi:\Lambda_{(k)}\rightarrow\mathbb{R}$ nonnegative on the $k$-Schur
functions and such that $\varphi(s_{R_{a}})=r_{a}$ for any $a=1,\ldots,k.$

\item To each $\vec{r}\in\mathbb{R}_{\geq0}^{k}$ corresponds a unique matrix
$M$ in $T_{\geq0}$ whose $k$ southwest initial minors are exactly
$r_{1},\ldots,r_{k}$.

\item Both previous one-to-one correspondences are homeomorphisms, moreover
$\varphi$ and $M$ can be explicitly computed from $\vec{r}$ by using Perron
Frobenius theorem.

\item The minimal boundary of $\mathcal{B}_{k}$ is homeomorphic to the simplex
$\mathcal{S}_{k}=\{(r_{1},\ldots,r_{k})\in\mathbb{R}_{\geq0}^{k}\mid r_{1}+\cdots
+r_{k}=1\}$.

\item To each $\vec{r}\in\mathcal{S}_{k}$ corresponds a central random walk
$(v_{n})_{n\geq0}$ on affine Grassmannian elements which verifies a law of
large numbers. The coordinates of its drift are the image by $\varphi$ of
rational fractions in the $k$-Schur functions. They are moreover rational on
$\mathcal{S}_{k}$.
\end{enumerate}
\end{theorem}

This note is organized as follows. In Section 2, we recall some background on
alcoves, partitions and $k$-Schur functions and prove the first four statements of \cref{main}.\ In Section 3, we explain briefly the theory of central random walks and establish the fifth statement of \cref{main}.

Although we restrict ourselves to type $A$ in this note, we expect to extend
our approach to other types in a future work by using notably the results of
\cite{LSS} and \cite{P}.

\section{Harmonic analysis on the lattice of $k$-bounded partitions}

\subsection{The lattice $\mathcal{B}_{k}$.}

\label{subset_Lattices}In this section, we refer to \cite{LLMSSZ} and
\cite{Mac} for the material which is not defined.\ Fix $l>1$ a nonnegative
integer and set $k=l-1$. Let $\widetilde{W}$ be the affine Weyl group of type
$A_{k}^{(1)}$. As a Coxeter group, $\widetilde{W}$ is generated by the
reflections $s_{0},s_{1},\ldots,s_{k}$ so that its subgroup generated by
$s_{1},\ldots,s_{k}$ is isomorphic to the symmetric group $S_{l}$. Write
$\ell$ for the length function on $\widetilde{W}$. The group $\widetilde{W}$
determines a Coxeter arrangement by considering the hyperplanes orthogonal to
the roots of type $A_{k}^{(1)}$. The connected components of this hyperplane
arrangement yield a tessellation of $\mathbb{R}^{k}$ by alcoves on which the
action of $\widetilde{W}$ is regular.\ We denote by $A^{(0)}$ the fundamental
alcove. Write $\widetilde{R}$ for the set of affine roots of type $A_{k}%
^{(1)}$ and $R$ for its subset of classical roots of type $A_{k}$. The simple
roots are denoted by $\alpha_{0},\ldots,\alpha_{k}$ and $P$ is the weight
lattice of type $A_{k}$ with fundamental weights $\Lambda_{1},\ldots
,\Lambda_{k}$.

A reduced alcove path is a sequence of alcoves $(A_{1},\ldots,A_{m})$ such
that $A_{1}=A^{(0)}$ and for any $i=1,\ldots,m-1$, the alcoves $A_{i+1}$ and
$A_{i}$ share a common face contained in a hyperplane $H_{i}$ so that the
sequence $H_{1},\ldots,H_{m-1}$ is without repetition (each hyperplane can be
crossed only once).\ In the sequel, all the alcove paths we shall consider
will be reduced. For any $i=1,\ldots,m-1$, let $w_{i}$ be the unique element
of $\widetilde{W}$ such that $A_{i}=w_{i}(A^{(0)})$.\ Write $\vartriangleleft$
for the weak Bruhat order on $\widetilde{W}$ and $\rightarrow$ for the
covering relation $w\rightarrow w^{\prime}$ if and only if $w\vartriangleleft
w^{\prime}$ and $\ell(w^{\prime})=\ell(w)+1$. We then have $w_{1}\rightarrow
w_{2}\rightarrow\cdots\rightarrow w_{m}$. The affine Grassmannian elements are the elements $w\in\widetilde{W}$ whose
associated alcoves are exactly those located in the fundamental Weyl chamber
(that is, in the Weyl chamber containing the fundamental alcove $A^{(0)}%
$).

We shall identify a partition and its Young diagram. A $k$-bounded partition is a partition $\lambda$ such that $\lambda_{1}\leq
k$. There is a simple bijection between the $k$-bounded partitions and the affine Grassmannian elements (see \cite{LLMSSZ}
pages 18 and 19). Hence, the weak Bruhat order on $\tilde{W}$ yields a lattice structure on the $k$-bounded partitions, and the corresponding lattice is denoted by $\mathcal{B}_{k}$. So reduced alcove paths in
the fundamental Weyl chamber, saturated chains of affine Grassmannian elements
and paths in $\mathcal{B}_{k}$ naturally correspond.



\subsection{The $k$-Schur functions}
Let $\Lambda$ be\ the algebra of symmetric functions in infinitely many
variables over $\mathbb{R}$. It is endowed with a scalar product $\langle
\cdot,\cdot\rangle$ such that $\langle s_{\lambda},s_{\mu}\rangle
=\delta_{\lambda,\mu}$ for any partitions $\lambda$ and $\mu$. Let
$\Lambda_{(k)}$ be the subalgebra of $\Lambda$ generated by the complete
homogeneous functions $h_{1},\ldots,h_{k}$. In particular, $\{h_{\lambda}%
\mid\lambda$ is $k$-bounded$\}$ is a basis of $\Lambda_{(k)}$.

We now define a distinguished basis of $\Lambda_{(k)}$ related to the graph
structures of $\mathcal{B}_{k}$. Consider $\lambda$ and
$\mu$ two $k$-bounded partitions with $\lambda\subset\mu$ and $r\leq k$ a
positive integer. Recall the involution $\omega_{k}$ on $\Lambda_{(k)}$ analogue to the conjugation of partitions (see \cite{LLMSSZ}).
\begin{definition}
\label{Def_Hori_Strip}We will say that $\mu/\lambda$ is a weak horizontal
strip of size $r$ when

\begin{enumerate}
\item $\mu/\lambda$ is an horizontal strip with $r$ boxes (i.e. the boxes in
$\mu/\lambda$ belong to different columns),

\item $\mu^{\omega_{k}}/\lambda^{\omega_{k}}$ is a vertical strip with $r$
boxes (i.e. the boxes in $\mu^{\omega_{k}}/\lambda^{\omega_{k}}$ belong to
different rows).
\end{enumerate}
\end{definition}


\begin{definition}
[Pieri rule for $k$-Schur functions, \cite{LLMSSZ}]The $k$-Schur functions $s_{\kappa}^{(k)},\kappa\in\mathcal{B}_{k}$ are the
unique functions in $\Lambda_{(k)}$ such that for any $r\leq k$, $s_{(r)}^{(k)}=h_{r}$ and for any $r\leq k$, any $\kappa
\in\mathcal{B}_{k}$ we have
\begin{equation}
h_{r}s_{\kappa}^{(k)}=\sum_{\varkappa\in\mathcal{B}_{k}}s_{\varkappa}^{(k)}
\label{PieriKSchur}%
\end{equation}
where the sum is over all the $k$-bounded partitions $\varkappa$ such that
$\varkappa/\kappa$ is a weak horizontal strip of size $r$ in $\mathcal{B}_{k}$.
\end{definition}

We write $\kappa \rightarrow \mu$ when $\varkappa/\kappa$ is a weak horizontal strip of size $1$. Thanks to a geometric interpretation of the $k$-Schur
functions in terms of the homology of affine Grassmannians, Lam showed that
the product of two $k$-Schur functions has nonnegative coefficients on the basis of $k$-Schur functions:

\begin{theorem}[\cite{LamLR}] Given $\kappa$ and $\delta$ two $k$-bounded partitions, we have $s_{\kappa}^{(k)}s_{\delta}^{(k)}=\sum_{\nu\in\mathcal{B}_{k}}c_{\lambda
,\delta}^{\nu(k)}s_{\nu}^{(k)}%
$ with $c_{\lambda,\delta}^{\nu(k)}\in\mathbb{Z}_{\geq0}$.
\end{theorem}

For any $a=1,\ldots,k$, let $R_{a}$ be the rectangle partition $(k-a+1)^{a}$.

\begin{proposition}
\label{reducedCase} (See \cite{LLMSSZ}.) For each $k$-bounded partition $\lambda$, there exists a
unique irreducible $k$-bounded partition\footnote{A $k$-bounded partition is
irreducible when it contains less than $a$ parts equal to $k-a$ for any
$a=0,\ldots,k-1$.} $\widetilde{\lambda}$ and a unique sequence of nonnegative
integers $p_{1},\ldots,p_{k}$ such that
\[
s_{\lambda}^{(k)}=\prod_{a=1}^{k}s_{R_{a}}^{p_{a}}s_{\widetilde{\lambda}%
}^{(k)}.
\]
In particular, the $k$-Schur functions are completely determined by the
$k$-Schur functions indexed by the irreducible $k$-bounded partitions and by
the ordinary Schur functions $s_{(k-a+1)^{a}}$, $a=1,\ldots,k$.
\end{proposition}

\subsection{Harmonic functions and minimal boundary of $\mathcal{B}_{k}$}

\begin{definition}
A function $f:\mathcal{B}_{k}\rightarrow\mathbb{R}$ is harmonic when $f(\lambda)=\sum_{\lambda\rightarrow\mu}f(\mu)\text{ for any }\lambda
\in\mathcal{B}_{k}.
$ We denote by $\mathcal{H}(\mathcal{B}_{k})$ the set of harmonic functions on
$\mathcal{B}_{k}$.
\end{definition}

The harmonic functions on
$\mathcal{B}_{k}$ correspond to the right eigenvectors associated to the eigenvalue $1$ of $\mathcal{M}$, the infinite adjacency matrix of the graph $\mathcal{B}_{k}$. One can also consider $t$-harmonic functions
which correspond to the right eigenvectors for $\mathcal{M}$ associated to the
eigenvalue $t$. Clearly $\mathcal{H}(\mathcal{B}_{k})$ is a vector space over
$\mathbb{R}$. In fact, we mostly restrict ourselves to the set $\mathcal{H}%
^{+}(\mathcal{B}_{k})$ of positive harmonic functions for which $f$ takes
values in $\mathbb{R}_{\geq0}$. Then, $\mathcal{H}^{+}(\mathcal{B}_{k})$ is a
cone since it is stable by addition and multiplication by a positive real. To
study $\mathcal{H}^{+}(\mathcal{B}_{k})$, we only have to consider its subset
$\mathcal{H}_{1}^{+}(\mathcal{B}_{k})$ of normalized harmonic functions such
that $f(1)=1$. In fact, $\mathcal{H}_{1}^{+}(\mathcal{B}_{k})$ is a convex set
and its structure is controlled by its extremal subset $\partial
\mathcal{H}^{+}(\mathcal{B}_{k})$.

\noindent We aim to characterize the extremal positive harmonic functions
defined on $\mathcal{B}_{k}$ and obtain a simple parametrization of
$\partial\mathcal{H}^{+}(\mathcal{B}_{k})$. By applying the Pieri rule on
$k$-Schur functions, we get%
\[
s_{\lambda}s_{(1)}=\sum_{\lambda\rightarrow\mu}s_{\mu}%
\]
for any $k$-bounded partitions $\lambda$ and $\mu$.\ This means that
$\mathcal{B}_{k}$ is a so-called multiplicative graph with associated algebra
$\Lambda_{(k)}$. Moreover, if we denote by $K$ the positive cone spanned by
the set of $k$-Schur functions, we can apply the ring theorem of Kerov and
Vershik (see for example \cite[Section 8.4]{LT}) which characterizes the
subset of extreme points $\partial\mathcal{H}^{+}(\mathcal{B}_{k})$.\ Denote
by $\mathrm{Mult}^{+}(\Lambda_{(k)})\subset\Lambda_{(k)}^{\ast}$ the set of
multiplicative functions on $\Lambda_{(k)}$ which are nonnegative on $K$ and
equal to $1$ on $s_{1}$. So a linear form $f\in \Lambda_{(k)}^{*}$ belongs to $\mathrm{Mult}%
^{+}(\Lambda_{(k)})$ when $f(K)\subset \mathbb{R}^{+}$, $f(s_{1})=1$ and it
satisfies $f(uv)=f(u)f(v)$ for any $u,v\in\Lambda_{(k)}$. Note that
$i:\mathcal{B}_{k}\longrightarrow\Lambda_{(k)}$ such that $i(\lambda
)=s_{\lambda}^{(k)}$ induces a map $i^{\ast}:\Lambda_{(k)}^{\ast
}\longrightarrow F(\mathcal{B}_{k},\mathbb{R})$. Then, the ring theorem yields the following algebraic characterization of $\partial\mathcal{H}%
^{+}(\mathcal{B}_{k})$.

\begin{proposition}
\label{multiGraphExtreme}The map $i^{\ast}$ yields an homeomorphism between
$\mathrm{Mult}^{+}(\Lambda_{(k)})$ and $\partial\mathcal{H}^{+}(\mathcal{B}%
_{k})$.\ 
\end{proposition}

Since $i(\mathcal{B}_{k})$ is a basis of $\Lambda_{(k)}$, this means that
$\partial\mathcal{H}(\mathcal{B}_{k})$ is completely determined by the
$\mathbb{R}$-algebra morphisms $\varphi:\Lambda_{(k)}\rightarrow\mathbb{R}$
such that $\varphi(s_{1})=1$ and $\varphi(s_{\lambda}^{(k)})\geq0$ for any
$k$-bounded partition $\lambda$. Each function $f\in\partial\mathcal{H}%
^{+}(\mathcal{B}_{k})$ can then be written $f=\varphi\circ i$.

\subsection{The matrix $\Phi$}

\label{subsec_MatrixFI}By \cref{reducedCase}, each morphism
$\varphi:\Lambda_{(k)}\rightarrow\mathbb{R}$ is uniquely determined by its
values on the rectangle Schur functions $s_{R_{a}},1\leq a\leq k$ and on each
$s_{\tilde{\lambda}}^{(k)}$ where $\tilde{\lambda}$ is an irreducible
$k$-bounded partition. Set $r_{a}=\varphi(s_{R_{a}}),a=1,\ldots,k$ and
$\vec{r}=(r_{1},\ldots,r_{k})$. Denote by $\mathcal{P}_{\mathrm{irr}}$
the set of irreducible $k$-bounded partitions (including the empty partition).
Then, for $\lambda\in\mathcal{P}_{\mathrm{irr}}$,
\begin{equation}
\varphi(s_{\lambda}^{(k)})\varphi(s_{(1)})=\sum_{\lambda\rightarrow\mu}%
\varphi(s_{\mu}^{(k)}). \label{PieriPhi}%
\end{equation}
By \cref{reducedCase}, for each $k$-bounded partition $\mu$ there
exists a sequence $\{p_{1}^{\mu},p_{2}^{\mu}\ldots,p_{k}^{\mu}\}$ of integers and an irreducible partition $\widetilde{\mu}$ such that
\begin{equation}
s_{\mu}^{(k)}=\prod_{a=1}^{k}s_{R_{a}}^{p_{a}^{\mu}}s_{\widetilde{\mu}}%
^{(k)}\text{ and thus }\varphi(s_{\mu}^{(k)})=\prod_{a=1}^{k}r_{a}^{p_{a}%
^{\mu}}\varphi(s_{\widetilde{\mu}}^{(k)}). \label{decompos}%
\end{equation}
Hence by setting
\[
\varphi_{\lambda\nu}=\sum_{\substack{\lambda\rightarrow\mu\\\widetilde{\mu
}=\nu}}\prod_{1\leq a\leq k}r_{a}^{p_{a}^{\mu}}%
\]
we get
\[
\varphi(s_{\lambda}^{(k)})=\sum_{\nu\in\mathcal{P}_{\mathrm{irr}}}%
\varphi_{\lambda\nu}\varphi(s_{\nu}^{(k)}).
\]
Let $\Phi_{(r_{1},\ldots,r_{k})}:=(\varphi_{\nu\lambda})_{\lambda,\nu
\in\mathcal{P}_{\mathrm{irr}}}$ \footnote{Observe we have defined
$\Phi_{(r_{1},\ldots,r_{k})}$ as the transpose of the matrix $(\varphi
_{\lambda\mu})_{\lambda,\nu\in\mathcal{P}_{\mathrm{irr}}}$ to make it
compatible with the multiplication by $s_{(1)}$ used in \cref{Secion_Field}.} and define $f\in\mathbb{R}^{\mathcal{P}_{\mathrm{irr}}}$
as the vector $(\varphi(s_{\lambda}^{(k)}))_{\lambda\in\mathcal{P}%
_{\mathrm{irr}}}$. When there is no risk of confusion, we simply write $\Phi$
instead of $\Phi_{(r_{1},\ldots,r_{k})}$. The vector $f$ is a left eigenvector
of $\Phi$ for the eigenvalue $\varphi(s_{1})$ with positive entries having
value $1$ on $\emptyset$ and $\varphi(s_{1})$ on $s_{1}$.

Recall that a matrix $M\in M_{n}(\mathbb{R})$ with nonnegative entries is
irreducible if and only if for each $1\leq i,j\leq n$ there exists $n\geq1$
such that $(M^{n})_{ij}>0$. In view of using the Perron-Frobenius theorem, we have the following result on the irreducibility of $\Phi$.

\begin{proposition}
\label{irredPhi}Consider $\vec{r}=(r_{1},\ldots,r_{k})\in\mathbb{R}_{\geq
0}^{k}.$ Then, the matrices $\Phi_{(r_{1},\ldots,r_{k})}$ and $\Phi
_{(r_{1},\ldots,r_{k})}^{t}$ associated to $\varphi$ are irreducible if and
only if for all $1\leq a\leq k-1$, $r_{a}$ or $r_{a+1}$ is positive.
\end{proposition}

\subsection{Field extensions and $k$-Schur functions}
\label{Secion_Field}

Recall that $\Lambda_{(k)}\mathbb{=R}[h_{1},\ldots,h_{k}]$.\ Since
$h_{1},\ldots,h_{k}$ are algebraically independent over $\mathbb{R}$, we can
consider the fraction field $\mathbb{L}=\mathbb{R}(h_{1},\ldots,h_{k}%
)$.\ Write $\mathbb{A}=\mathbb{R}[s_{R_{1}},\ldots s_{R_{k}}]$ the subalgebra
of $\Lambda_{(k)}$ generated by the rectangle Schur functions $s_{R_{a}%
},a=1,\ldots,k$. By standard arguments one can prove that $s_{R_{1}},\ldots s_{R_{k}}$ are algebraically
independent over $\mathbb{R}$, so that we can consider the fraction field $\mathbb{K}=\mathbb{R}(s_{R_{1}%
},\ldots s_{R_{k}})$ of the algebra $\mathbb{A}$. By classical results from \cite{KR} and \cref{reducedCase}, $s_{1}=h_{1}$ is
algebraic over $\mathbb{K}$. The following result is a main step in the proof of \cref{main}.

\begin{theorem}\label{Cor_Delta}
We have $\mathbb{L}=\mathbb{K}(s_{(1)})$, that is $s_{(1)}$ is a primitive
element for $\mathbb{L}$ regarded as an extension of $\mathbb{K}$.
In particular, there exist $\Delta\in\mathbb{A}$ and for each irreducible
$k$-bounded partition $\kappa$ a polynomial $P_{\kappa}\in\mathbb{A}[T]$ such
that
\[
s_{\kappa}^{(k)}=\frac{1}{\Delta}P_{\kappa}(s_{(1)}).
\]
In particular, for any morphism $\varphi:\Lambda_{(k)}\rightarrow\mathbb{R}$
such that $\varphi(\Delta)\neq0$ we have%
\[
\varphi(s_{\kappa}^{(k)})=\frac{1}{\varphi(\Delta)}\varphi(P_{\kappa}%
)\varphi(s_{(1)}).
\]

\end{theorem}
\subsection{Minimal boundary of $\mathcal{B}_{k}$} 
We give here a sketch of the proof of the first part of \cref{main}. Set $\mathcal{R}=\lbrace s_{R_{i}},1\leq i\leq k\rbrace$. Recall that by \cref{multiGraphExtreme}, we have $\partial
\mathcal{H}^{+}(\mathcal{B}_{k})\simeq\mathrm{Mult}^{+}(\Lambda_{(k)})$. Using the particular structure of morphisms on $\Lambda_{(k)}$ given by \eqref{decompos} and the Perron-Frobenius Theorem, one can prove the following crucial step:
\begin{theorem}
\label{principalDomain} \ 

\begin{enumerate}
\item Let $\varphi:\mathbb{A}\rightarrow\mathbb{R}$ be a morphism, nonnegative
on $\mathcal{R}$ and such that its associated matrix $\Phi$ is irreducible.
Then there exists a unique morphism $\tilde{\varphi}:\Lambda_{(k)}%
\longrightarrow\mathbb{R}_{\geq0}$ extending $\varphi$, nonnegative on the
$k$-Schur functions and positive on $\mathcal{I}=\{s_{\lambda}\mid\lambda\in\mathcal{P}_{\mathrm{irr}}\}$.

\item A positive morphism $\varphi:\Lambda_{(k)}\longrightarrow\mathbb{R}$ is
uniquely determined by its values on $\mathcal{R}$.
\end{enumerate}
\end{theorem} 
Remark that this solves almost all the cases thanks to \cref{irredPhi}. To conclude the proof of the first part of \cref{main}, we need to extend the previous result to the case when $\Phi$ is not irreducible. This extension is proven by classical continuity arguments using the algebraic structure of $\Lambda_{(k)}$ given in \cref{Secion_Field}. Since morphisms on $\mathbb{A}$ which are nonnegative on $\mathcal{R}$ are homeomorphic to $\mathbb{R}^{k}_{\geq 0}$, simple computations yield the following result.
\begin{proposition}
$\mathrm{Mult}^{+}(\Lambda_{(k)})$ is homeomorphic to $\mathbb{R}_{\geq 0}^{k}$ and $\partial\mathcal{H}^{+}(\mathcal{B}_{k})$ is homeomorphic to $\mathcal{S}%
_{k}=\{(r_{1},\ldots,r_{k})\in\mathbb{R}_{\geq0}^{k}\mid r_{1}+\cdots
+r_{k}=1\}$.
\end{proposition}

\subsection{Rietsch parametrization of Toeplitz matrices}

Consider the variety $T_{\geq0}\subset\mathbb{R}_{>0}^{k}$ of totally
nonnegative unitriangular Toeplitz $(k+1)\times(k+1)$ matrices

\[
M=\left[
\begin{array}
[c]{ccccc}%
1 &  &  &   \\
h_{1} & 1 &  &  &  \\
\vdots & h_{1} & \ddots &  & \\

h_{k-1} & \vdots & \vdots  & \ddots & \\
h_{k} & h_{k-1} & \cdots  & h_{1} & 1
\end{array}
\right]  .
\]

The set $T_{>0}$ of totally positive unitriangular Toeplitz $(k+1)\times(k+1)$
matrices is defined as the subset of $T_{\geq0}$ of matrices $M$ whose minors
with no row and no column in the upper part of $M$ are positive. By Theorem
3.2.1 in \cite{BFZ}, $M$ is totally positive if and only if for $a=1,\ldots
,k$, the $a\times a$ initial minors obtained by selecting $a$ rows of $M$
arbitrary and then the first $a$ columns of $M$ are positive.

\begin{lemma}
\ \label{Lemma_closed}

\begin{enumerate}
\item The previous initial minors are equal to Schur functions $s_{\lambda}$,
where the maximal hook of the partition $\lambda$ has length less or equal to
$k$.

\item We have $\overline{T}_{>0}=T_{\geq0}$ that is, each totally nonnegative
unitriangular Toeplitz matrix is the limit of a sequence of totally positive
unitriangular Toeplitz matrices.
\end{enumerate}

\end{lemma}

Observe in particular that for any $a=1,\ldots,k$, the initial minor
$\Delta_{\lbrack k-a+1,k]}$ gives the value $r_{a}$ of the rectangle Schur
function $s_{R_{a}}$ evaluated in $(h_{1},\ldots,h_{k})$. In \cite{Ri},
Rietsch obtained the parametrization of $T_{\geq0}$ by using the
quantum cohomology of partial flag varieties. Our approach permits to reprove this theorem and makes both the associated homeomorphism and its converse explicit.

\begin{theorem}
\label{Th_RiRef}The map $g:T_{\geq0}\rightarrow\mathbb{R}_{\geq0}^{k}:(h_{1},\ldots,h_{k})\longmapsto(r_{1},\ldots,r_{k})$ is a homeomorphism. Moreover, $g^{-1}$ can be made explicit from the Perron-Frobenius theorem applied on $\Phi$ and \cref{Cor_Delta}.
\end{theorem}

%\begin{theorem}
%\label{Th_RiRef}We have $T_{>0}=V$ and $T_{\geq0}=\overline{V}$, in particular
%the map $g:T_{\geq0}\rightarrow\mathbb{R}_{\geq0}^{k}$ is a homeomorphism$.$
%\end{theorem}
So our \cref{Th_RiRef}
permits in fact to compute the nonnegative Toeplitz matrix associated to any
point of $\mathbb{R}_{\geq0}^{k}$ (i.e. to reconstruct $M$ from the datum of the minors
$(r_{1},\ldots,r_{k})$). 
When $\Phi$ is irreducible this is achieved directly by applying the Perron-Frobenius theorem which gives the  $\varphi(s_{\kappa})$'s 
with $\kappa$ any irreducible $k$-bounded partition of \cref{Cor_Delta}. In particular, one so gets $\varphi(s_{(a)})=h_{a}$ for $a=1,\ldots
,k$. In general we show that the formulas of \cref{Cor_Delta} can be extended by continuity when $\Delta=0$.

\section{Markov chains on alcoves}

\subsection{Central Markov chains on alcoves from harmonic functions}

\label{definitionCentral_Markov}

Recall the notation of \cref{subset_Lattices} for the notion of reduced
alcove paths.\ A probability distribution on reduced alcove paths is said
central when the probability $p_{\pi}$ of the path $\pi=(A_{1}=A^{(0)}%
,A_{2},\ldots,A_{m})$ only depends on $m,A_{1}$ and $A_{m}$, that is only on
its length and its alcoves ends. In the situation we consider, central random reduced alcove paths in the Weyl chamber correspond to central random paths on
$\mathcal{B}_{k}$. They are determined by the positive harmonic
functions on $\mathcal{B}_{k}$ (see \cite{Ker}).

More precisely any central probability distribution on the affine Grassmannian
alcove paths can be written
\[
p_{\pi}=\frac{h(\mu)}{h(\lambda)}%
\]
where $h\in\mathcal{H}^{+}(\mathcal{B}_{k})$ is positive and for any path
$\pi=(A_{1},\ldots,A_{m}),$ $\mu$ and $\lambda$ are the $k$-bounded partitions
associated to $A_{1}$ and $A_{m}$. Also we then get a Markov chain on
$\mathcal{B}_{k}$ (or equivalently on the affine Grassmannian elements) with
transition matrix
\[
\Pi(\lambda,\mu)=\frac{h(\mu)}{h(\lambda)}.
\]
When the harmonic function $h$ is extremal, it corresponds to a morphism $\varphi$ on $\Lambda
_{(k)}$ with $\varphi(s_{(1)})=1$, nonnegative on the $k$-Schur functions. We
get an extremal central distribution on the trajectories starting at $A^{(0)}$
verifying $p_{\pi}=\frac{\varphi(s_{\mu}^{(k)})}{\varphi(s_{\lambda}^{(k)})}%
$.\ The associated Markov chain has then the transition matrix $\Pi
(\lambda,\mu)=\frac{\varphi(s_{\mu}^{(k)})}{\varphi(s_{\lambda}^{(k)})}$.

\subsection*{Involutions on the reduced walk}

\label{SubSec_Invol}
Let $\mathcal{A}_{k}$ be the set of alcoves in the dominant Weyl chamber. We describe here two important involutions on the set of irreducible $k$-bounded partitions. The first symmetry is due to the action of $\omega_{k}$ on $\Lambda_{(k)}$
which sends $s_{R_{a}}$ to $s_{R_{k-a}}$ for any $a=1,\ldots,k$. By bijection between alcoves in the Weyl chamber and $k$-bounded partitions, this symmetry yields an involution $\Omega$ on $\mathcal{A}_{k}$, which corresponds to the involution on
the Dynkin diagram of affine type $A_{k}^{(1)}$ fixing the node $0$ and
sending each node $i\in\{1,\ldots,k\}$ to $i^{\ast}=k+1-i$. 

For the second symmetry, we need some basic facts about the affine Coxeter
arrangement of type $A_{k}^{(1)}$. For any root $\alpha$ and any integer, let
$H_{\alpha,r}$ be the affine hyperplane
\[
H_{\alpha,r}=\{v\in\mathbb{R}^{k},\langle v,\alpha\rangle=r\}.
\]
We denote by $s_{\alpha,r}$ the reflection with respect to this hyperplane and
for $\beta$ in the weight lattice $P$, we write $t_{\beta}$ for the
translation by $\beta$. We have then $s_{\alpha,r}=t_{r\alpha}s_{\alpha,0}$. Affine Grassmannian elements are in bijection with alcoves in the dominant
Weyl chamber through a map $w\mapsto A_{w}$ such that $w\rightarrow w^{\prime
}$ (that is we have a covering relation for the weak order from $w$ to
$w^{\prime}$) if and only if there is a hyperplane $H_{\alpha,r}$ such that
$A_{w^{\prime}}=s_{\alpha,r}(A_{w})$. In this case, we write
$w\xrightarrow{\alpha,r}w^{\prime}$.

Write $v_{w}$ for the center of the alcove $A_{w}$ (defined as the mean of the
its extreme weights). Any alcove $A_{w}$ is completely determined by its center $v_{w}$, and with this notation $w\xrightarrow{\alpha,r}w^{\prime
}$ if and only if $v_{w^{\prime}}=s_{\alpha,r}(v_{w})$ and $r<\langle
\alpha,v_{w^{\prime}}\rangle<r+1$.

\ Let $\mathsf{B}$ be the set of alcoves which are included in the fundamental parallelepiped $\lbrace v\in \mathbb{R}^{k}| \langle v,\alpha_{i}\rangle\in]0,1[\text{ for } i=1,\ldots,k\rbrace$. This set is in bijection with the set of irreducible $k$-bounded partitions. For example, after identifying an alcove with its center, we have $\mathsf{B}=\lbrace \Lambda_{1}/3+\Lambda_{2}/3,2\Lambda_{1}/3+2\Lambda_{2}/3\rbrace$ for $k=2$.
Consider now the
involution $I:\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ defined by
\[
I=t_{\rho}\circ w_{0},
\]
where $\rho=\sum_{i=1}^{k}\Lambda_{i}$ and $w_{0}$ is the longest element of
$W.$ Observe we indeed get an involution because $w_{0}\circ t_{\rho}\circ
w_{0}=t_{-\rho}$. The involution $I$ has the simple expression $I(l_{1}\Lambda_{1}+\dots +l_{k}\Lambda_{k})=(1-l_{k})\Lambda_{1}+\dots+(1-l_{1})\Lambda_{k}$ for $l_{1},\ldots,l_{k}\in \mathbb{R}$.

\begin{lemma}
\label{Lem_IonB}The involution $I$ restricts to an involution on the set
$\mathsf{B}$, and $A\rightarrow A^{\prime}$
if and only if $I(A^{\prime})\rightarrow I(A)$.
\end{lemma}
Hence, $I$ yields a bijection on $\mathsf{B}$ which preserves the graph structure coming from $\mathcal{B}_{k}$.

\subsection*{Drift under harmonic measures}

We
denote by $\Gamma_{f}(\mathcal{A}_{k})$ the set of reduced finite alcove paths
which start at $A^{(0)}$ and remain in the dominant Weyl chamber. For any $A$
in $\mathcal{A}_{k}$, write $\lambda_{A}\in\mathcal{B}_{k}$ its corresponding
$k$-bounded partition. Conversely recall that for any $\lambda\in
\mathcal{B}_{k}$, $A_{w_{\lambda}}\in\mathcal{A}_{k}$ is the alcove associated
to $\lambda$. Let $\varphi$ be an extremal harmonic measure on $\mathcal{B}%
_{k}$ associated to $\vec{r}=(r_{1},\ldots,r_{k})\in\mathbb{R}^{k}$, and let
$(A_{n})_{n\geq1}$ be the central Markov chain on $\mathcal{A}_{k}$ defined in
\cref{definitionCentral_Markov}. By considering for each $n\geq1$ the
center $v_{n}$ of the alcove $A_{n}$, we get a genuine random walk
$(v_{n})_{n\geq1}$ on $\mathbb{R}^{k}$. Our goal is now to prove the law of
large numbers for this random walk. For any alcove $A$, set $\overline{A}=\mathsf{I}\Omega(A)$.

\begin{theorem}
\label{result_drift} \ 
As $n$ goes to infinity, the normalized random walk $\left(  \frac{1}%
{n}v_{n}\right)  _{n\geq1}$ converges almost surely to a vector $v_{\varphi
}\in\mathbb{R}^{k}$, whose coordinate on
$\Lambda_{i}$ satisfies
\[
v_{\varphi}(i)=\varphi\left(  \frac{s_{R_{i}}}{\sum_{A\in\mathsf{B}}%
s_{\lambda_{A}}^{(k)}s_{\lambda_{\overline{A}}}^{(k)}}\sum
_{\substack{e:A\rightarrow A^{\prime}\\c(e)=i}}s_{\lambda_{\overline{A}}%
}^{(k)}s_{\lambda_{A^{\prime}}}^{(k)}\right)
\]
which is a rational function on $\mathbb{R}^{k}$.
\end{theorem}
\bigskip

\acknowledgments{Both authors thank the international
French-Mexican laboratory LAISLA, the CIMAT and the IDP for their support and
hospitality. We are also grateful to J. Guilhot for numerous discussions on
the combinatorics of alcoves and T. Lam for indicating us references potentially useful to extend results of the paper beyond type $A$.}

\vspace{-2ex}
\printbibliography

\end{document}
