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\begin{document}
\title[Conditioned one-way random walk and representation theory]{Conditioned one-way simple random walk and combinatorial representation theory}
\date{}
\author{C\'{e}dric Lecouvey, Emmanuel Lesigne and Marc Peign\'{e}}
\address{\noindent\hskip-12pt Laboratoire de Math\'ematiques et Physique Th\'eorique
(UMR CNRS 6083)\\
Universit\'e Fran\c{c}ois-Rabelais, Tours \\
F\'ed\'eration de Recherche Denis Poisson - CNRS\\
Parc de Grandmont, 37200 Tours, France. }
\subjclass[2010]{05E05, 05E10, 60G50, 60J10, 60J22}

\begin{abstract}
A one-way simple random walk is a random walk in the quadrant $\mathbb{Z}%
_{+}^{n}$ whose increments are elements of the canonical base. In relation
with representation theory of Lie algebras and superalgebras, we describe the
law of such a random walk conditioned to stay in a closed octant, a semi-open
octant, or other types of semi-groups. The combinatorial representation theory
of these algebras allows us to describe a generalized Pitman transformation
which realizes the conditioning on the set of paths of the walk. We pursue
here a direction initiated by O'Connell and his coauthors, and also developed
by the authors.

\end{abstract}
\maketitle


\thispagestyle{myheadings} \font\rms=cmr8 \font\its=cmti8 \font\bfs=cmbx8

\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 70 \rms (2014), Article~B70b\hfill}
\def\thepage{}

\section{Introduction}

Let $B=(\varepsilon_{1},\ldots,\varepsilon_{n})$ be the standard basis of
$\mathbb{R}^{n}$. The one-way simple walk is defined as the random walk
$\mathcal{W}=(\mathcal{W}_{\ell}=X_{1}+\cdots+X_{\ell})_{\ell\geq1}$, where
$(X_{k})_{k\geq1}$ is a sequence of independent and identically distributed
random variables with values in the base $B$ and with common mean vector
$\mathbf{m}${.} In this paper, we generalize some results due to O'Connell
\cite{OC1,OC2} giving the law of the random walk $\mathcal{W}$ conditioned to
never exit the cone $\mathcal{C}^{\emptyset}=\{x=(x_{1},\ldots,x_{n}%
)\in\mathbb{R}^{n}\mid x_{1}\geq\cdots\geq x_{n}\geq0\}$\footnote{We will
discuss the cases of the Lie algebras $\mathfrak{gl}(n),\mathfrak{gl}(m,n)$
and $\mathfrak{q}(n)$ in parallel. The exponents $\emptyset$, $h$ or $s$ will
refer to these three cases, respectively, in this order. See \S~\ref{Wsc}.}.
This is achieved in \cite{OC1} by considering first a natural transformation
$\mathfrak{P}$ derived from the 
Robinson--Schensted--Knuth correspondence which associates to any path with
steps in $B$ a path in the cone $\mathcal{C}^{\emptyset}$, next by checking
that the image of the random walk $\mathcal{W}$ by this transformation is a
Markov chain which has the same law as $\mathcal{W}$ conditioned to never exit
$\mathcal{C}^{\emptyset}$. The entries of the transition matrix of the Markov
chain so obtained can be expressed as quotients of Schur functions (the Weyl
characters of $\mathfrak{sl}_{n}$) with variables specialized to the
coordinates of $\mathbf{m}$.

One can introduce a similar transformation $\mathfrak{P}$ for a wide class of
random walks {$(X_{1}+\cdots+X_{\ell})_{\ell\geq1}$ }for which the variables
$X_{k}$ take values in the set of weights of a fixed representation $V$ of a
simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. This was done in
\cite{BBOC1} in the case of equidistributed random variables $X_{k}$ and in
\cite{LLP} in general. The transformation $\mathfrak{P}$ is then defined by
using Kashiwara's crystal basis theory \cite{Kashi} (or, equivalently, by the
Littelmann path model). When $V$ is a minuscule representation, we also obtain
in \cite{LLP} the law of the associated random walk conditioned to never exit
the dominant Weyl chamber $\mathcal{C}$ of $\mathfrak{g}$ under the crucial
assumption (also required in \cite{OC1} and \cite{OC2}) that $\mathbf{m}%
=\mathbb{E}(X_{k})$ lies in the interior of $\mathcal{C}$. The transition
matrix obtained has a simple expression in terms of the Weyl characters of the
irreducible representations of $\mathfrak{g}$.

It is then a natural question to try to extend the results of \cite{OC1} and
\cite{LLP} to other conditionings. In this paper, we generalize the results of
\cite{OC1} to the one-way simple walk conditioned to stay in some semigroups
$\mathcal{C}^{h}$ and $\mathcal{C}^{s}$ which are the analogues of
$\mathcal{C}^{\emptyset}$ for the Lie superalgebras $\mathfrak{gl}(m,n)$ and
$\mathfrak{q}(n),$ respectively. In particular, the points with integer
coordinates of $\mathcal{C}^{h}$ are parametrized by hook partitions and those
of $\mathcal{C}^{s}$ by strict partitions with at most $n$ parts. Here we must
also assume that the drift $\mathbf{m}$ lies in the interior of the cone
generated by the semigroups considered.

%First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire
%--restoring the headers and pagenumbering
\pagenumbering{arabic} \addtocounter{page}{1}
\markboth{\SMALL C\'EDRIC LECOUVEY, EMMANUEL LEDIGNE AND MARC PEIGN\'E}{\SMALL
CONDITIONED ONE-WAY RANDOM WALK AND REPRESENTATION THEORY}

In both cases, this is also achieved by introducing a Pitman type transform
$\mathfrak{P}$. There are nevertheless important differences with the Pitman
transforms used in \cite{OC1}, \cite{BBOC1}, and \cite{LLP}; indeed, these
transforms can be obtained by interpreting each path as a vertex in a
Kashiwara crystal and by then applying raising crystal operators until the
highest weight (source) vertex is reached. A contrario, the situation is more
complicated for $\mathfrak{gl}(m,n)$ and $\mathfrak{q}(n)$, essentially
because the highest weight vertices in the associated crystals are not so
easily tractable. To overcome these complications, we will define the
transformation $\mathfrak{P}$ by using analogues of the
Robinson--Schensted--Knuth insertion procedure on tableaux introduced in
\cite{BKK} and more recently in \cite{GHSKM}.

Entries of the transition matrix of the one-way simple walk $\mathcal{W}$
conditioned to stay in $\mathcal{C}^{s}$ (respectively in $\mathcal{C}^{h}$)
can be expressed in terms of $P$-Schur functions (respectively supersymmetric
functions).
A key ingredient to compute this transition matrix is the asymptotic behavior
of the tensor multiplicities corresponding to the vector representation.
Although these limits could also be deduced from results by Kerov \cite{K} and
Nazarov \cite{N} on asymptotic representation theory of the symmetric and spin
symmetric groups, we obtain them here by different methods, probabilistic in
nature, which moreover generalize to other Lie algebras (see \cite{LLP}).
Namely, we derive these asymptotic behaviors from an extension of a quotient
local limit theorem established in \cite{LLP} for random walks conditioned to
stay in semigroups.

We study the three conditionings of the one-way simple walk in $\mathcal{C}%
^{\emptyset},\mathcal{C}^{h}$, and $\mathcal{C}^{s}$ simultaneously, by using
representation theory of $\mathfrak{gl}(n),\mathfrak{gl}(m,n)$, and
$\mathfrak{q}(n)$, respectively. We introduce the generalized Pitman
transformations by using insertion procedures on tableaux and
Robinson--Schensted--Knuth correspondences (as in \cite{OC1}) rather than
crystal basis theory (as in \cite{BBOC1} and \cite{LLP}). We thus avoid
technical difficulties inherent to crystal basis theory of the superalgebras
$\mathfrak{gl}(m,n)$ and $\mathfrak{q}(n)$. Throughout the paper, we provide
explicit examples for each situation, in order to help the reader to
understand the used tools and to visualize the Pitman transform.

\bigskip

The paper is organized as follows. Sections~\ref{Sec_Markov}, \ref{Sec_RT} and
\ref{sec_ComTab} are review sections. They are devoted to basics on Markov
chains, on representation theory, and on combinatorial representation theory
of the algebras $\mathfrak{gl}(n),\mathfrak{gl}(m,n)$ and $\mathfrak{q}(n)$,
respectively. In particular, we completely recall the relevant notions of
Robinson--Schensted--Knuth correspondences. Section~\ref{Sec_Pitman} introduces
the generalized Pitman transform. Without any hypothesis on the drift, we show
that it maps the one-way simple walk onto a Markov chain whose transition
matrix is computed. The main result of the paper (Theorem~\ref{Th_coincide}),
giving the law of the conditioned one-way simple walk with suitable drift, is
stated in Section~\ref{Sec_restric}. The appendix is devoted to the proof of
Proposition~\ref{Cor_RSK} for which complements on crystal basis theory for
superalgebras are required. In conclusion and briefly speaking, this article
simultaneously revisits O'Connell's results in the case of the Lie algebra
$\mathfrak{g}(n)$ and extends them to the super Lie algebras $\mathfrak{gl}%
(m,n)$ and $\mathfrak{q}(n)$, hence to new semi-groups. As in \cite{LLP} a
novelty is the use of a local limit theorem for the conditioned random walk.
We also put generalizations of the Robinson--Schensted--Knuth correspondence
to the fore rather than Kashiwara's crystal basis theory, which becomes far
more complicated in the super Lie algebra case.

\section{Markov chains}

\label{Sec_Markov}We now recall the background on Markov chains and their
conditioning that we use in the sequel.

\subsection{Markov chains and conditioning}

\label{mcc}

Consider a probability space $(\Omega,\mathcal{T},\mathbb{P})$ and a countable
set $M$. Let $Y=(Y_{\ell})_{\ell\geq0}$ be a sequence of random variables
defined on $\Omega$ with values in $M$. The sequence $Y$ is a Markov chain
when
\[
\mathbb{P}[Y_{\ell+1}=y_{\ell+1}\mid Y_{\ell}=y_{\ell},\ldots,Y_{0}%
=y_{0}]=\mathbb{P}[Y_{\ell+1}=y_{\ell+1}\mid Y_{\ell}=y_{\ell}]
\]
for all $\ell\geq0$ and $y_{0},\ldots,y_{\ell},y_{\ell+1}\in M$. The Markov
chains considered in the sequel will also be assumed time homogeneous, that
is,
\[
\mathbb{P}[Y_{\ell+1}=y\mid Y_{\ell}=x]=\mathbb{P}[Y_{\ell}=y\mid Y_{\ell
-1}=x]
\]
for all $\ell\geq1$ and $x,y\in M$; the transition probability from $x$ to $y$
is then defined by $\Pi(x,y):=\mathbb{P}[Y_{1}=y\mid Y_{0}=x]$ and we refer to
$\Pi$ as the transition matrix of the Markov chain $Y$. The distribution of
$Y_{0}$ is called the initial distribution of the chain $Y$. It is well known
that the initial distribution and the transition probability determine the law
of the Markov chain and that, given a probability distribution and a
transition matrix on $M$, there exists an associated Markov chain.

Let $Y$ be a Markov chain on $(\Omega,\mathcal{T},\mathbb{P})$ whose initial
distribution has full support, i.e., $\mathbb{P}[Y_{0}=x]= \mathbb{P}%
\{\omega\in\Omega\mid Y_{0}(\omega)=x\}>0$ for all $x\in M$. Let $\mathcal{C}$
be a nonempty subset of $M$ and consider the event $S=\{\omega\in\Omega\mid
Y_{\ell}(\omega)\in\mathcal{C}$ for all $\ell\geq0\}$. Assume that $\mathbb{P}
[S\mid Y_{0}=\lambda]>0$ for all $\lambda\in\mathcal{C}$. This implies that
$\mathbb{P}[S]>0$, and we can consider the conditional probability
$\mathbb{Q}$ relative to this event: $\mathbb{Q}[\,\cdot\,] =\mathbb{P}%
[\,\cdot\,|S]$.

It is easy to verify that, under this new probability $\mathbb{Q}$, the
sequence $(Y_{\ell})_{\ell\geq0}$ is still a Markov chain, with values in
$\mathcal{C}$, and with transition probabilities given by
\begin{equation}
\mathbb{Q}[Y_{\ell+1}=\lambda\mid Y_{\ell}=\mu]=\mathbb{P}[Y_{\ell+1}
=\lambda\mid Y_{\ell}=\mu]\ \frac{\mathbb{P}[S\mid Y_{0}=\lambda]}
{\mathbb{P}[S\mid Y_{0}=\mu]}. \label{reco}%
\end{equation}
We will denote this Markov chain by $Y^{\mathcal{C}}$, and the restriction of
the transition matrix $\Pi$ to the entries which are in $\mathcal{C}$ by
$\Pi^{\mathcal{C}}$ (in other words, $\displaystyle
\Pi^{\mathcal{C}}=\left(  \Pi(\lambda,\mu)\right)  _{\lambda,\mu\in
\mathcal{C}}$.)

\subsection{The Doob $h$-transform}

\label{sub_sec_Doobh}

A \emph{substochastic matrix} on the set $M$ is a map $\Pi:M\times
M\rightarrow\lbrack0,1]$ such that $\sum_{y\in M}\Pi(x,y)\leq1$ for all $x\in
M$. If $\Pi$ and $\Pi^{\prime}$ are substochastic matrices on $M$, we define
their product $\Pi\times\Pi^{\prime}$ as the substochastic matrix given by the
ordinary product of matrices, that is, $(\Pi\times\Pi^{\prime})(x,y)=\sum
_{z\in M}\Pi(x,z)\Pi^{\prime}(z,y).$ The matrix $\Pi^{\mathcal{C}}$ defined in
the previous subsection is an example of a substochastic matrix.

A function $h:M\rightarrow\mathbb{R}$ is \emph{harmonic} for $\Pi$ if
$\sum_{y\in M}\Pi(x,y)h(y)=h(x)$ for all $x\in M$. When $h$ is positive, we
can define the Doob transform of $\Pi$ by $h$ (also called the $h$-transform
of $\Pi$) setting $\Pi_{h}(x,y)=\frac{h(y)}{h(x)}\Pi(x,y).$ We then have
$\sum_{y\in M}\Pi_{h}(x,y)=1$ for all $x\in M$, and $\Pi_{h}$ can be
interpreted as the transition matrix of a certain Markov chain. An example is
given in the second part of the previous subsection (see \eqref{reco}): the
state space is $\mathcal{C}$, the substochastic matrix is $\Pi^{\mathcal{C}}$,
the harmonic function is $h_{\mathcal{C}} (\lambda):=\mathbb{P}[S\mid
Y_{0}=\lambda]$, and the transition matrix $\Pi_{h_{\mathcal{C}}}%
^{\mathcal{C}}$ is the one of the Markov chain $Y^{C}$.

\subsection{Green function and Martin kernel}

Let $\Pi$ be a substochastic matrix on the set $M$. Its Green function is
defined as the series $\Gamma(x,y)=\sum_{\ell\geq0}\Pi^{\ell}(x,y)$; when
$\Pi$ is the transition matrix of a Markov chain, the quantity $\Gamma(x,y)$
is the expected value of the number of passages of the Markov chain from $x$
to $y$.

Assume that there exists $x^{\ast}$ in $M$ such that $0<\Gamma(x^{\ast
},y)<\infty$ for all $y\in M$. The Martin kernel associated with $\Pi$ (with
reference point $x^{\ast}$) is then defined by $K(x,y)=\frac{\Gamma
(x,y)}{\Gamma(x^{\ast},y)}.$ Consider a positive harmonic function $h$, let
$\Pi_{h}$ be the $h$-transform of $\Pi$, and consider the Markov chain
$Y^{h}=\left(  Y_{\ell}^{h}\right)  _{\ell\geq1}$ starting at $x^{\ast}$ and
whose transition matrix is $\Pi_{h}$. The following theorem is due to Doob
(see for instance \cite{LLP} for a detailed proof).

\begin{theorem}
[\textsc{Doob}]\label{Th_Doob} Assume that there exists a function $f:
M\rightarrow\mathbb{R}$ such that, for all $x\in M$, $\displaystyle \lim
_{\ell\rightarrow+\infty}K\left(  x,Y_{\ell}^{h}(\omega)\right)  =f(x)$ for
$\mathbb{P}$-almost all $\omega\in\Omega$. Then there exists a positive real
constant $c$ such that $f=c\cdot h$.
\end{theorem}

\subsection{Quotient local limit theorem for a random walk in a semigroup}

\label{stayC} We now recall some results on random walks similar to those
established in \cite{LLP}. However, the notion of random walk in a cone which
appears in \cite{LLP} will be replaced by the notion of random walk in a
semigroup. For the sake of brevity, we omit the proofs: they require
extensions of arguments used in \cite{LLP} to the case of semigroups. We refer
the reader to \cite{LLP2} (which is an extended version of the present paper)
for a complete exposition.

Let $\mathcal{C}$ be a subsemigroup of $(\mathbb{R}^{N},+)$ whose interior
$\mathring{\mathcal C}$ is non empty; we denote by $\mathcal{C}_{c}$ the cone
of $\mathbb{R}^{N}$ generated by $\mathcal{C}$.

Here are three examples of additive subsemigroups of $\mathbb{R}^{N}$ that
will appear in the sequel:

\begin{itemize}
\item[(e1)] $N=n>0$ and $\mathcal{C}^{\emptyset}=\left\{  (x_{1},x_{2}%
,\ldots,x_{n})\in\mathbb{R}^{n}\mid x_{1}\geq x_{2}\geq\dots\geq x_{n}%
\geq0\right\}  $,

\item[(e2)] $N=n>0$ and $\mathcal{C}^{s}=\left\{  (x_{1},x_{2},\ldots
,x_{n})\in\mathcal{C}^{\emptyset}\mid x_{i+1}\neq x_{i}\ \text{if}\ x_{i}%
\neq0\right\}  ,$

\item[(e3)] Let $m,n$ be two positive integers. The set
\begin{multline*}
\mathcal{C}^{h}=\big\{(x_{\overline{m}},x_{\overline{m-1}},\ldots
,x_{\overline{1}},x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n+m}\mid\\
x_{\overline{m}}\geq x_{\overline{m-1}}\geq\dots\geq x_{\overline{1}}%
\geq0,\ x_{1}\geq x_{2}\geq\dots\geq x_{n}\geq0\ \text{and }x_{i}=0\ \text{for
all } i>x_{\overline{1}}\big\}
\end{multline*}
is a semigroup in $\mathbb{R}^{n+m}$, but is not a cone, and we have
\begin{align*}
\mathring{\mathcal C^{h}}  &  =\big\{x_{\overline{m}}>x_{\overline{m-1}}%
>\dots>x_{\overline{1}}>m,\ x_{1}>x_{2}>\dots>x_{n}>0\big\},\\
\mathcal{C}_{c}^{h}  &  \subset\big\{x_{\overline{m}}\geq x_{\overline{m-1}%
}\geq\dots\geq x_{\overline{1}}\geq0,\ x_{1}\geq x_{2}\geq\dots\geq x_{n}%
\geq0\big\}.
\end{align*}

\end{itemize}

The subsemigroups $\mathcal{C}$ we consider are assumed to satisfy the
geometric hypothesis

\begin{enumerate}
\item[(h1)] $\mathring{\mathcal C}+\mathcal{C}_{c}\subset\mathcal{C}$.
\end{enumerate}

Observe this is in particular the case for our three examples (e1), (e2) and (e3).

Now, let $(X_{\ell})_{{\ell}\geq1}$ be a sequence of independent and
identically distributed random variables defined on a probability space
$(\Omega,{\mathcal{T}},\mathbb{P})$ with values in the Euclidean space
${\mathbb{R}}^{n}$; the associated random walk is defined by $S_{0}=0$ and
$S_{\ell}:=X_{1}+\cdots+X_{\ell}$ for $\ell\geq1$. We assume that
$\mathbb{E}|X_{\ell}|<+\infty$, and we set $\mathbf{m}:=\mathbb{E}(X_{\ell})$.
Consider a semigroup $\mathcal{C}$ in $\mathbb{R}^{n}$, with interior
$\mathring{\mathcal C}\neq\emptyset$ satisfying (h1), and introduce the
following additional hypotheses:

\begin{enumerate}
\item[(h2)] there exists $\ell_{0}>0$ such that $\mathbb{P}\left[  S_{1}%
\in\mathcal{C},S_{2}\in\mathcal{C},\ldots,S_{\ell_{0}-1}\in\mathcal{C}%
,S_{\ell_{0}}\in\mathring{\mathcal C}\right]  >0,$

\item[(h3)] there exists $t>0$ such that $t\mathbf{m}\in\mathring{\mathcal C}$.
\end{enumerate}

Under hypotheses (h2) and (h3), we get $\mathbb{P}\left[  S_{\ell}%
\in\mathcal{C} \text{ for all }\ell\geq0\right]  >0$.

The quotient local limit theorem stated in \cite{LLP} can be extended to our
situation. We limit our statement to random walks in the discrete lattice
$\mathbb{Z}^{n}$, and thus make an aperiodicity hypothesis, assuming that the
support $S_{\mu}$ of the law $\mu$ of the random variables $X_{\ell}$ is a
subset of $\mathbb{Z}^{n}$ which is not contained in a coset of a proper
subgroup of $\mathbb{Z}^{n}$. If this aperiodicity hypothesis is not
satisfied, the problem is not well posed: by subtracting a well chosen
constant vector to each $X_{\ell}$, we see that the new walk lives in a proper
subgroup of $\mathbb{Z}^{n}$.

\begin{theorem}
\label{RQT} Assume that the random variables $X_{\ell}$ are almost surely
bounded. Let $\mathcal{C}$ be an additive subsemigroup of $\mathbb{R}^{n}$
satisfying hypotheses \emph{(h1)}, \emph{(h2)} and \emph{(h3)}. Let $(g_{\ell
})_{\ell\ge1}$ and $(h_{\ell})_{\ell\ge1}$ be two sequences in $\mathbb{Z}%
^{n}$, and $\alpha<2/3$ such that
\[
\lim_{\ell\to\infty}{\ell}^{-\alpha}\Vert g_{\ell}-{\ell}\mathbf{m}%
\Vert=0\quad\text{and}\quad\lim_{\ell\to\infty}{\ell}^{-1/2}\Vert h_{\ell
}\Vert=0.
\]
Then, as ${\ell}$ tends to infinity, we have
\[
\mathbb{P}\left[  S_{1}\in\mathcal{C},\ldots,S_{\ell}\in\mathcal{C},S_{\ell
}=g_{\ell}+h_{\ell}\right]  \sim\mathbb{P}\left[  S_{1}\in\mathcal{C}%
,\ldots,S_{\ell}\in\mathcal{C},S_{\ell}=g_{\ell}\right]  .
\]

\end{theorem}

\section{Basics on representation theory}

\label{Sec_RT} We recall in the following paragraphs some classical material
on representation theory of classical Lie algebras and superalgebras. For a
complete review, the reader is referred to \cite{Bour}, \cite{Hal}, and
\cite{Kac}.

\subsection{Weights and roots}

The root and weight lattices of the Lie algebra $\mathfrak{gl}(n)$ over
$\mathbb{C}$ are realized in the Euclidean space $\mathbb{R}^{n}$ with
standard basis $\mathcal{B}=(\varepsilon_{1},\ldots,\varepsilon_{n})$. The
root lattice is $Q=\bigoplus_{i=1}^{n-1}\mathbb{Z\alpha}_{i}$, where
$\alpha_{i}=\varepsilon_{i}-\varepsilon_{i+1}$ for $i=1,\ldots,n-1$. The
weight lattice is $P=\mathbb{Z}^{n}$, and we denote by $P_{+}=\{x=(x_{1}%
,\ldots,x_{n})\in\mathbb{Z}^{n}\mid x_{1}\geq\cdots\geq x_{n}\}$ the cone of
dominant positive weights.

The Weyl group of $\mathfrak{gl}(n)$ can be identified with the symmetric
group $S_{n}$ acting on $\mathbb{Z}^{n}$ by permutation of the coordinates.
The Cartan Lie subalgebra $\mathfrak{h}$ of $\mathfrak{gl}(n)$ is the
subalgebra of diagonal matrices. The triangular decomposition of
\[
\mathfrak{gl}(n)=\mathfrak{gl}(n)_{+}\oplus\mathfrak{h}\oplus\mathfrak{gl}%
(n)_{-}%
\]
is the usual one obtained by considering strictly upper diagonal, diagonal,
and strictly lower diagonal matrices.

The Lie superalgebra $\mathfrak{gl}(m,n)$ can be regarded as the graded
$\mathbb{Z}/2\mathbb{Z}$ algebra of the matrices of the form
\[
\left(
\begin{array}
[c]{ll}%
A & C\\
D & B
\end{array}
\right) , \quad A\in\mathfrak{gl}(m),B\in\mathfrak{gl}(n),C\in M(m,n),D\in
M(n,m),
\]
where $M(m,n)$ is the set of complex $m\times n$ matrices. It decomposes into
the sum of its even and odd parts $\mathfrak{gl}(m,n)=\mathfrak{gl}%
(m,n)_{0}\oplus\mathfrak{gl}(m,n)_{1}$, the term $\mathfrak{gl}(m,n)_{0}$
being the set of matrices with $C=D=0$, and the term $\mathfrak{gl}(m,n)_{1}$
being the set of matrices with $A=B=0$. The ordinary Lie bracket is replaced
by its super version, that is, $[X,Y]=XY-(-1)^{ij}YX$ for $X\in\mathfrak{gl}%
(n)_{i}$ and $Y\in\mathfrak{gl}(n)_{j}$. Here $i$ and $j$ are regarded as
elements of $\mathbb{Z}/2\mathbb{Z}$.

The Cartan subalgebra $\mathfrak{h},$ the Weyl group $W$, the weight lattice
$P$, and the set $P_{+}$ of positive dominant weights of $\mathfrak{gl} (m,n)$
coincide with those of the even part $\mathfrak{gl}(m,n)_{0}$. In particular,
$P\simeq\mathbb{Z}^{m+n}$ and $W=S_{m}\times S_{n}$. Writing each weight
$\beta\in\mathbb{Z}^{m+n}$ in the form $\beta=(\beta_{\overline{m}}%
,\ldots,\beta_{\overline{1}}\mid\beta_{1},\ldots,\beta_{n})$, we have
$P_{+}=\{(\beta_{\overline{m}},\ldots,\beta_{\overline{1}}\mid\beta_{1}%
,\ldots,\beta_{n})\in\mathbb{Z}^{n+m}$ with $\beta_{\overline{m}}\geq
\cdots\geq\beta_{\overline{1}}$ and $\beta_{1}\geq\cdots\geq\beta_{n}\}$.

The superalgebra $\mathfrak{gl}(m,n)$ admits the triangular decomposition
\[
\mathfrak{gl}(m,n)=\mathfrak{gl}(m,n)_{+}\oplus\mathfrak{h}\oplus
\mathfrak{gl}(m,n)_{-},
\]
where $\mathfrak{gl}(m,n)_{+}$ and $\mathfrak{gl}(m,n)_{-}$ are the sets of
strictly upper and lower matrices in $\mathfrak{gl}(m,n)$, respectively, and
$\mathfrak{h}$ is the subalgebra of diagonal matrices.

We denote by $\mathfrak{q}(n)$ the Lie superalgebra of all matrices of the
form
\[
\left(
\begin{array}
[c]{ll}%
A & A^{\prime}\\
A^{\prime} & A
\end{array}
\right)  \ ,\quad A,A^{\prime}\in\mathfrak{gl}(n),
\]
endowed with the previous super Lie bracket. This superalgebra decomposes into
the sum $\mathfrak{q}(n)=\mathfrak{q}_{0}(n)\oplus\mathfrak{q}_{1}(n)$ of even
($A^{\prime}=0$) and odd ($A=0)$ parts. We denote by $e_{r,s}$ (respectively
$e_{r,s}^{\prime}$), $1\leq r,s\leq n$, the element of $\mathfrak{q}_{0}(n)$
(respectively of $\mathfrak{q}_{1}(n)$) in which $A$ (respectively $A^{\prime
}$) has $(r,s)$-entry equal to $1$ and the other entries equal $0$. Let
\[
\mathfrak{h}_{q}=\bigoplus_{r=1}^{n}\mathbb{C}e_{r,r}\oplus\bigoplus_{r=1}%
^{n}\mathbb{C}e_{r,r}^{\prime}%
\]
be the Cartan subalgebra of $\mathfrak{q}(n)$. The superalgebra $\mathfrak{q}%
(n)$ admits the triangular decomposition
\[
\mathfrak{q}(n)=\mathfrak{q}_{+}(n)\oplus\mathfrak{h}_{q}\oplus\mathfrak{q}%
_{-}(n),
\]
where $\mathfrak{q}_{+}(n)$ and $\mathfrak{q}_{-}(n)$ are the subalgebras
generated over $\mathbb{C}$ by $\{e_{r,s},e_{r,s}^{\prime}\mid1\leq r<s\leq
n\}$ and $\{e_{r,s},e_{r,s}^{\prime}\mid1\leq s<r\leq n\}.$ We also set
$\mathfrak{h}=\bigoplus_{r=1}^{n}\mathbb{C}e_{r,r}$. The weight lattice $P$ of
$\mathfrak{q}(n)$ can be identified with $\mathbb{Z}^{n}$ and its Weyl group
with $S_{n}$. Both coincide with those of the even part $\mathfrak{q}_{0}(n)$,
and we set $P_{+}=\mathbb{Z}_{\geq0}^{n}.$

\subsection{Weight spaces and characters}

\label{Wsc}

Assume $\mathfrak{g}=\mathfrak{gl}(n),\mathfrak{gl}(m,n)$, or $\mathfrak{q}
(n)$. For short, we set $N=n$ when $\mathfrak{g}=\mathfrak{gl}(n)$ or
$\mathfrak{q}(n)$ and $N=m+n$ when $\mathfrak{g}=\mathfrak{gl}(m,n)$. It will
be convenient to associate the symbol $\diamondsuit\in\{\emptyset,h,s\}$ to
the objects attached to the algebras $\mathfrak{gl} (n),\mathfrak{gl}%
(m,n),\mathfrak{q}(n),$ respectively.

In the sequel, we will only consider finite dimensional weight $\mathfrak{g}%
$-modules. Such a module $M$ admits a decomposition in weight spaces
$M=\bigoplus_{\mu\in P}M_{\mu}$, where $M_{\mu}:=\{v\in M\mid h\cdot
v=\mu(h)v$ for all $h\in\mathfrak{h}\}$, and $P$ is embedded in the dual of
$\mathfrak{h.}$ The space $M_{\mu}$ is thus an $\mathfrak{h}$-module. If $M$
and $M^{\prime}$ are finite-dimensional weight $\mathfrak{g}$-modules and
$\mu\in P$, we get $(M\oplus M^{\prime})_{\mu}=M_{\mu}\oplus M_{\mu}^{\prime}%
$. In particular, the weight spaces associated to any $\mathfrak{gl}(m,n)$-
(respectively $\mathfrak{q}(n)$-) module are defined as the weight spaces of
its restriction to $\mathfrak{gl}(m,n)_{0}$ (respectively $\mathfrak{q}%
(n)_{0}$). The character of $M$ is the Laurent polynomial in $N$ variables
$\mathrm{char}(M)(x):=\sum_{\mu\in P}\dim(M_{\mu})x^{\mu}$, where $\dim
(M_{\mu})$ is the dimension of the weight space $M_{\mu}$ and $x^{\mu}$ is a
formal exponential such that
\[
x^{\mu}=\left\{
\begin{array}
[c]{lll}%
x_{1}^{\mu_{1}}\cdots x_{n}^{\mu_{n}} & \text{ for } & \diamondsuit
\in\{\emptyset,s\},\\
x_{\overline{m}}^{\mu_{\overline{m}}}\cdots x_{\overline{1}}^{\mu
_{\overline{1}}}x_{1}^{\mu_{1}}\cdots x_{n}^{\mu_{n}} & \text{ for } &
\diamondsuit=h.
\end{array}
\right.
\]
For any $\sigma$ in the Weyl group, we have $\dim(M_{\mu})=\dim(M_{\sigma
(\mu)})$ so that $\mathrm{char}(M)(x)$ is a symmetric polynomial for
$\diamondsuit\in\{\emptyset,s\}$ and is invariant under the action of
$S_{m}\times S_{n}$ for $\diamondsuit=h$.

A weight $\mathfrak{g}$-module $M$ is called a highest weight module with
highest weight $\lambda$ if $M$ is generated by $M_{\lambda}$ and
$\mathfrak{g}_{+}\cdot v=0$ for all $v$ in $M_{\lambda}$. To each dominant
weight $\lambda\in P_{+}$ corresponds a unique (up to isomorphism) irreducible
finite dimensional representation of $\mathfrak{g}$ of highest weight
$\lambda$. We denote it by $V^{\diamondsuit}(\lambda)$; it decomposes as
$\bigoplus_{\mu\in P}V^{\diamondsuit}(\lambda)_{\mu}$. Letting $K_{\lambda
,\mu}^{\diamondsuit}=\mathrm{dim(}V^{\diamondsuit} (\lambda)_{\mu})$, we get
\begin{equation}
\mathrm{char}(V^{\diamondsuit}(\lambda))(x)=\sum_{\mu\in P}K_{\lambda,\mu
}^{\diamondsuit}x^{\mu}. \label{Kostka}%
\end{equation}


\subsection{Partitions and Young diagrams}

\label{subsec_YD} A partition of length $k\geq1$ is a $k$-tuple $\lambda
=(\lambda_{1},\ldots,\lambda_{k})\in\mathbb{Z}_{\geq0}^{k}$ such that
$\lambda_{1}\geq\cdots\geq\lambda_{k}$; we set $\left\vert \lambda\right\vert
=\lambda_{1} +\cdots+\lambda_{k}$ and denote by $\mathcal{P}_{k} $ the set of
partitions of length $k$. The Young diagram $Y(\lambda)$ associated with
$\lambda$ is the juxtaposition of rows of lengths $\lambda_{1},\ldots
,\lambda_{k}$, respectively, arranged from top to bottom. For $i=1,\ldots,k$,
the $i$-th row is divided into $\lambda_{i}$ boxes, and the rows are left
justified. (See the example below). The partition $\lambda^{\prime}$ obtained
by counting the number of boxes in each column of $Y(\lambda)$ is the
conjugate partition of $\lambda$.

$\bullet$ We let $\mathcal{P}^{\emptyset}=\mathcal{P}_{n}$. To each partition
$\lambda\in\mathcal{P}^{\emptyset}$, we associate the weight
$\displaystyle \pi(\lambda)=\sum_{i=1}^{n}\lambda_{i}\varepsilon_{i}$ which is
dominant for $\mathfrak{g}=\mathfrak{gl}(n)$. We say that \textit{$Y(\lambda)$
is a $\emptyset$-diagram for $\mathfrak{gl} (n)$.}

$\bullet$ Let $m,n$ be positive integers. We define the set $\mathcal{P}^{h}$
of \textit{hook partitions} as the set of partitions $\lambda$ of arbitrary
length such that $\lambda_{i}\leq n$ for all $i>m$. For any $\lambda
\in\mathcal{P}^{h}$, we denote by $\lambda^{(1)}\in\mathcal{P}_{m}$ the
partition corresponding to the Young diagram obtained by considering the $m$
longest rows of $\lambda$. We also denote by $\nu_{n}(\lambda)$ the partition
which remains after deleting the boxes corresponding to $\lambda^{(1)}$ from
the Young diagram of $\lambda$. By definition of the hook partition $\lambda$,
the conjugate partition of $\nu_{n}(\lambda)$ is an element of $\mathcal{P}%
_{n}$, it is denoted $\lambda^{(2)}$. In symbols, $\lambda^{(2)}=\nu
_{n}(\lambda)^{\prime}$. We will write $\lambda=(\lambda^{(1)}\mid
\lambda^{(2)})$ for short, with $\lambda^{(1)}=(\lambda_{\overline{m}}%
^{(1)},\ldots,\lambda_{\overline{1}}^{(1)})$ and $\lambda^{(2)}=(\lambda
_{1}^{(2)},\ldots,\lambda_{n}^{(2)})$; note that the weight $\displaystyle \pi
(\lambda):=\sum_{i=1}^{m}\lambda_{\overline{i}}^{(1)}\varepsilon_{i}%
+\sum_{j=1}^{n}\lambda_{j}^{(2)} \varepsilon_{j+m}$ is dominant for
$\mathfrak{gl}(m,n)$. We say that \textit{$Y(\lambda)$ is an $h$-diagram for
$\mathfrak{gl}(m,n)$.}

$\bullet$ We similarly define the set $\mathcal{P}^{s}$ of \textit{strict
partitions} as the set of partitions $\lambda\in\mathcal{P}^{\emptyset}$ with
the property that, if $\lambda_{i+1}>0$, then $\lambda_{i}>\lambda_{i+1}$, for
$i=1,\ldots,n-1$. We then define the shifted Young diagram $Y(\lambda)$
associated with $\lambda\in\mathcal{P}^{s}$ as the juxtaposition of rows of
lengths $\lambda_{1},\ldots,\lambda_{n}$ arranged from top to bottom. Each row
$i=1,\ldots,n$ is divided into $\lambda_{i}$ boxes, but the $i$-th row is
shifted $i-1$ units to the right with respect to the top row. We also denote
by $\displaystyle \pi(\lambda)=\sum_{i=1}^{n}\lambda_{i}\varepsilon_{i}$ the
dominant weight of $\mathfrak{q}(n)$ associated with $\lambda$. We say that
\textit{$Y(\lambda)$ is an $s$-diagram for $\mathfrak{q}(n)$.}

\begin{example}
(1) The diagram $\displaystyle\quad%
\begin{tabular}
[c]{|l|ll}\hline
\ \  & \ \  & \multicolumn{1}{|l|}{\ \ }\\\hline
&  & \multicolumn{1}{|l|}{}\\\hline
&  & \multicolumn{1}{|l}{}\\\cline{1-2}
&  & \\\cline{1-1}%
\end{tabular}
\quad$ is a $\emptyset$-diagram for $\mathfrak{gl}(4)$ with\break
$\lambda=(3,3,2,1).$

\medskip(2) The diagram $\displaystyle\quad%
\begin{tabular}
[c]{|l|ll}\hline
\ \  & \ \  & \multicolumn{1}{|l|}{\ \ }\\\hline
&  & \multicolumn{1}{|l|}{}\\\hline
&  & \multicolumn{1}{|l}{}\\\cline{1-2}
&  & \multicolumn{1}{|l}{}\\\cline{1-2}
&  & \multicolumn{1}{|l}{}\\\cline{1-2}
&  & \\\cline{1-1}%
\end{tabular}
\quad$ is an $h$-diagram for $\mathfrak{gl}(2,2)$ with $\lambda
=(3,3,2,2,2,1).$

We have $\lambda^{(1)}=(3,3)$ and $\lambda^{(2)}=(4,3)$; as an element of
$\mathbb{Z}^{4}$, we thus have\break$\lambda=(3,3\mid4,3)$ and $\pi
(\lambda)=3\varepsilon_{\overline{2}}+3\varepsilon_{\overline{1}}%
+4\varepsilon_{1}+3\varepsilon_{2}$.

\medskip(3) The diagram $\displaystyle\quad%
\begin{tabular}
[c]{ll|l|l}\hline
\multicolumn{1}{|l}{} & \multicolumn{1}{|l|}{} &  & \multicolumn{1}{|l|}{\ \ }%
\\\hline
& \multicolumn{1}{|l|}{} &  & \\\cline{2-3}\cline{3-3}%
\ \  & \ \  & \ \  & \\\cline{3-3}%
\end{tabular}
\quad$ is an $s$-diagram for $\mathfrak{q}(3)$ with $\lambda=(4,2,1)$.
\end{example}

\begin{notation}
To simplify notation, in the sequel we shall identify the partition $\lambda$
with its associated dominant weight and simply write $V^{\diamondsuit}%
(\lambda)$ for the highest weight module with highest weight $\lambda$ rather
than $V^{\diamondsuit}(\pi(\lambda)).$
\end{notation}

\begin{remark}
For any $\diamondsuit\in\{\emptyset,h,s\},$ the set $\mathcal{P}%
^{\diamondsuit}$ is naturally associated with one of the connected
subsemigroups $\mathcal{C}^{\diamondsuit}$ of \S ~\ref{stayC}; more
precisely,
%\begin{itemize}\item
$\mathcal{P}^{\diamondsuit}$ is the intersection of $\mathcal{C}%
^{\diamondsuit}$ with the integral lattice
%\item
and $\mathcal{C}^{\diamondsuit}$ satisfies hypothesis (h1) of \S ~\ref{stayC}%
.
%\end{itemize}

\end{remark}

Assume $\lambda,\mu$ are elements of $\mathcal{P}^{\diamondsuit}$. We write
$\mu\subset\lambda$ when the Young diagram of $\mu$ is contained in the one of
$\lambda$. In that case, the skew Young diagram $\lambda/\mu$ is obtained from
$\lambda$ by deleting the boxes appearing in $\mu$.

\subsection{Tensor powers of the natural representation}

Each algebra $\mathfrak{g}\mathfrak{=}\mathfrak{gl}(n)$,\break $\mathfrak{gl}(m,n)$,
and $\mathfrak{q}(n)$ can be realized as a matrix algebra. They thus admit a
natural representation $V^{\diamondsuit}$ which is the vector representation
of the underlying matrix algebra. For any $\ell\geq0,$ the tensor power
$(V^{\diamondsuit})^{\otimes\ell}$ is a semisimple representation of
$\mathfrak{g}$. This means that $(V^{\diamondsuit})^{\otimes\ell}$ decomposes
into a direct sum of irreducible representations
\[
(V^{\diamondsuit})^{\otimes\ell}\simeq\bigoplus_{\lambda\in P_{+}%
}V^{\diamondsuit}(\lambda)^{\oplus f_{\lambda}^{\diamondsuit}},
\]
where, for any $\lambda\in P_{+},$ the module $V^{\diamondsuit}(\lambda)$ is
the irreducible module with highest weight $\lambda$ and multiplicity
$f_{\lambda}^{\diamondsuit}$ in $(V^{\diamondsuit})^{\otimes\ell}$. In
general, we cannot realize all the irreducible highest weight modules as
irreducible components in a tensor product $(V^{\diamondsuit})^{\otimes\ell}$,
but we have the following statement.

\begin{proposition}
For any $\lambda\in P_{+},$ the module $V^{\diamondsuit}(\lambda)$ appears as
an irreducible component in a tensor product $(V^{\diamondsuit})^{\otimes\ell
}$ if and only if $\lambda\in\mathcal{P}^{\diamondsuit}$ and $\left|
\lambda\right|  =\ell$.
\end{proposition}

When $\mu\in\mathcal{P}^{\diamondsuit}$, we define the multiplicities
$f_{\lambda/\mu}^{\diamondsuit}$ by
\begin{equation}
V^{\diamondsuit}(\mu)\otimes(V^{\diamondsuit})^{\otimes\ell}\simeq
\bigoplus_{\lambda\in P_{+}}V^{\diamondsuit}(\lambda)^{\oplus f_{\lambda/\mu
}^{\diamondsuit}}. \label{skew_multi}%
\end{equation}
Set $\ell^{\prime}=\left\vert \mu\right\vert $. Since $V^{\diamondsuit}(\mu)$
appears as an irreducible component of $(V^{\diamondsuit})^{\otimes
\ell^{\prime}},$ we get $f_{\lambda/\mu}^{\diamondsuit}\neq0$ if and only if
$\lambda$ appears as an irreducible component of $(V^{\diamondsuit}
)^{\otimes\ell+\ell^{\prime}}$. In this situation, $\lambda\in\mathcal{P}%
^{\diamondsuit}$ and $\left\vert \lambda\right\vert =\ell+\ell^{\prime}$. When
$\ell=1$ we have
\begin{equation}
V^{\diamondsuit}(\mu)\otimes V^{\diamondsuit}\simeq\bigoplus_{\mu
\rightsquigarrow\lambda}V^{\diamondsuit}(\lambda) ,\label{ten_by_V}%
\end{equation}
where $\mu\rightsquigarrow\lambda$ means that the sum is over all the
partitions $\lambda\in\mathcal{P}^{\diamondsuit}$ obtained from $\mu
\in\mathcal{P}^{\diamondsuit}$ by adding one box. More generally, if
$\kappa\in\mathcal{P}^{\diamondsuit},$ we set
\begin{equation}
V^{\diamondsuit}(\mu)\otimes V^{\diamondsuit}(\kappa)=\bigoplus_{\lambda
\in\mathcal{P}^{\diamondsuit}}V^{\diamondsuit}(\lambda)^{\oplus m_{\kappa,\mu
}^{\lambda,\diamondsuit}}. \label{LRcoef}%
\end{equation}
Observe that $m_{\kappa,\mu}^{\lambda,\diamondsuit}=m_{\mu,\kappa}%
^{\lambda,\diamondsuit}$ since $V^{\diamondsuit}(\mu)\otimes V^{\diamondsuit
}(\kappa)$ and $V^{\diamondsuit}(\kappa)\otimes V^{\diamondsuit}(\mu)$ are
isomorphic, and that $m_{\mu,(1)}^{\lambda,\diamondsuit}=1$ if and only if
$\mu\rightsquigarrow\lambda.$

\section{Combinatorics of tableaux}

\label{sec_ComTab}In this section, we review the different notions of tableaux
which are relevant for the representation theory of $\mathfrak{gl} (n)$,
$\mathfrak{gl}(m,n)$, and $\mathfrak{q}(n).$ We also recall the associated
insertion schemes which are essential to define the generalized Pitman
transform in an elementary way. These notions are very classical for
$\mathfrak{gl}(n)$ (see \cite{Fu}); for $\mathfrak{gl}(m,n),$ they were
introduced by Benkart, Kang and Kashiwara in \cite{BKK}; for $\mathfrak{q}(n)
$ this has been developed very recently in \cite{GHSKM}.

\pagebreak

\subsection{Characters and tableaux}

\subsubsection{Semistandard $\mathfrak{gl}(n)$-tableaux}

Consider $\lambda\in\mathcal{P}^{\emptyset}$. A (semistandard) $\mathfrak{gl}
(n)$-tableau of shape $\lambda$ is a filling (let us call it $T $) of the
Young diagram associated with $\lambda$ by letters of the ordered alphabet
$\mathcal{A}_{n}=\{1<2<\cdots<n\}$ such that entries along rows of $T$
increase weakly from left to right and entries along columns increase strictly
from top to bottom (see Example~\ref{ex_A}). We denote by $T^{\emptyset
}(\lambda)$ the set of all $\mathfrak{gl}(n)$-tableaux of shape $\lambda$. We
define the reading of $T\in T^{\emptyset}(\lambda)$ as the word $\mathrm{w}%
(T)$ of $\mathcal{A} _{n}^{\ast}$ obtained by reading the rows of $T$ from
right to left and then top to bottom.

The weight of a word $w\in\mathcal{A}_{n}^{\ast}$ is the $n$-tuple
$\mathrm{wt}(w)=(\mu_{1},\ldots,\mu_{n})$, where for $i=1,\ldots, n $ the
nonnegative integer $\mu_{i}$ is the number of letters $i$ in $w$. The weight
$\mathrm{wt}(T)$ of $T\in T^{\emptyset} (\lambda)$ is then defined as the
weight of its reading $\mathrm{w}(T)$. The Schur function $s_{\lambda
}^{\emptyset}$ is the character of $V^{\emptyset}(\lambda)$. This is a
symmetric polynomial in the variables $x_{1} ,\ldots,x_{n}$ which can be
expressed as a generating function over $T^{\emptyset}(\lambda)$, namely as
\begin{equation}
\label{schur-funct-gn}s_{\lambda}^{\emptyset}(x)=\sum_{T\in T^{\emptyset
}(\lambda)}x^{\mathrm{wt} (T)}.
\end{equation}
According to the Weyl character formula, we have
\begin{equation}
s_{\lambda}^{\emptyset}(x)=\frac{1}{\prod_{1\leq i<j\leq n}(1-\frac{x_{j}
}{x_{i}})}\sum_{\sigma\in S_{n}}\varepsilon(\sigma)x^{\sigma(\lambda
+\rho)-\rho}=\frac{1}{\prod_{1\leq i<j\leq n}(x_{i}-x_{j})}\sum_{\sigma\in
S_{n}}\varepsilon(\sigma)x^{\sigma(\lambda+\rho)} ,\label{Weyl}%
\end{equation}
where $S_{n}$ is the symmetric group of rank $n,$ $\varepsilon(\sigma)$ is the
sign of $\sigma,$\break$\rho=(n-1,n-2,\ldots,0)\in\mathbb{Z}^{n}$, and $S_{n}$
acts on $\mathbb{Z}^{n}$ by permutation of the coordinates.

\subsubsection{Semistandard $\mathfrak{gl}(m,n)$-tableaux}

Consider $\lambda\in\mathcal{P}^{h}$. A (semistandard) $\mathfrak{gl}%
(m,n)$-tableau of shape $\lambda$ is a filling $T$ of the hook Young diagram
associated with $\lambda$ by letters of the ordered alphabet $\mathcal{A}%
_{m,n}=\{\overline{m}<\overline{m-1}<\cdots<\overline{1}<1<2<\cdots<n\}$ such
that entries along rows of $T$ increase weakly from left to right with no
repetition of unbarred letters permitted and entries along columns increase
weakly from top to bottom with no repetition of barred letters permitted (see
Example~\ref{ex_h}). We denote by $T^{h}(\lambda)$ the set of all
$\mathfrak{gl}(m,n)$-tableaux of shape $\lambda$. We define the reading of
$T\in T^{h}(\lambda)$ as the word $\mathrm{w}(T)$ of $\mathcal{A}_{m,n}^{\ast
}$ obtained by reading the rows of $T$ from right to left and top to bottom.

The weight of a word $w\in\mathcal{A}_{m,n}$ is the $(m+n)$-tuple
$\mathrm{wt}(w)=(\mu_{\overline{m}},\ldots,\mu_{\overline{1}}\mid\nu
_{1},\ldots,\nu_{n})$, where, for $i=1,\ldots,m$ and $j=1,\ldots,n,$ the
nonnegative integer $\mu_{\overline{i}}$ is the number of letters
$\overline{i}$ in $w$ and $\nu_{j}$ is the number of letters $j$ in $w$. The
weight of $T\in T^{h}(\lambda)$ is then defined as the weight of its reading
$\mathrm{w}(T)$. The Schur function $s_{\lambda}^{h}$ is the character of the
irreducible representation $V^{h}(\lambda)$ of $\mathfrak{gl}(m,n)$. This is a
polynomial in the variables $x_{\overline{m}},\ldots,x_{x_{\overline{1}}%
},x_{1},\ldots,x_{n}$. It admits a nice expression as a generating function
over $T^{h}(\lambda)$, namely as
\begin{equation}
s_{\lambda}^{h}(x)=\sum_{T\in T^{h}(\lambda)}x^{\mathrm{wt}(T)}.
\label{schur-funct-gmn}%
\end{equation}
For a general highest weight $\mathfrak{gl}(m,n)$-module there is no simple
Weyl character formula. Nevertheless, for the irreducible modules
$V^{h}(\lambda)$ with $\lambda\in\mathcal{P}^{h}$, such a formula exists due
to Berele, Regev, and Serge'ev (see \cite{Hu}). Consider $\lambda
\in\mathcal{P}^{h}$: the character $s_{\lambda}^{h}$ of $V^{h}(\lambda)$ is
given by
\begin{equation}
s_{\lambda}^{h}(x)=\frac{\prod_{(i,j)\in\lambda}(1+\frac{x_{j}}{x_{\overline
{i}}})}{\prod_{\overline{m}\leq\overline{i}<\overline{j}\leq\overline{1}%
}(1-\frac{x_{\overline{j}}}{x_{\overline{i}}})\prod_{1\leq r<s\leq n}%
(1-\frac{x_{s}}{x_{r}})}\sum_{w\in S_{m}\times S_{n}}\varepsilon
(w)\,x^{w(\lambda+\rho_{+})-\rho_{+}} ,\label{superWeyl}%
\end{equation}
where $\rho_{+}=(m-1,\ldots,1,0\mid n-1,\ldots,1,0)$, and $(i,j)\in\lambda$
means that the hook Young diagram associated with $\lambda$ has a box in the
$i$-th row, $i\in\{1,\ldots,m\}$, and the $j$-th column, $j\in\{1,\ldots,n\}.$

\subsubsection{Semistandard $\mathfrak{q}(n)$-tableaux}

Let us first give a definition. We say that a non\-empty word $w=x_{1}\cdots
x_{\ell}\in\mathcal{A}_{n}^{\ast}$ is a \textit{hook word} if there exists $k$
with $1\leq k\leq\ell$ such that $x_{1}\geq x_{2}\geq\cdots\geq x_{k}%
<x_{k+1}<\cdots<x_{\ell}$. Each hook word can be decomposed as
$w=w_{\downarrow}w_{\uparrow}$, where, by convention, the \emph{decreasing
part} $w_{\downarrow}=x_{1}\geq x_{2}\geq\cdots\geq x_{k}$ is nonempty. The
\emph{increasing part} $w_{\uparrow}=x_{k+1}<\cdots<x_{\ell}$ is possibly
empty. In particular, when $w=x_{1}\cdots x_{\ell}$ is such that $x_{1}%
<\cdots<x_{\ell}$, then we have $w_{\downarrow}=x_{1}$ and $w_{\uparrow}%
=x_{2}\cdots x_{\ell}$.

Consider a strict partition $\lambda\in\mathcal{P}^{s}$. A (semistandard)
$\mathfrak{q}(n)$-tableau of shape $\lambda$ is a filling $T$ of the shifted
Young diagram associated with $\lambda$ by letters of $\mathcal{A}
_{n}=\{1<2<\cdots<n\}$ such that, for $i=1,\ldots,n,$

\begin{enumerate}
\item the word $w_{i}$ formed by reading the $i$-th row of $T$ \emph{from left
to right} is a hook word (of length $\lambda_{i}$,

\item $w_{i}$ is a hook subword of maximal length in $w_{i+1}w_{i}$ (see
Example~\ref{ex_s}).
\end{enumerate}

We denote by $T^{s}(\lambda)$ the set of all $\mathfrak{q}(n)$-tableaux of
shape $\lambda$. The reading of $T\in T^{s}(\lambda)$ is the word
$\mathrm{w}(T)=w_{n}\cdots w_{2}w_{1}$ of $\mathcal{A}_{n}^{\ast}.$ We define
the weight of $T$ as the weight of its reading. The Schur function
$s_{\lambda}^{s}$ is defined as the generating function
\begin{equation}
s_{\lambda}^{s}(x)=\sum_{T\in T^{s}(\lambda)}x^{\mathrm{wt}(T)}.
\label{schur-funct-qn}%
\end{equation}
This is not the original combinatorial definition of the Schur function given
in terms of different tableaux called \emph{shifted Young tableaux}.
Nevertheless, according to Theorem~2.17 in \cite{Sera} and Remark~2.6 in
\cite{GHSKM} there exists a weight-preserving bijection between the set of
shifted Young tableaux of shape $\lambda$ and the set of $\mathfrak{q}%
(n)$-tableaux with the same shape. Set $d(\lambda)$ for the depth of $\lambda
$, that is, for the number of nonzero coordinates in $\lambda$. The Schur
function admits a Weyl type expression, namely we have.
\[
s_{\lambda}^{s}(x)
%=\sum_{\sigma\in S_{n}/S_{\lambda}}\sigma\left(  x^{\lambda
%}\prod_{1\leq i\leq d(\lambda)}\prod_{i<j\leq n}\left(  \dfrac{1+\frac{x_{j}%
%}{x_{i}}}{1-\frac{x_{j}}{x_{i}}}\right)  \right)
=\sum_{\sigma\in S_{n}/S_{\lambda}}\sigma\left(  x^{\lambda}\prod_{1\leq i\leq
d(\lambda)}\prod_{i<j\leq n}\left(  \dfrac{x_{i}+x_{j}}{x_{i}-x_{j}}\right)
\right) ,
\]
where $S_{\lambda}$ is the stabilizer of (the vector) $\lambda$ under the
action of $S_{n}$. Thus $S_{\lambda}$ is either isomorphic to $S_{n-d(\lambda
)}$ when $d(\lambda)<n-1$ or it reduces to $\{id\}$ otherwise.

\subsection{Insertion schemes}

\label{sub_sec_insert}To make our notation consistent, we set $\mathcal{A}%
^{\emptyset}=\mathcal{A}^{s}=\mathcal{A}_{n}$ and $\mathcal{A}^{h}%
=\mathcal{A}_{m,n}.$

\subsubsection{Insertion in $\mathfrak{gl}(n)$-tableaux}

Let $T$ be a $\mathfrak{gl}(n)$-tableau of shape $\lambda\in\mathcal{P}
^{\emptyset}$. We write $T=C_{1}\cdots C_{s}$ as the juxtaposition of its
columns. Consider $x\in\mathcal{A}_{n}$. We denote by $x\rightarrow T$ the
tableau obtained by applying the following recursive procedure:

\begin{enumerate}
\item If $T=\emptyset,$ then $x\rightarrow T$ is the tableau with one box
filled by $x$.

\item Assume $C_{1}$ is nonempty.

\begin{enumerate}
\item If all the letters of $C_{1}$ are less than $x,$ the tableau
$x\rightarrow T$ is obtained from $T$ by adding one box filled by $x$ at the
bottom of $C_{1}.$

\item Otherwise, let $y=\min\{t\in C_{1}\mid t\geq x\}$. Write $C_{1}
^{\prime}$ for the column obtained by replacing $y$ by $x$ in $C_{1}.$ Then
$x\rightarrow T=C_{1}^{\prime}(y\rightarrow C_{2}\cdots C_{s})$ is defined as
the juxtaposition of $C_{1}^{\prime}$ with the tableau obtained by inserting
$y$ in the remaining columns.
\end{enumerate}
\end{enumerate}

One easily checks that $x\rightarrow T$ is a $\mathfrak{gl} (n)$-tableau. More
generally, for a word $w=x_{1}x_{2}\cdots x_{\ell} \in\mathcal{A}_{n}^{\ast}$,
we define the $\mathfrak{gl}(n)$-tableau $P^{\emptyset}(w)$ by
\begin{equation}
P^{\emptyset}(w)=x_{\ell}\rightarrow(\cdots(x_{2}\rightarrow(x_{1}
\rightarrow\emptyset))). \label{defP(w)}%
\end{equation}


\begin{example}
\label{ex_A}With $n\geq4$ and $w=232143,$ we obtain the following sequences of
tableaux:
\[%
\begin{tabular}
[c]{|l|}\hline
$\mathtt{2}$\\\hline
\end{tabular}
\ \text{, }
\begin{tabular}
[c]{|l|}\hline
$\mathtt{2}$\\\hline
$\mathtt{3}$\\\hline
\end{tabular}
\ \text{, }
\begin{tabular}
[c]{|l|l}\hline
$\mathtt{2}$ & \multicolumn{1}{|l|}{$\mathtt{2}$}\\\hline
$\mathtt{3}$ & \\\cline{1-1}%
\end{tabular}
\ \text{,
\begin{tabular}
[c]{|l|ll}\hline
$\mathtt{1}$ & $\mathtt{2}$ & \multicolumn{1}{|l|}{$\mathtt{2}$}\\\hline
$\mathtt{3}$ &  & \\\cline{1-1}%
\end{tabular}
, }
\begin{tabular}
[c]{|l|ll}\hline
$\mathtt{1}$ & $\mathtt{2}$ & \multicolumn{1}{|l|}{$\mathtt{2}$}\\\hline
$\mathtt{3}$ &  & \\\cline{1-1}%
$\mathtt{4}$ &  & \\\cline{1-1}%
\end{tabular}
\ \text{, }
\begin{tabular}
[c]{|l|ll}\hline
$\mathtt{1}$ & $\mathtt{2}$ & \multicolumn{1}{|l|}{$\mathtt{2}$}\\\hline
$\mathtt{3}$ & $\mathtt{3}$ & \multicolumn{1}{|l}{}\\\cline{1-2}%
$\mathtt{4}$ &  & \\\cline{1-1}%
\end{tabular}
\ =P^{\emptyset}(w).
\]

\end{example}

\subsubsection{Insertion in $\mathfrak{gl}(m,n)$-tableaux}

Let $T=C_{1}\cdots C_{s}$ be a $\mathfrak{gl}(m,n)$-tableau of shape
$\lambda\in\mathcal{P}^{h}$. Consider $x\in\mathcal{A}_{m,n}$. We denote by
$x\rightarrow T$ the tableau obtained by applying the following procedure:

\begin{enumerate}
\item If $T=\emptyset,$ then $x\rightarrow T$ is the tableau with one box
filled by $x$.

\item Assume $C_{1}$ is nonempty and $x$ is a barred letter.

\begin{enumerate}
\item If all the letters of $C_{1}$ are less than $x,$ the tableau
$x\rightarrow T$ is obtained from $T$ by adding one box filled by $x$ at the
bottom of $C_{1}.$

\item Otherwise, let $y=\min\{t\in C_{1}\mid t\geq x\}$. Write $C_{1}
^{\prime}$ for the column obtained by replacing the topmost occurrence of $y$
in $C_{1}$ by $x$. (If $y$ is unbarred, it may happen that $y$ appears several
times in $C_{1}$.) The insertion $x\rightarrow T$ is then defined recursively
as $x\rightarrow T=C_{1}^{\prime}(y\rightarrow C_{2}\cdots C_{s}).$
\end{enumerate}

\item Assume $C_{1}$ is nonempty and $x$ is an unbarred letter.

\begin{enumerate}
\item If all the letters of $C_{1}$ are less than or equal to $x,$ the tableau
$x\rightarrow T$ is obtained from $T$ by adding one box filled by $x$ at the
bottom of $C_{1}.$

\item Otherwise, let $y=\min\{t\in C_{1}\mid t>x\}$. Write $C_{1}^{\prime}$
for the column obtained by replacing the topmost occurrence of $y$ in $C_{1}$
by $x$. Again, we define $x\rightarrow T=C_{1}^{\prime}(y\rightarrow
C_{2}\cdots C_{s}).$
\end{enumerate}
\end{enumerate}

One verifies that $x\rightarrow T$ is a $\mathfrak{gl} (m,n)$-tableau. For any
word $w=x_{1}x_{2}\cdots x_{\ell}\in\mathcal{A} _{m,n}^{\ast}$, we define the
$\mathfrak{gl}(m,n)$-tableau $P^{h}(w)$ recursively as in \eqref{defP(w)}.

\begin{example}
\label{ex_h}With $(m,n)=(2,3)$ and $w=\bar{2}3\bar{2}\bar{1}32\bar{1}2,$ we
obtain the following sequence of tableaux:
\[%
\begin{tabular}
[c]{|l|}\hline
$\mathtt{\bar{2}}$\\\hline
\end{tabular}
\text{, }
\begin{tabular}
[c]{|l|}\hline
$\mathtt{\bar{2}}$\\\hline
$\mathtt{3}$\\\hline
\end{tabular}
\text{, }
\begin{tabular}
[c]{|l|l}\hline
$\mathtt{\bar{2}}$ & \multicolumn{1}{|l|}{$\mathtt{\bar{2}}$}\\\hline
$\mathtt{3}$ & \\\cline{1-1}%
\end{tabular}
\text{, }
\begin{tabular}
[c]{|l|l|}\hline
$\mathtt{\bar{2}}$ & $\mathtt{\bar{2}}$\\\hline
$\mathtt{\bar{1}}$ & $\mathtt{3}$\\\hline
\end{tabular}
\text{, }
\begin{tabular}
[c]{|l|l}\hline
$\mathtt{\bar{2}}$ & \multicolumn{1}{|l|}{$\mathtt{\bar{2}}$}\\\hline
$\mathtt{\bar{1}}$ & \multicolumn{1}{|l|}{$\mathtt{3}$}\\\hline
$\mathtt{3}$ & \\\cline{1-1}%
\end{tabular}
\text{, }
\begin{tabular}
[c]{|l|l|}\hline
$\mathtt{\bar{2}}$ & $\mathtt{\bar{2}}$\\\hline
$\mathtt{\bar{1}}$ & $\mathtt{3}$\\\hline
$\mathtt{2}$ & $\mathtt{3}$\\\hline
\end{tabular}
\text{, }
\begin{tabular}
[c]{|l|l|l}\hline
$\mathtt{\bar{2}}$ & $\mathtt{\bar{2}}$ & \multicolumn{1}{|l|}{$\mathtt{3}$%
}\\\hline
$\mathtt{\bar{1}}$ & $\mathtt{\bar{1}}$ & \\\cline{1-2}\cline{2-2}%
$\mathtt{2}$ & $\mathtt{3}$ & \\\cline{1-2}%
\end{tabular}
\text{, }
\begin{tabular}
[c]{|l|ll}\hline
$\mathtt{\bar{2}}$ & $\mathtt{\bar{2}}$ & \multicolumn{1}{|l|}{$\mathtt{3}$%
}\\\hline
$\mathtt{\bar{1}}$ & $\mathtt{\bar{1}}$ & \multicolumn{1}{|l}{}\\\cline{1-2}%
\cline{2-2}%
$\mathtt{2}$ & $\mathtt{3}$ & \multicolumn{1}{|l}{}\\\cline{1-2}%
$\mathtt{2}$ &  & \\\cline{1-1}%
\end{tabular}
=P^{h}(w).
\]

\end{example}

\subsubsection{Insertion in $\mathfrak{q}(n)$-tableaux}

Let $T=L_{1}\cdots L_{k}$ be a $\mathfrak{q}(n)$-tableau of shape $\lambda
\in\mathcal{P}^{s}$. Here we regard $T$ as the juxtaposition of its rows
(written by decreasing lengths) rather than as the juxtaposition of its
columns. Consider $x\in\mathcal{A}_{n}$. We denote by $x\rightarrow T$ the
tableau obtained by applying the following procedure (which implies in
particular that, at each step and for any row $L_{i}$, the word $w(L_{i})$ is
a hook word):

\begin{enumerate}
\item If $T=\emptyset,$ then $x\rightarrow T$ is the tableau with one box
filled by $x$.

\item Assume $L_{1}$ is nonempty and write $w=w_{\downarrow}w_{\uparrow}$ for
the decomposition of $\mathrm{w}(L_{1})$ as decreasing and increasing subwords.

\begin{enumerate}
\item If $wx$ is a hook word, then $x\rightarrow T$ is the tableau obtained
from $T$ by adding one box filled by $x$ at the right end of $L_{1}.$

\item Otherwise, $w_{\uparrow}\neq\emptyset$ and $y=\min\{t\in w_{\uparrow
}\mid t\geq x\}$ exists. We first replace $y$ by $x$ in $w_{\uparrow}.$ Now
let $z=\max\{t\in w_{\downarrow}\mid t<y\}$. We replace $z$ by $y$ in
$w_{\downarrow}.$ Write $L_{1}^{\prime}$ for the row obtained in this manner.
The insertion $x\rightarrow T$ is then again defined recursively by
$x\rightarrow T=L_{1}^{\prime}(z\rightarrow L_{2}\cdots L_{k}).$
\end{enumerate}
\end{enumerate}

One also verifies that this gives a $\mathfrak{q}(n)$-tableau. For any word
$w=x_{1}x_{2}\cdots x_{\ell}\in\mathcal{A}_{n}^{\ast}$, we define the
$\mathfrak{q}(n)$-tableau $P^{s}(w)$ recursively as in \eqref{defP(w)}.

\begin{example}
\label{ex_s}With $n=4$ and $w=232145331,$ we obtain the following sequence of
tableaux:
\begin{gather*}%
\begin{tabular}
[c]{|l|}\hline
$2$\\\hline
\end{tabular}
\ \text{, }
\begin{tabular}
[c]{|l|l|}\hline
$2$ & $\mathbf{3}$\\\hline
\end{tabular}
\ \text{, }
\begin{tabular}
[c]{c|c|}\hline
\multicolumn{1}{|c|}{$3$} & $2$\\\hline
& $2$\\\cline{2-2}%
\end{tabular}
\ \text{,
\begin{tabular}
[c]{c|c|c}\hline
\multicolumn{1}{|c|}{$3$} & $2$ & \multicolumn{1}{|c|}{$1$}\\\hline
& $2$ & \\\cline{2-2}%
\end{tabular}
,
\begin{tabular}
[c]{c|c|cc}\hline
\multicolumn{1}{|c|}{$3$} & $2$ & $1$ & \multicolumn{1}{|c|}{$\mathbf{4}$%
}\\\hline
& $2$ &  & \\\cline{2-2}%
\end{tabular}
, }
\begin{tabular}
[c]{c|c|ccc}\hline
\multicolumn{1}{|c|}{$3$} & $2$ & $1$ & \multicolumn{1}{|c}{$\mathbf{4}$} &
\multicolumn{1}{|c|}{$\mathbf{5}$}\\\hline
& $2$ &  &  & \\\cline{2-2}%
\end{tabular}
\ \text{, }\\%
\begin{tabular}
[c]{c|c|c|cc}\hline
\multicolumn{1}{|c|}{$4$} & $2$ & $1$ & $\mathbf{3}$ &
\multicolumn{1}{|c|}{$\mathbf{5}$}\\\hline
& $2$ & $\mathbf{3}$ &  & \\\cline{2-3}%
\end{tabular}
\ \text{, }
\begin{tabular}
[c]{cc|c|cc}\hline
\multicolumn{1}{|c}{$4$} & \multicolumn{1}{|c|}{$3$} & $1$ & $\mathbf{3}$ &
\multicolumn{1}{|c|}{$\mathbf{5}$}\\\hline
& \multicolumn{1}{|c|}{$3$} & $2$ &  & \\\cline{2-3}\cline{3-3}
&  & $2$ &  & \\\cline{3-3}%
\end{tabular}
\ \text{, }
\begin{tabular}
[c]{cc|c|cc}\hline
\multicolumn{1}{|c}{$4$} & \multicolumn{1}{|c|}{$3$} & $3$ & $1$ &
\multicolumn{1}{|c|}{$\mathbf{5}$}\\\hline
& \multicolumn{1}{|c|}{$3$} & $2$ & $1$ & \multicolumn{1}{|c}{}\\\cline{2-4}%
\cline{3-3}
&  & $2$ &  & \\\cline{3-3}%
\end{tabular}
\ =P^{s}(w),
\end{gather*}
where we have indicated in bold type the increasing part of each row.
\end{example}

\subsection{Robinson--Schensted--Knuth correspondence}

\label{subsec_RSK} For any word $w=x_{1}\cdots x_{\ell}\in(\mathcal{A}%
^{\diamondsuit})^{\ell}$ and $k=1,\ldots\ell,$ let $\lambda^{(k)}$ be the
shape of the tableau $P^{\diamondsuit}(x_{1}\cdots x_{k})$; it is obtained
from $\lambda^{(k-1)}$ by adding one box $b_{k}$. The tableau $Q^{\diamondsuit
}(w)$ of shape $\lambda^{(\ell)}$ is obtained by filling each box $b_{k}$ with
the letter $k$; observe that $Q^{\diamondsuit}(w)$ is a standard tableau: it
contains exactly once all the integers $1,\ldots,\ell$, its rows strictly
increase from left to right and its columns strictly increase from top to
bottom. Note also that the datum of a standard tableau with $\ell$ boxes is
equivalent to that of a sequence of shapes $(\lambda^{(1)},\ldots
,\lambda^{(\ell)})\in(\mathcal{P}^{\diamondsuit})^{\ell}$ such that
$\lambda^{(1)}=(1)$ and, for $k=1,\ldots,\ell$, the shape $\lambda^{(k)}$ is
obtained by adding one box to $\lambda^{(k-1)}$.

\begin{examples}
From the previous examples, we deduce

\noindent$Q^{\emptyset}(232143)=
\begin{tabular}
[c]{|l|ll}\hline
$\mathtt{1}$ & $\mathtt{3}$ & \multicolumn{1}{|l|}{$\mathtt{4}$}\\\hline
$\mathtt{2}$ & $\mathtt{6}$ & \multicolumn{1}{|l}{}\\\cline{1-2}%
$\mathtt{5}$ &  & \\\cline{1-1}%
\end{tabular}
,$ \quad$Q^{h}(\bar{2}3\bar{2}\bar{1}32\bar{1}2)=
\begin{tabular}
[c]{|l|ll}\hline
$\mathtt{1}$ & $\mathtt{3}$ & \multicolumn{1}{|l|}{$\mathtt{7}$}\\\hline
$\mathtt{2}$ & $\mathtt{4}$ & \multicolumn{1}{|l}{}\\\cline{1-2}\cline{2-2}%
$\mathtt{5}$ & $\mathtt{6}$ & \multicolumn{1}{|l}{}\\\cline{1-2}%
$\mathtt{8}$ &  & \\\cline{1-1}%
\end{tabular}
\ $\newline and $Q^{s}(23214433)=
\begin{tabular}
[c]{cc|c|cc}\hline
\multicolumn{1}{|c}{$1$} & \multicolumn{1}{|c|}{$2$} & $4$ & $5$ &
\multicolumn{1}{|c|}{$6$}\\\hline
& \multicolumn{1}{|c|}{$3$} & $7$ & $9$ & \multicolumn{1}{|c}{}\\\cline{2-4}%
\cline{3-3}
&  & $8$ &  & \\\cline{3-3}%
\end{tabular}
\ .$
\end{examples}

By a $\diamondsuit$-tableau, with $\diamondsuit\in\{\emptyset,h,s\}$, we mean
a tableau for $\mathfrak{gl}(n)$, $\mathfrak{gl}(m,n)$, or $\mathfrak{q}(n)$,
respectively. We can now state the Robinson--Schensted--Knuth correspondence
for $\mathfrak{gl}(n)$ (see \cite{Fu}) and its generalizations for
$\mathfrak{gl}(m,n)$ and $\mathfrak{q}(n)$ obtained in \cite{BKK} and
\cite{GHSKM}, respectively. For any $\ell\geq0$ and $\diamondsuit
\in\{\emptyset,h,s\}$, let $\mathcal{U}_{\ell}^{\diamondsuit}$ be the set of
pairs $(P,Q)$, where $P$ is a $\diamondsuit$-tableau with $\ell$ boxes and $Q$
a standard tableau of the same shape as $P$. Given a standard tableau $T$ of
shape $\lambda\in\mathcal{P}^{\diamondsuit}$ with $\left\vert \lambda
\right\vert =\ell$, we set $B^{\diamondsuit}(T)=\{w\in(\mathcal{A}%
^{\diamondsuit})^{\ell}\mid Q^{\diamondsuit}(w)=T\}.$ We have the following result.

\begin{theorem}
\ \label{Th_RSK} \cite{GHSKM} Fix $\diamondsuit\in\{\emptyset,h,s\}$.

\begin{enumerate}
\item The map
\[
\displaystyle
\left\{  \theta_{\ell}^{\diamondsuit}:
\begin{array}
[c]{ccl}%
(\mathcal{A}^{\diamondsuit})^{\ell} & \rightarrow & \mathcal{U}_{\ell}
^{\diamondsuit}\\
w & \mapsto & (P^{\diamondsuit}(w),Q^{\diamondsuit}(w))
\end{array}
\right.
\]
is a one-to-one correspondence. In particular, for any standard tableau $T$,
the map $P^{\diamondsuit}$ restricts to a weight preserving bijection
$P^{\diamondsuit}:B^{\diamondsuit}(T)\longleftrightarrow T^{\diamondsuit
}(\lambda).$

\item For any $\lambda\in\mathcal{P}^{\diamondsuit}$, the multiplicity
$f_{\lambda}^{\diamondsuit}$ is equal to the number of standard $\diamondsuit
$-tableaux of shape~$\lambda.$
\end{enumerate}
\end{theorem}

Given $\lambda,\mu$ in $\mathcal{P}^{\diamondsuit}$ regarded as Young diagrams
such that $\mu_{i}\leq\lambda_{i}$ for all $i$ with $\mu_{i}>0$, we denote by
$\lambda/\mu$ the skew Young diagram obtained by deleting in $\lambda$ the
boxes of $\mu$. By a standard tableau of shape $\lambda/\mu$ with $\ell$
boxes, we mean a filling of $\lambda/\mu$ by the letters of $\{1,\ldots
,\ell\}$ whose rows and columns strictly increase from left to right and top
to bottom, respectively. By a skew $\diamondsuit$-tableau of shape
$\lambda/\mu,$ we mean a filling of $\lambda/\mu$ by letters of $\mathcal{A}%
^{\diamondsuit}$ whose rows and columns satisfy the same conditions as for the
ordinary $\diamondsuit$-tableaux. The next proposition will follow from the
Littlewood--Richardson rules proved in \cite{Fu} for $\mathfrak{gl}(n)$, in
\cite{KK} for $\mathfrak{gl}(m,n)$, and in \cite{GHSKM} for $\mathfrak{q}(n)$.
We postpone its proof to the appendix.

\begin{proposition}
\label{Cor_RSK} \ 

\begin{enumerate}
\item Given $\lambda,\mu$ in $\mathcal{P}^{\diamondsuit}$ such that
$\mu\subset\lambda,$ the multiplicity $f_{\lambda/\mu}^{\diamondsuit}$ defined
in \eqref{skew_multi} is equal to the number of standard tableaux of shape
$\lambda/\mu$.

\item Given $\lambda,\kappa,\mu$ in $\mathcal{P}^{\diamondsuit}$, we have
$m_{\kappa,\mu}^{\lambda,\diamondsuit}\leq K_{\mu,\lambda-\kappa
}^{\diamondsuit}$, where $K_{\mu,\lambda-\kappa}^{\diamondsuit}$ is the weight
multiplicity defined in \eqref{Kostka} and $m_{\kappa,\mu}^{\lambda
,\diamondsuit}$ the tensor multiplicity defined in \eqref{LRcoef}.
\end{enumerate}
\end{proposition}

\begin{remark}
Using Kashiwara crystal basis theory, we can obtain a stronger version of the
previous theorem. For any $\diamondsuit\in\{\emptyset,h,s\}$, the set
$B^{\diamondsuit}(T)$ has a crystal structure: namely, it can be endowed with
the structure of an oriented graph (depending on $\diamondsuit$) with arrows
colored by integers. Such a structure can also be defined on the set of words
$(\mathcal{A}^{\diamondsuit})^{\ell}$; one then shows that the sets
$B^{\diamondsuit}(T)$, where $T$ runs over the set of standard tableaux with
$\ell$ boxes, are the connected components of $(\mathcal{A}^{\diamondsuit
})^{\ell}.$ Moreover, the bijection of Assertion~(1) of the theorem is a graph
isomorphism that is compatible with this crystal structure. For $\mathfrak{gl}%
(n)$, the crystal basis theory is now a classical tool in representation
theory (see \cite{Kashi}). For $\mathfrak{gl}(m,n)$ and $\mathfrak{q}(n),$ it
becomes more complicated. We postpone the background needed for the proof of
Proposition~\ref{Cor_RSK} to the Appendix.
\end{remark}

\section{Pitman transform on the space of paths}

\label{Sec_Pitman}

\subsection{Paths and random walks in $\mathbb{Z}^{N}$}

Denote by $\mathcal{B}^{\diamondsuit}=\{e_{i},i\in\mathcal{A}^{\diamondsuit
}\}$ the standard basis of $\mathbb{Z}^{N}$ and fix a point $A\in
\mathbb{Z}^{N}$. We only consider paths in $\mathbb{Z}^{N}$ with steps in
$\mathcal{B}^{\diamondsuit}$; observe that there is a straightforward
bijection between the set of such paths of length $\ell$ starting from $A$ and
the set $(\mathcal{A}^{\diamondsuit})^{\ell}$ of words of length $\ell$ on the
alphabet $\mathcal{A} ^{\diamondsuit}$: the word $x_{1}\cdots x_{\ell}
\in(\mathcal{A}^{\diamondsuit})^{\ell}$ of length $\ell$ corresponds to the
path starting at $A$ whose $k$-th step is the translation by $e_{x_{k}}.$

Fix a sequence $(p_{i})_{i \in\mathcal{A}^{\diamondsuit}}$ of nonnegative
reals such that $\sum_{i\in\mathcal{A}^{\diamondsuit}}p_{i}=1$; this defines a
probability measure $p=(p_{i})$ on $\mathcal{A}^{\diamondsuit}$ with mean
vector $\mathbf{m}:= \sum_{i\in\mathcal{A}^{\diamondsuit}}p_{i}e_{i}$. Let
$(\mathcal{A}^{\diamondsuit})^{\mathbb{N}}$ be the set of sequences on the
alphabet $\mathcal{A}^{\diamondsuit}$ endowed with the product $\sigma
$-algebra, that is, the smallest $\sigma$-algebra containing all subsets of
$(\mathcal{A}^{\diamondsuit})^{\mathbb{N}}$ defined by a condition concerning
only finitely many coordinates. We fix on this measurable space the product
probability measure $\mathbb{P}=p^{\otimes\mathbb{N}}$; by construction, the
\emph{coordinates} $X_{\ell}$ defined by
\[
X_{\ell}:w=(x_{i})_{i\geq1}\in(\mathcal{A}^{\diamondsuit})^{\mathbb{N}
}\mapsto x_{\ell}
\]
for all $\ell\geq1$, are independent and identically distributed with law $p$
and mean vector $\mathbf{m}$.

For any $\ell\geq1$, we denote by $\pi_{\ell}$ the canonical projection from
$(\mathcal{A}^{\diamondsuit})^{\mathbb{N}}$ onto $(\mathcal{A}^{\diamondsuit
})^{\ell}$ defined by $\pi_{\ell}(\omega)=\omega^{(\ell)}:=x_{1}x_{2}\ldots
x_{\ell}$ for any $\omega=x_{1}x_{2}\ldots\in(\mathcal{A}^{\diamondsuit
})^{\mathbb{N}}$, and we set
\[
\mathcal{W}_{\ell}(\omega):=\mathcal{W}_{0}(\omega)+\mathrm{wt}\circ\pi_{\ell
}(\omega)=\mathcal{W}_{0}(\omega)+\mathrm{wt}\left(  \omega^{(\ell)}\right)
\text{\label{def_Wl}},
\]
where $\mathcal{W}_{0}$ is a fixed random variable defined on $(\mathcal{A}%
^{\diamondsuit})^{\mathbb{N}}$ with values in $\mathbb{Z}^{N}$, and
$\mathrm{wt}\left(  \omega^{(\ell)}\right)  $ is the weight of the word
$\omega^{(\ell)}$. The random process $\mathcal{W}=\left(  \mathcal{W}_{\ell
}\right)  _{\ell\geq0}$ is a random walk on $\mathbb{Z}^{N}$ since
$\mathcal{W}_{\ell}=\mathcal{W}_{0}+X_{1}+\cdots+X_{\ell}$ for all $\ell\geq
1$; this is in particular a Markov chain on $\mathbb{Z}^{N}$ with transition
matrix $\Pi_{\mathcal{W}}$ given by
\begin{equation}
\Pi_{\mathcal{W}}(\alpha,\beta)=\left\{
\begin{array}
[c]{ll}%
p_{i} & \text{ if }\beta-\alpha=e_{i}\text{ with }i\in\mathcal{A}%
^{\diamondsuit},\\
0 & \text{ otherwise,}%
\end{array}
\right. \quad\quad\text{for all }\alpha, \beta\in\mathbb{Z}^{N} .
\label{Pi(W)}%
\end{equation}


\subsection{Pitman transform of paths}

Our aim is to define a \emph{Pitman transform} $\mathfrak{P}^{\diamondsuit
},\diamondsuit\in\{\emptyset,h,s\}$, on the set $(\mathcal{A}^{\diamondsuit
})^{\mathbb{N}}$ of paths we have considered in the previous section; this
will be a generalization of the classical Pitman transform in the same spirit
as in \cite{BBOC1}.

For any $\omega\in(\mathcal{A}^{\diamondsuit})^{\mathbb{N}}$, set
$P^{\diamondsuit}(\omega):=\left(  P^{\diamondsuit}(\omega^{(\ell)})\right)
_{\ell}$. Since we identified Young diagrams with dominant weights, the shape
$sh(T)$ of a $\diamondsuit$-tableau $T$ is an element of $\mathcal{P}%
^{\diamondsuit}$; the shape of a sequence $(T_{\ell})_{\ell}$ of tableaux will
be the sequence $\left(  sh(T_{\ell})\right)  _{\ell}$, it defines a path in
$\mathcal{P}^{\diamondsuit}$. We now set%
\[
\mathfrak{P}^{\diamondsuit}(\omega)=sh\left(  P^{\diamondsuit}(\omega)\right)
=sh\left(  Q^{\diamondsuit}(\omega)\right)  \text{ for all }\omega
\in(\mathcal{A}^{\diamondsuit})^{\mathbb{N}}.
\]


\begin{example}
Consider $\omega=1121231212\cdots$. The path in $\mathbb{Z}^{3}$ associated
with $\omega$ remains in $\mathcal{P}^{\emptyset}$; we obtain
\[\kern-30pt
\begin{tabular}
[c]{c|ccccccccccc}%
$\ell$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $\cdots
$\\\cline{1-12}%
\  &  &  &  &  &  &  &  &  &  &  & \\
$sh\left(  P^{\emptyset}(\omega^{(\ell)})\right)  $ & $(1)$ & $(2)$ & $(2,1)$
& $(3,1)$ & $(3,2)$ & $(3,2,1)$ & $(4,2,1)$ & $(4,3,1)$ & $(5,3,1)$ &
$(5,4,1)$ & $\cdots$ \ \\
&  &  &  &  &  &  &  &  &  &  & \\\cline{1-12}%
\  &  &  &  &  &  &  &  &  &  &  & \\
$sh\left(  P^{s}(\omega^{(\ell)})\right)  $ & $(1)$ & $(2)$ & $(3)$ & $(3,1)$
& $(4,1)$ & $(5,1)$ & $(5,2)$ & $(5,3)$ & $(5,3,1)$ & $(6,3,1)$ & $\cdots$%
\end{tabular}
\]

\end{example}

\bigskip

\begin{remarks}
\noindent$\mathrm{(i)}$ In \cite{LLP}, for a simple Lie algebra $\mathfrak{g}%
$, the generalized Pitman transform of $\omega\in(\mathcal{A}^{\diamondsuit
})^{\mathbb{N}}$ was defined from the crystal structure on the set of words as
the sequence of dominants weights corresponding to the highest weight vertices
(source vertices) of the connected components containing the words
$\omega^{(\ell)}$, $\ell\geq1.$ For $\mathfrak{g}=\mathfrak{gl}(n)$ this
definition agrees with the one we have just introduced in terms of the
insertion algorithm on tableaux. For $\mathfrak{gl}(m,n)$ and $\mathfrak{q}%
(n),$ there is also a crystal structure on the set of words, but it is more
complicated to describe; in particular there may exist several highest weight
vertices for a given connected component. It becomes thus easier to define the
generalized Pitman transform with the help of insertion algorithms on
tableaux. This is what we do here, in the spirit of \cite{OC1}.

\smallskip$\mathrm{(ii)}$ In view of the previous example, we see that the
Pitman transform $\mathfrak{P}^{s}$ does not fix the paths contained in
$\mathcal{P}^{s}$ (but $\mathfrak{P}^{\emptyset}$ does). This phenomenon can
be explained by the special behavior of the crystal tensor product when
$\diamondsuit\in\{h, s\}$ (see Lemma~\ref{Lem_split} and the remark following it).
\end{remarks}

We then consider the random variable $\mathcal{H}_{\ell}^{\diamondsuit}$
\[
\mathcal{H}_{\ell}^{\diamondsuit}:\left\{
\begin{array}
[c]{ccl}%
(\mathcal{A}^{\diamondsuit})^{\mathbb{N}} & \rightarrow & \mathcal{P}%
^{\diamondsuit}\\
\omega & \mapsto & \mathfrak{P}^{\diamondsuit}(\omega^{(\ell)})
\end{array}
\right.
\]
This yields a stochastic process $\mathcal{H}^{\diamondsuit}\mathcal{=(H}%
_{\ell}^{\diamondsuit}\mathcal{)}_{\ell\geq0}$.

\begin{proposition}
\label{Prop_laws} For any $\diamondsuit\in\{\emptyset,h,s\},$ any $\ell
\in\mathbb{N}$, and $\lambda\in P_{+}$, we get\break$\mathbb{P}[\mathcal{H}%
_{\ell}^{\diamondsuit}=\lambda]=f_{\lambda}^{\diamondsuit}\cdot s_{\lambda
}^{\diamondsuit}(p).$
\end{proposition}

\begin{proof}
By definition of the random variable $\mathcal{H}_{\ell}^{\diamondsuit},$ we
have
\[
\mathbb{P}[\mathcal{H}_{\ell}^{\diamondsuit}=\lambda]=\sum_{T\text{ tableau of
shape }\lambda}\left(  \sum_{\omega\in B(T)}p_{\omega}\right)  =\sum_{T\text{
tableau of shape }\lambda}\mathbb{P}[B(T)].
\]
By (2) of Theorem~\ref{Th_RSK}, we have $\mathbb{P}[B(T)]=s_{\lambda
}^{\diamondsuit}(p)$, and, in particular, it does not depend on $T$ but only
on $\lambda$. By (3) of Theorem~\ref{Th_RSK}, we then deduce $\mathbb{P}%
\left[  \mathcal{H}_{\ell}^{\diamondsuit}=\lambda\right]  =f_{\lambda
}^{\diamondsuit}\cdot s_{\lambda}^{\diamondsuit}(p)$.
\end{proof}

\bigskip

We can now state the main result of this section. For any $\lambda,\mu
\in\mathcal{P}^{\diamondsuit},$ we write $\delta_{\mu\rightsquigarrow\lambda
}^{\diamondsuit}=1$ when $\lambda$ is obtained by adding one box to $\mu$ as
in \eqref{ten_by_V} and $\delta_{\mu\rightsquigarrow\lambda}^{\diamondsuit}=0$ otherwise.

\begin{theorem}
\label{Th_main}The stochastic process $\mathcal{H}^{\diamondsuit}$ is a Markov
chain with transition probabilities
\begin{equation}
\Pi_{\mathcal{H}^{\diamondsuit}}(\mu,\lambda)=\frac{s_{\lambda}^{\diamondsuit
}(p)}{s_{\mu}^{\diamondsuit}(p)}\delta_{\mu\rightsquigarrow\lambda
}^{\diamondsuit}\quad\lambda,\mu\in\mathcal{P}^{\diamondsuit}. \label{Pi(H)}%
\end{equation}

\end{theorem}

\begin{proof}
Consider a sequence of dominant weights $\lambda^{(1)},\ldots,\lambda^{(\ell
)},\lambda^{(\ell+1)}$ in $\mathcal{P}^{\diamondsuit}$ such that
$\lambda^{(k)}\leadsto\lambda^{(k+1)}$ for $k=1,\ldots,\ell$. We have seen in
\S ~\ref{subsec_RSK} that this determines a unique standard tableau $T$, and
we have
\[
\mathbb{P}[\mathcal{H}_{\ell+1}^{\diamondsuit}=\lambda^{\ell+1},\mathcal{H}%
_{k}^{\diamondsuit}=\lambda^{(k)}\text{ for }k=1,\ldots,\ell]=\sum_{w\in
B^{\diamondsuit}(T)}p_{w}=s_{\lambda^{(\ell+1)}}^{\diamondsuit}(p).
\]
Similarly, we have $\mathbb{P}[\mathcal{H}_{k}^{\diamondsuit}=\lambda^{(k)}$
for $k=1,\ldots,\ell]=s_{\lambda^{(\ell)}}^{\diamondsuit}(p).$Hence
\[
\mathbb{P}[\mathcal{H}_{\ell+1}^{\diamondsuit}=\lambda^{(\ell+1)}%
\mid\mathcal{H}_{k}^{\diamondsuit}=\lambda^{(k)}\text{ for }k=1,\ldots
,\ell]=\frac{s_{\lambda^{(\ell+1)}}^{\diamondsuit}(p)}{s_{\lambda^{(\ell)}%
}^{\diamondsuit}(p)}.
\]
In particular, $\mathbb{P}[\mathcal{H}_{\ell+1}^{\diamondsuit}=\lambda
^{(\ell+1)}\mid\mathcal{H}_{k}^{\diamondsuit}=\lambda^{(k)}$ for
$k=1,\ldots,\ell]$ depends only on $\lambda^{(\ell+1)}$ and $\mu
=\lambda^{(\ell)}$, this is the Markov property.
\end{proof}

\bigskip

\begin{remarks}
\noindent$\mathrm{(i)}$ By Proposition~\ref{Cor_RSK}, for all $\lambda,\mu$ in
$\mathcal{P}^{\diamondsuit}$ with $\mu\subset\lambda$, the multiplicity
$f_{\lambda/\mu}^{\diamondsuit}$ is equal to the number of standard tableaux
of shape $\lambda/\mu$. The datum of such a tableau $T$ is equivalent to that
of the sequence $(\mu,\lambda^{(1)},\ldots,\lambda^{(\ell)})$ of dominant
weights in $\mathcal{P}^{\diamondsuit}$, where $\ell=\left\vert \lambda
\right\vert -\left\vert \mu\right\vert $ and $\lambda^{(\ell)}=\lambda$;
furthermore, for $k=1,\ldots,\ell$, the shape of the $\diamondsuit$-diagram
$\lambda^{(k)}$ is obtained by adding to $\mu$ the boxes of $T$ filled by the
letters in $\{1,\ldots,k\}$. Therefore there is a bijection between the
standard tableaux of shape $\lambda/\mu$ and the paths from $\mu$ to $\lambda$
which remain in $\mathcal{P}^{\diamondsuit}$.

\smallskip$\mathrm{(ii)}$ Let $\lambda\in\mathcal{P}^{\diamondsuit}$ and
consider the irreducible highest weight representation $V^{\diamondsuit
}(\lambda)$. Assume $\left\vert \lambda\right\vert =\ell$. Let $T$ be a
standard tableau of shape $\lambda$ and define $B^{\diamondsuit}(T)$ as in
Theorem~\ref{Th_RSK}. Then, for any weight $\mu,$ the dimension $K_{\lambda
,\mu}$ of the $\mu$-weight space in $V(\lambda)$ is equal to the number of
words in $B^{\diamondsuit}(T)$ of weight $\mu$. This follows from the
bijection between $B^{\diamondsuit}(T)$ and $T^{\diamondsuit}(\lambda)$
obtained in Theorem~\ref{Th_RSK}. Since we have identified paths and words,
the integer $K_{\lambda,\mu}$ is equal to the number of paths from $0$ to
$\mu$ which remain in $B^{\diamondsuit}(T)$.
\end{remarks}

%Consider $\kappa,\mu,\lambda$ in $\mathcal{P}^{\diamondsuit}$. Let $T$ be a
%standard $\diamondsuit$-tableau of shape $\mu$.
%\begin{lemma}
%\label{Lem_util}The multiplicity $m_{\kappa,\mu}^{\lambda,\diamondsuit}$ is
%equal to the number of paths from $\kappa$ to $\lambda$ which remains in
%$\mathcal{P}^{\diamondsuit}$ and which are the $\kappa$-translates of a path
%in $B^{\diamondsuit}(T)$. In particular $m_{\kappa,\mu}^{\lambda
%,\diamondsuit}\leq K_{\lambda,\kappa-\mu}^{\diamondsuit}$.
%\end{lemma}
%\begin{proof}
%Recall that $m_{\kappa,\mu}^{\lambda,\diamondsuit}$ is the multiplicity of
%$V^{\diamondsuit}(\lambda)$ in the tensor product $V^{\diamondsuit}%
%(\kappa)\otimes V^{\diamondsuit}(\mu)$.
%This observation is in fact essentially contained in the Littelmann path model
%(at least for $\mathfrak{gl}_{n}$ for which such a theory is well-known).
%\end{proof}
%\bigskip


Recall that $\mathbf{m}=\mathbb{E}(X)$ is the drift of the random walk defined
in \eqref{def_Wl}. We have $\displaystyle\mathbf{m}=\sum_{i=1}^{n}p_{i} e_{i}$
for $\diamondsuit=\emptyset, s$ and $\displaystyle\mathbf{m}=\sum_{i=1}%
^{m}p_{\overline{i}}e_{i}+\sum_{j=1}^{n}p_{j}e_{j+m}$ for $\diamondsuit=h$. In
the sequel, we will assume that the following condition is satisfied:

\begin{condition}
\label{Cond}

\begin{enumerate}
\item For $\diamondsuit\in\{\emptyset,s\}$, there holds $p_{1}>\cdots>p_{n}>0$,

\item For $\diamondsuit=h$, there hold $p_{\overline{m}}>\cdots>p_{\overline
{1}}>0$ and $p_{1}>\cdots>p_{n}>0$.
\end{enumerate}
\end{condition}

%In both cases, there exists in particular $k\in\mathbb{R}^{*+}$ such that $km$
%lies in the interior of the cone generated by $\mathcal{P}^{\diamondsuit}
%$.
%For $\mathfrak{g}=\mathfrak{gl}(n)$ or $\mathfrak{q}(n),$ this simply means
%that $m=(p_{1},\ldots,p_{n})$ with $p_{1}>\cdots>p_{n}>0$.


In Section~\ref{Sec_restric}, we will need the following result.

\begin{proposition}
\label{Prop_dec_skew}

\begin{enumerate}
\item For all $\lambda,\nu\in\mathcal{P}^{\diamondsuit}$ such that $\nu
\subset\lambda$, we have
\[
\displaystyle f_{\lambda/\nu}^{\diamondsuit} =\sum_{\mu\in\mathcal{P}%
^{\diamondsuit}}f_{\mu}^{\diamondsuit}\cdot m_{\mu,\nu}^{\lambda,\diamondsuit
}.
\]


\item Assume that $\mathbf{m}$ satisfies Condition~\ref{Cond}. Consider a
sequence of weights\break $(\lambda^{(a)})_{a\in\mathbb{N}}$ of the form
$\lambda^{(a)}=a\mathbf{m}+o(a)$ and fix a nonnegative integer $\ell$. Then,
for large enough $a$, the weight $\lambda^{(a)}$ is an element of
$\mathcal{P}^{\diamondsuit}$. Moreover, for all $\kappa,\mu\in\mathcal{P}
^{\diamondsuit}$ such that $\left\vert \mu\right\vert =\ell$ and $\left\vert
\lambda^{(a)}\right\vert =\left\vert \kappa\right\vert +\ell$, we have
$m_{\kappa,\mu}^{\lambda^{(a)},\diamondsuit}=K_{\mu,\lambda^{(a)}-\kappa
}^{\diamondsuit}$.

\item For $(\lambda^{(a)})_{a\in\mathbb{N}}$ as above and all $\mu
\in\mathcal{P}^{\diamondsuit}$, we have, for all large enough $a$,
\begin{equation}
f_{\lambda^{(a)}/\mu}^{\diamondsuit}=\sum_{\kappa\in\mathcal{P}^{\diamondsuit
}}f_{\kappa}^{\diamondsuit}\cdot K_{\mu,\lambda^{(a)}-\kappa}^{\diamondsuit}
=\sum_{\gamma\in P}f_{\lambda^{(a)}-\gamma}^{\diamondsuit}\cdot K_{\mu,\gamma
}^{\diamondsuit}. \label{dec_skew_coef}%
\end{equation}
%for all $\ell\geq1.$

\end{enumerate}
\end{proposition}

\begin{proof}
To prove Assertion~(1), write $L=\left\vert \lambda\right\vert -\left\vert
\nu\right\vert $; by the definition of the Schur functions, we get, using
\eqref{skew_multi} and \eqref{LRcoef},
\[
s_{\nu}^{\diamondsuit}\cdot(s_{(1)}^{\diamondsuit})^{L}=\sum_{\mu}f_{\mu
}^{\diamondsuit}\cdot s_{\mu}^{\diamondsuit}\cdot s_{\nu}^{\diamondsuit}%
=\sum_{\mu}\sum_{\lambda}f_{\mu}^{\diamondsuit}\cdot m_{\mu,\nu}%
^{\lambda,\diamondsuit}\cdot s_{\lambda}^{\diamondsuit}=\sum_{\lambda
}f_{\lambda/\nu}^{\diamondsuit}\cdot s_{\lambda}^{\diamondsuit},
\]
where all the sums run over $\mathcal{P}^{\diamondsuit}.$ The assertion
immediately follows by comparing the two last expressions.

To prove Assertion~(2), observe first that $\lambda^{(a)}$ is an element of
$\mathcal{P}^{\diamondsuit}$ for $a$ sufficiently large because $\mathbf{m}$
satisfies Condition~\ref{Cond}. For any $\kappa\in\mathcal{P}^{\diamondsuit}$
such that $\left\vert \kappa\right\vert =\left\vert \lambda^{(a)}\right\vert
-\ell$, by Assertion~(1) we have
\[
f_{\lambda^{(a)}/\kappa}^{\diamondsuit}=\sum_{\mu\in\mathcal{P}^{\diamondsuit
},\left\vert \mu\right\vert =\ell}f_{\mu}^{\diamondsuit}\cdot m_{\mu,\kappa
}^{\lambda^{(a)},\diamondsuit}=\sum_{\mu\in\mathcal{P}^{\diamondsuit
},\left\vert \mu\right\vert =\ell}f_{\mu}^{\diamondsuit}\cdot m_{\kappa,\mu
}^{\lambda^{(a)},\diamondsuit},
\]
since $m_{\mu,\kappa}^{\lambda^{(a)},\diamondsuit}=m_{\kappa,\mu}%
^{\lambda^{(a)},\diamondsuit}$. Decomposing $(V^{\diamondsuit})^{\otimes\ell}$
into its irreducible components, one checks that the dimension of the weight
space $\lambda^{(a)}-\kappa$ in $(V^{\diamondsuit})^{\otimes\ell}$ is equal
to
\[
K_{\otimes\ell,\lambda^{(a)}-\kappa}^{\diamondsuit}:=\sum_{\mu\in
\mathcal{P}^{\diamondsuit},\left\vert \mu\right\vert =\ell}f_{\mu
}^{\diamondsuit}\cdot K_{\mu,\lambda^{(a)}-\kappa}^{\diamondsuit}.
\]
By (2) of Proposition~\ref{Cor_RSK}, we have $0\leq m_{\kappa,\mu}%
^{\lambda^{(a)},\diamondsuit}\leq K_{\mu,\lambda^{(a)}-\kappa}^{\diamondsuit}$
for all $\mu\in\mathcal{P}^{\diamondsuit}$. It thus suffices to show that
$f_{\lambda^{(a)}/\kappa}^{\diamondsuit}=K_{\otimes\ell,\lambda^{(a)}-\kappa
}^{\diamondsuit}$ for $a$ large enough. Observe that $K_{\otimes\ell
,\lambda^{(a)}-\kappa}^{\diamondsuit}$ is equal to the number of words of
length $\ell$ and weight $\lambda^{(a)}-\kappa$ on $\mathcal{A}^{\diamondsuit
}$. On the other hand, by the first assertion of Proposition~\ref{Cor_RSK}, we
deduce that $f_{\lambda^{(a)}/\kappa}^{\diamondsuit}$ is the number of
sequences of diagrams $(\delta^{(0)},\ldots,\delta^{(\ell)})$ in
$\mathcal{P}^{\diamondsuit}$ of length $\ell$ such that $\delta^{(0)}%
=\kappa,\delta^{(\ell)}=\lambda^{(a)}$ and $\delta^{(k+1)}/\delta^{(k)}$
consists of a single box for $k=0,\ldots,\ell-1$. For $\mathfrak{g}%
=\mathfrak{gl}(n)$ or $\mathfrak{q}(n),$ we associate to $(\delta^{(0)}%
,\ldots,\delta^{(\ell)})$ the word $w=x_{1}\cdots x_{\ell}$, where $x_{k}=i$
if the box $\delta^{(k)}/\delta^{(k-1)}$ appears in row~$i$ of $\delta^{(k)}$,
$i\in\{1,\ldots,n\}$, for $k=1,\ldots,\ell-1$. For $\mathfrak{g}%
=\mathfrak{gl}(m,n),$ we associate similarly to $(\delta^{(0)},\ldots
,\delta^{(\ell)})$ the word $w=x_{1}\cdots x_{\ell}$, where $x_{k}%
=\overline{i}$ if the box $\delta^{(k)}/\delta^{(k-1)}$ appears in row~$i$ of
$\delta^{(k)}$, $i\in\{1,\ldots,m\}$, and $x_{k}=j$ if this box appears in
column~$j$ of $\delta^{(k)}$, $j\in\{1,\ldots,n\}$, for $k=1,\ldots,\ell-1$.
In both cases, this yields an injective map from the set of sequences of
diagrams $(\delta^{(0)},\ldots,\delta^{(\ell)})$ with $\delta^{(k+1)}%
/\delta^{(k)}$ consisting of a single box into the set of words of length
$\ell$ and weight $\lambda^{(a)}-\kappa$ on $\mathcal{A}^{\diamondsuit}$.
Moreover this map is surjective; indeed, the tableaux $\kappa$ and
$\lambda^{(a)}$ differ by at most $\ell=\left\vert \mu\right\vert $ boxes, and
adding $\ell$ boxes in any order to the rows of $\kappa$ thus yields a diagram
in $\mathcal{P}^{\diamondsuit}$. Therefore $f_{\lambda^{(a)}/\kappa
}^{\diamondsuit}=K_{\otimes\ell,\lambda^{(a)}-\kappa}^{\diamondsuit}$.

Assertion~(3) is a direct consequence of (1) and (2).
\end{proof}

\section{Conditioning to stay in $\mathcal{P}^{\diamondsuit}$}

\label{Sec_restric}

Throughout this section, we assume that Condition~\ref{Cond} is satisfied, and
we denote by $\mathcal{W}^{\diamondsuit}$ the random walk $\mathcal{W}$
conditioned to stay in $\mathcal{P}^{\diamondsuit}.$ By Section~\ref{mcc}, the
process $\mathcal{W}^{\diamondsuit}$ is a Markov chain, with transition matrix
$\Pi_{\mathcal{W}^{\diamondsuit}}$.

We have explicit formulas for the transition matrices $\Pi_{\mathcal{H}%
^{\diamondsuit}}$ and $\Pi_{\mathcal{W}}$ of the Markov chains $\mathcal{W}$
and $\mathcal{H}^{\diamondsuit}$. By \eqref{Pi(H)} and \eqref{Pi(W)}, the
transition matrix $\Pi_{\mathcal{H}^{\diamondsuit}}$ of $\mathcal{H}%
^{\diamondsuit}$ is the Doob $\psi$-transform of $\Pi_{\mathcal{W}}$, where
$\psi$ is the harmonic function
\begin{equation}
\psi:\left\{
\begin{array}
[c]{lll}%
\mathcal{P}^{\diamondsuit} & \rightarrow & \mathbb{R}_{>0}\\
\lambda & \mapsto & p^{-\lambda}s_{\lambda}^{\diamondsuit}(p)
\end{array}
\right.  . \label{psi}%
\end{equation}
On the other hand, setting $h_{\mathcal{P}^{\diamondsuit} }(\lambda
):=\mathbb{P}\left[  \mathcal{W}_{\ell} \in\mathcal{P} \text{ for all }
\ell\geq0^{\diamondsuit}\mid\mathcal{W}_{0}=\lambda\right]  $ for all
$\lambda\in\mathcal{P}^{\diamondsuit}$, we know by \S ~\ref{sub_sec_Doobh}
that $\Pi_{\mathcal{W}^{\diamondsuit}}$ is the Doob $h_{\mathcal{P}%
^{\diamondsuit}} $-transform of $\Pi_{\mathcal{W}}$. We are going to prove
that $\psi$ and $h_{\mathcal{P}^{\diamondsuit}}$ coincide.

\subsection{Limit of $\psi$ along a drift}

Since Condition~\ref{Cond} holds, the products
\begin{gather}
\nabla^{\emptyset}=\frac{1}{\prod_{1\leq i<j\leq n}(1-\frac{p_{j}}{p_{i}}
)},\quad\nabla^{h}=\frac{\prod_{i=1}^{m}\prod_{j=1}^{n}(1+\frac{p_{j}
}{p_{\overline{i}}})}{\prod_{\overline{m}\leq\overline{i}<\overline{j}
\leq\overline{1}}(1-\frac{p_{\overline{j}}}{p_{\overline{i}}})\prod_{1\leq
r<s\leq n}(1-\frac{p_{s}}{p_{r}})},\label{def_nabla}\\
\text{ and}\quad\nabla^{s}=\prod_{1\leq i<j\leq n}\dfrac{p_{i}+p_{j}}%
{p_{i}-p_{j}}\nonumber
\end{gather}
are well-defined, and we have the following proposition.

\begin{proposition}
\label{prop-_imit-psi}Assume Condition~\emph{\ref{Cond}} is satisfied and
consider a sequence\break $(\lambda^{(a)})_{a\in\mathbb{N}}$ in $\mathcal{P}%
^{\diamondsuit}$ such that $\lambda^{(a)}=a\mathbf{m}+o(a).$ Then
$\lim_{a\rightarrow+\infty}p^{-\lambda^{(a)}}s_{\lambda^{(a)}}^{\diamondsuit
}(p)=\nabla^{\diamondsuit}.$
\end{proposition}

\begin{proof}
Assume first that $\diamondsuit=\emptyset$. We have $s_{\lambda^{(a)}%
}^{\emptyset}(p)=\nabla^{\emptyset}\sum_{\sigma\in S_{n}}\varepsilon
(\sigma)p^{\sigma(\lambda^{(a)}+\rho)-\rho}$ by the Weyl character formula;
this gives $p^{-\lambda^{(a)}}s_{\lambda^{(a)}}^{\emptyset}(p)=\nabla
^{\emptyset}\sum_{\sigma\in S_{n}}\varepsilon(\sigma)p^{\sigma(\lambda
^{(a)}+\rho)-\lambda^{(a)}-\rho}.$ When $\sigma=id$, we get $\varepsilon
(\sigma)p^{\sigma(\lambda^{(a)}+\rho)-\lambda^{(a)}-\rho}=1$. So it suffices
to prove that\break$\lim_{a\rightarrow+\infty}\varepsilon(\sigma
)p^{\sigma(\lambda^{(a)}+\rho)-\lambda^{(a)}-\rho}=0$ for all $\sigma\neq id$;
in this case, observe that
\[
\lambda^{(a)}+\rho-\sigma(\lambda^{(a)}+\rho)=\lambda^{(a)}-\sigma
(\lambda^{(a)})+\rho-\sigma(\rho)=a(\mathbf{m}-\sigma(\mathbf{m}))+\rho
-\sigma(\rho)+o(a).
\]
Since $\mathbf{m}$ satisfies Condition~\ref{Cond}, the coordinates of
$\mathbf{m}$ strictly decrease and are positive; this implies that
$\displaystyle p^{\mathbf{m}-\sigma(\mathbf{m})}>1$.

Assume $\diamondsuit=h$. Since \ref{Cond} is satisfied, for $a$ large enough
the Young diagram of $\lambda^{(a)}$ has a box at position $(i,j)$ for all
$i\in\{1,\ldots,m\}$ and $j\in\{1,\ldots,n\}$. For such an integer $a,$ by
\eqref{superWeyl} we thus have $s_{\lambda^{(a)}}^{h}(p)=\nabla^{h}\sum_{w\in
S_{m}\times S_{n}}\varepsilon(w)p^{w(\lambda^{(a)}+\rho_{+})-\rho_{+}}$, and
we conclude as above.

Finally, assume $\diamondsuit=s$. Since $\lambda^{(a)}$ has only positive
coordinates, we obtain, using the case $\diamondsuit=\emptyset$ with the
sequence $\lambda^{(a)}-\rho$,
\[
\lim_{a\rightarrow+\infty}p^{-\lambda^{(a)}}s_{\lambda^{(a)}}^{s}%
(p)=\lim_{a\rightarrow+\infty}p^{-\lambda^{(a)}+\rho}s_{\lambda^{(a)}-\rho
}^{\emptyset}(p)\left(  \prod_{1\leq i<j\leq n}p_{i}+p_{j}\right)  p^{-\rho
}=\nabla^{s}.
\]
%for the stabilizer of $\lambda^{(a)}$ reduced to $\{1\}$.

\end{proof}

\subsection{The transition matrix $\Pi_{\mathcal{W}^{\diamondsuit}}$}

\label{tmP} Let $\Pi^{\diamondsuit}$ be the restriction of $\Pi_{\mathcal{W}}$
to $\mathcal{P}^{\diamondsuit}$, and denote by $\Gamma$ the Green function
associated with $\Pi^{\diamondsuit}$. For all $\mu, \lambda\in P^{\diamondsuit
}$, we have
\[
\Gamma(\mu,\lambda)=
\begin{cases}
(\Pi^{\diamondsuit})^{\ell}(\mu,\lambda), & \text{ if } \ell=\left\vert
\lambda\right\vert -\left\vert \mu\right\vert \geq0,\\
0, & \text{ if } \left\vert \lambda\right\vert <\left\vert \mu\right\vert .
\end{cases}
\]
In particular $\Gamma(\mu,\lambda)=0$ if $\lambda\notin\mathcal{P}
^{\diamondsuit}$. We consider the Martin kernel $K(\mu,\lambda)=\frac
{\Gamma(\mu,\lambda)}{\Gamma(0,\lambda)}.$

In order to apply Theorem~\ref{Th_Doob}, we want to prove that, almost surely,
$K(\cdot,\mathcal{W}_{\ell})$ converges everywhere to the harmonic function
$\psi(\cdot)$ defined in \eqref{psi}. By definition of $\Pi^{\diamondsuit}$,
we have $(\Pi^{\diamondsuit})^{\ell}(\mu,\lambda)=\mathrm{card}%
(ST^{\diamondsuit}(\lambda/\mu))p^{\lambda-\mu}$, where $ST^{\diamondsuit
}(\lambda/\mu)$ is the set of standard $\diamondsuit$-tableaux of shape
$\lambda/\mu$ (see the remark after Theorem~\ref{Th_main}). Indeed, all the
paths from $\mu$ to $\lambda$ have the same probability $p^{\lambda-\mu}$, and
we have seen that there are $\mathrm{card}(ST^{\diamondsuit}(\lambda/\mu))$
such tableaux. By Proposition~\ref{Cor_RSK}, we have $\mathrm{card}%
(ST^{\diamondsuit}(\lambda/\mu))=f_{\lambda/\mu}^{\diamondsuit}.$ According to
Proposition~\ref{Prop_dec_skew}, given any sequence $\lambda^{(a)}$ of weights
of the form $\lambda^{(a)}=a\mathbf{m}+o(a)$, for large enough $a$ we can
write
\[
\Gamma(\mu,\lambda^{(a)})=f_{\lambda^{(a)}/\mu}^{\diamondsuit}\cdot
p^{\lambda^{(a)}-\mu}=p^{\lambda^{(a)}-\mu}\sum_{\gamma\in P}f_{\lambda
^{(a)}-\gamma}^{\diamondsuit}\cdot K_{\mu,\gamma}^{\diamondsuit}.
\]
Since $\displaystyle\Gamma(0,\lambda^{(a)})=f_{\lambda^{(a)}}^{\diamondsuit
}p^{\lambda^{(a)}},$ for $a$ large enough this yields
\[
K(\mu,\lambda^{(a)})=p^{-\mu}\sum_{\gamma\in P}K_{\mu,\gamma}^{\diamondsuit
}\frac{f_{\lambda^{(a)}-\gamma}^{\diamondsuit}}{f_{\lambda^{(a)}%
}^{\diamondsuit}}=p^{-\mu}\sum_{\gamma\in P}K_{\mu,\gamma}^{\diamondsuit
}p^{\gamma}\frac{f_{\lambda^{(a)}-\gamma}^{\diamondsuit}\cdot p^{\lambda
^{(a)}-\gamma}}{f_{\lambda^{(a)}}^{\diamondsuit}\cdot p^{\lambda^{(a)}}},
\]
so that
\begin{equation}
K(\mu,\lambda^{(a)})=p^{-\mu}\sum_{\gamma\text{ weight of }V^{\diamondsuit
}(\mu)}K_{\mu,\gamma}^{\diamondsuit}p^{\gamma}\frac{\Gamma(0,\lambda
^{(a)}-\gamma)}{\Gamma(0,\lambda^{(a)})}. \label{expand_K}%
\end{equation}
Now we have the following proposition.

\begin{proposition}
\label{Prop_quotient}Assume Condition~\ref{Cond} is satisfied, and consider
sequences\break $(\lambda^{(a)})_{a\in\mathbb{N}}$ and $(\mu^{(a)})_{a\in\mathbb{N}%
}$ in $\mathcal{P}^{\diamondsuit}$ such that $\lambda^{(a)}=a\mathbf{m}%
+o(a^{\delta+{\frac{1}{2}}})$ for some $\delta>0$ and $\mu^{(a)}=o(a^{1/2}).$
Then
\[
\lim_{a\rightarrow+\infty}\frac{\Gamma(0,\lambda^{(a)}-\mu^{(a)})}%
{\Gamma(0,\lambda^{(a)})}=1.
\]

\end{proposition}

\begin{proof}
For $\diamondsuit=\emptyset, s, h$, we get $\mathcal{P}^{\diamondsuit
}=\mathcal{C}^{\diamondsuit}\cap\mathbb{Z}^{N}$, and the semigroup
$\mathcal{C}^{\diamondsuit}$ satisfies hypothesis (h1) of \S ~\ref{stayC}.
Moreover, the random walk $\mathcal{W}$ satisfies (h2) and (h3). The
proposition is thus a direct consequence of Theorem~\ref{RQT}.
\end{proof}

The strong law of large numbers for square integrable independent and
identically distributed random variables tells us that for all $\delta>0$ we
get $\mathcal{W}_{a}=a\mathbf{m}+o(a^{\delta+\frac{1}{2}})$ almost surely.
This leads to the following corollary.

\begin{corollary}
\label{coro_quotient} Assume Condition~\emph{\ref{Cond}} is satisfied, and fix
$\gamma\in P$. Then
\[
\lim_{a\rightarrow+\infty}\frac{\Gamma(0,\mathcal{W}_{a}-\gamma)}%
{\Gamma(0,\mathcal{W}_{a})}=1\quad\text{\emph{(a.s.)}}.
\]

\end{corollary}

We now may state the main result of this section. We denote by $\left(
\mathcal{W}_{\ell}^{\diamondsuit}\right)  _{\ell\geq0}$ the random walk
$(\mathcal{W}_{\ell})_{\ell\geq0}$ conditioned to never exit $\mathcal{P}%
^{\diamondsuit}$.

\begin{theorem}
\label{Th_coincide}Assume Condition~\emph{\ref{Cond}} is satisfied. Then the
Markov chains $(\mathcal{H}_{\ell}^{\diamondsuit})_{\ell\geq0}$ and $\left(
\mathcal{W}_{\ell}^{\diamondsuit}\right)  _{\ell\geq0}$ have the same
transition matrix.
\end{theorem}

\begin{proof}
The strong law of large numbers says that $\mathcal{W}_{a}=a\mathbf{m}+o(a)$
a.s. Combined with \eqref{expand_K}, this implies that
\[
\lim_{a \to+\infty}K(\mu,\mathcal{W}_{a})=\lim_{a \to+\infty} p^{-\mu}%
\sum_{\gamma\text{ weight of }V^{\diamond suit }(\mu)}p^{\gamma}\cdot
K_{\mu,\gamma}^{\diamondsuit}\frac{\Gamma(0,\mathcal{W}_{a}-\gamma)}%
{\Gamma(0,\mathcal{W}_{a})}\qquad\mathrm{(a.s.).}%
\]
The set $V^{\diamondsuit}(\mu)$ being finite, by Corollary~\ref{coro_quotient}
we thus get
\begin{equation}
L:=\lim_{a\rightarrow+\infty}K(\mu,\mathcal{W}_{a})=p^{-\mu}\sum_{\gamma\text{
weight of }V^{\diamondsuit}(\mu)}p^{\gamma K_{\mu,\gamma}^{\diamondsuit}%
}=p^{-\mu}s_{\mu}^{\diamondsuit}(p)=\psi^{\diamondsuit}(\mu)\quad\text{
(a.s.).} \label{K_tends_psi}%
\end{equation}
Hence, by Theorem~\ref{Th_Doob}, we obtain $\psi^{\diamondsuit}%
=ch_{\mathcal{P}^{\diamondsuit}}$ for some constant $c>0$, where
$h_{\mathcal{P}^{\diamondsuit}}$ is the harmonic function associated with the
restriction of $(\mathcal{W}_{\ell})_{\ell\geq0}$ to $\mathcal{P}%
^{\diamondsuit}$ and defined in Subsection~\ref{sub_sec_Doobh}. By
Theorem~\ref{Th_main}, we thus deduce
\[
\Pi_{h_{\mathcal{P}^{\diamondsuit}}}(\mu,\lambda)=\Pi_{\psi}(\mu,\lambda
)=\Pi_{\mathcal{H}^{\diamondsuit}}(\mu,\lambda)=\frac{s_{\lambda
}^{\diamondsuit}(p)}{s_{\mu}^{\diamondsuit}(p)}\delta_{\mu\leadsto\lambda
}^{\diamondsuit}.
\]

\end{proof}

\subsection{Some consequences}

As a direct consequence of Theorem~\ref{Th_coincide}, we obtain the following corollary.

\begin{corollary}
\label{yrester} \label{Cor_StayinC} Under the assumptions of
Theorem~\emph{\ref{Th_coincide}}, for all $\lambda\in\mathcal{P}%
^{\diamondsuit}$, we have
\[
\mathbb{P}[\mathcal{W}_{\ell}\in\mathcal{P}^{\diamondsuit}\text{ \emph{for all
}}\ell\geq0\mid\mathcal{W}_{0}=\lambda]=\frac{p^{-\lambda}s_{\lambda
}^{\diamondsuit}(p)}{\nabla^{\diamondsuit}},
\]
where $\nabla^{\diamondsuit}$ was defined in \eqref{def_nabla}.
\end{corollary}

\begin{proof}
Recall that the function $h_{\mathcal{P}^{\diamondsuit}}:\lambda
\longmapsto\mathbb{P}\left[  \mathcal{W}_{\ell}\in\mathcal{P}^{\diamondsuit
}\text{ for all }\ell\geq0\mid\mathcal{W}_{0}=\lambda\right]  $ is harmonic.
By Theorem~\ref{Th_coincide}, there is a positive constant $c$ such that
$h_{\mathcal{P}^{\diamondsuit}}(\lambda)=cp^{-\lambda}s_{\lambda
}^{\diamondsuit}(p)$. Now, for any sequence of dominant weights $(\lambda
^{(a)})_{a}$ with $\lambda^{(a)}=a\mathbf{m}+o(a)$, we get
\begin{multline*}
\lim_{a\rightarrow+\infty}\mathbb{P}\left[  \mathcal{W}_{\ell}\in
\mathcal{P}^{\diamondsuit}\ \text{for all }\ell\geq0\mid\mathcal{W}%
_{0}=\lambda^{(a)}\right] \\
= \lim_{a\rightarrow+\infty}\mathbb{P}\left[  \lambda^{(a)}+X_{1}%
+\cdots+X_{\ell}\in\mathcal{P}^{\diamondsuit}\ \text{for all }\ell
\geq1\right]  =1.
\end{multline*}
On the other hand, by Proposition~\ref{prop-_imit-psi}, we get
$\displaystyle\lim_{a\rightarrow+\infty}p^{-\lambda(a)}s_{\lambda
(a)}(p)=\nabla^{\diamondsuit}$ so that $c=\frac{1}{\nabla^{\diamondsuit}}$.
\end{proof}

We can also recover the asymptotic behaviors of $f_{\lambda/\mu}^{\emptyset},$
$f_{\lambda/\mu}^{s}$ and $f_{\lambda/\mu}^{h}$ without using the asymptotic
representation theory of the symmetric and spin symmetric groups (see \cite{K}
and \cite{N}).

\begin{theorem}
\label{Th_Asympt}Suppose that the vector $\mathbf{m}$ satisfies
Condition~\emph{\ref{Cond}}. If $\lambda^{(\ell)}=\ell\mathbf{m}%
+o(\ell^{\alpha})$ with $\alpha<2/3$, then for all $\mu\in\mathcal{P}%
^{\diamondsuit}$ we have
\begin{equation}
\lim_{\ell\rightarrow\infty}\frac{f_{\lambda^{(\ell)}/\mu}^{\diamondsuit}
}{f_{\lambda^{(\ell)}}^{\diamondsuit}}=s_{\mu}^{\diamondsuit}(p).
\label{asympt}%
\end{equation}

\end{theorem}

\begin{proof}
Consider a sequence of dominant weights of the form $\lambda^{(\ell)}%
=\ell\mathbf{m}+o(\ell^{\alpha})$. By Proposition~\ref{Prop_dec_skew}, we
have
\begin{equation}
\frac{f_{\lambda^{(\ell)}/\mu}^{\diamondsuit}}{f_{\lambda^{(\ell)}%
}^{\diamondsuit}}=\sum_{\gamma\in P}K_{\mu,\gamma}^{\diamondsuit}%
\frac{f_{\lambda^{(\ell)}-\gamma}^{\diamondsuit}}{f_{\lambda^{(\ell)}%
}^{\diamondsuit}}=\sum_{\gamma\in P}K_{\mu,\gamma}^{\diamondsuit}%
\frac{f_{\lambda^{(\ell)}-\gamma}^{\diamondsuit}\cdot p^{\lambda^{(\ell
)}-\gamma}}{f_{\lambda^{(\ell)}}^{\diamondsuit}\cdot p^{\lambda^{(\ell)}}%
}p^{\gamma},\label{quo-f}%
\end{equation}
where the sums have finitely many terms since $V^{\diamondsuit}(\mu)$ is
finite. For all $\gamma\in P$, we have
\[
\frac{f_{\lambda^{(\ell)}-\gamma}^{\diamondsuit}\cdot p^{\lambda^{(\ell
)}-\gamma}}{f_{\lambda^{(\ell)}}^{\diamondsuit}p^{\lambda^{(\ell)}}}%
=\frac{\mathbb{P}[\mathcal{W}_{1}\in\mathcal{P}^{\diamondsuit},\ldots
,\mathcal{W}_{\ell}\in\mathcal{P}^{\diamondsuit},\mathcal{W}_{\ell}%
=\lambda^{(\ell)}-\gamma]}{\mathbb{P}[\mathcal{W}_{1}\in\mathcal{P}%
^{\diamondsuit},\ldots,\mathcal{W}_{\ell}\in\mathcal{P}^{\diamondsuit
},\mathcal{W}_{\ell}=\lambda^{(\ell)}]},
\]
and this quotient tends to $1$ when $\ell\to+\infty$, by Theorem~\ref{RQT}.
This implies
\[
\lim_{\ell\rightarrow+\infty}\frac{f_{\lambda^{(\ell)}/\mu}^{\diamondsuit}%
}{f_{\lambda^{(\ell)}}^{\diamondsuit}}=\sum_{\gamma\in P}K_{\mu,\gamma
}^{\diamondsuit}p^{\gamma}=s_{\mu}^{\diamondsuit}(p).
\]

\end{proof}

%\subsection{Conditioning in dimension 2}
%Let us illustrate our results by considering the situation in dimension
%$2$. The relevant algebras are then $\mathfrak{g}=\mathfrak{gl}
%(2),\mathfrak{gl}(1,1)$ or $\mathfrak{q}(2)$. In this very simple case, we have
%\begin{enumerate}
%\item $\mathcal{P}^{\emptyset}=\{\lambda=(\lambda_{1},\lambda_{2}
%)\in\mathbb{Z}^{2}\mid\lambda_{1}\geq\lambda_{2}\geq0\},$
%\item $\mathcal{P}^{s}=\{\lambda=(\lambda_{1},\lambda_{2})\in\mathcal{P}%
%^{\emptyset}\mid\lambda_{1}=\lambda_{2}\Longrightarrow\lambda=(0,0)\},$
%\item $\mathcal{P}^{h}=\{\lambda=(\lambda_{\overline{1}},\lambda_{1}
%)\in\mathbb{Z}_{\geq0}^{2}\mid\lambda_{\overline{1}}=0\Longrightarrow
%\lambda_{1}=0\}.$
%\end{enumerate}
%By applying Corollary \ref{Cor_StayinC}, we get
%\[
%\mathbb{P}[\mathcal{W}_{\ell}\in\mathcal{P}^{\diamondsuit}\text{ for all }%
%\ell\geq0\mid\mathcal{W}_{0}=(0,0)]=\left\{
%\begin{array}
%[c]{lll}%
%\displaystyle1-\frac{p_{2}}{p_{1}} & \text{ for } & \diamondsuit=\emptyset,\\
%\displaystyle p_{1}\left(  1-\frac{p_{2}}{p_{1}}\right)   & \text{ for } &
%\diamondsuit=s,\\
%\displaystyle p_{\overline{1}} & \text{ for } & \diamondsuit=h.
%\end{array}
%\right.
%\]
%These formulas are not surprising and can be obtained in an elementary way by
%interpreting the one-way simple walk as the random walk on $\mathbb{Z}$ with
%transitions $\pm1$. Conditioning to stay in $\mathcal{P}^{s}$ starting from
%$(0,0)$ is equivalent to impose first that $\mathcal{W}_{1}=e_{1}$, next that
%$\mathcal{W}_{\ell},\ell\geq2,$ remains in $(1,0)+\mathcal{P}^{\emptyset}$.
%Conditioning to stay in $\mathcal{P}^{h}$ simply means that $\mathcal{W}%
%_{1}=e_{\overline{1}}$.
%\noindent Observe the reduction of the random walk $(\mathcal{W}_{\ell}
%)_{\ell\geq0}$ in dimension $2$ to a random walk on $\mathbb{Z}$ cannot be
%generalized to dimensions greater than $2$.
%\noindent In dimension $2$, there is also a simple relation between the Pitman
%transforms $\mathfrak{P}^{\emptyset}$ and $\mathfrak{P}^{s}$. For any
%$\ell\geq1$ and $\omega=x_{1}\cdots x_{\ell}x_{\ell+1}\in(\mathcal{A}%
%^{\diamondsuit})^{\ell+1}$, set $\omega^{\flat}=x_{1}\cdots x_{\ell}$ and
%$\mathfrak{P}^{\emptyset}(\omega^{\flat})=(\lambda_{1},\lambda_{2}%
%)\in\mathcal{P}^{\emptyset}$; then
%\begin{equation}
%\mathfrak{P}^{s}(\omega)=(\lambda_{1}+1,\lambda_{2}).\label{Pit2}%
%\end{equation}
%This means that $(\mathcal{H}_{\ell}^{s})_{\ell\geq0}$ is obtained from
%$(\mathcal{H}_{\ell}^{\emptyset})_{\ell\geq0}$ by translation by $(1,0)$ in
%space and translation by $1$ in time. Let $\omega^{s}$ and $\omega^{\emptyset
%}$ be the highest weight vertices associated to $\omega$ for the
%$\mathfrak{q}(2)$ and $\mathfrak{gl}(2)$-structures, respectively. Write $x$
%for the rightmost letter in $\omega^{\emptyset}$. By using crystal basis
%theory and relations (1.7) and (1.8) in \cite{GHSKM}, one can prove that
%$\omega^{s}=\omega^{\emptyset}$ if $x=1$. If $x=2$, the weight $\omega^{s}$ is
%obtained from $\omega^{\emptyset}$ by changing $x=2$ in $1$. This gives
%(\ref{Pit2}).
%\noindent For $n>2$, there is no simple relation between the generalized
%Pitman transforms $\mathfrak{P}^{\emptyset}$ and $\mathfrak{P}^{s}$ as shown
%by the following example.
%\begin{examples}
%In view of example \ref{ex_s}, we have for $\omega=232145331\cdots$
%\[%
%\begin{tabular}
%[c]{c|cccccccccc}%
%$\ell$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $\cdots
%$\\\cline{2-11}%
%$\mathcal{H}_{\ell}^{\emptyset}(\omega)$ & $(1)$ & $(1,1)$ & $(2,1)$ & $(3,1)$
%& $(3,1,1)$ & $(3,1,1,1)$ & $(3,2,1,1)$ & $(3,3,1,1)$ & $(4,3,1,1)$ & $\cdots
%$\\
%$\mathcal{H}_{\ell}^{s}(\omega)$ & $(1)$ & $(2)$ & $(2,1)$ & $(3,1)$ & $(4,1)$
%& $(5,1)$ & $(5,2)$ & $(5,2,1)$ & $(5,3,1)$ & $\cdots$%
%\end{tabular}
%\]
%where the middle row is obtained by computing $P^{\emptyset}(\omega)$.
%\end{examples}


\section{Appendix: RSK correspondence and crystal basis theory}

\label{Sec-append}

\subsection{RSK correspondence}

%\subsection{Proof of Theorem~\ref{RQT}}
%\label{subsec_miscell}
%We first need some elementary property of semigroups and cones; we have found
%no reference of the following fact in the literature:
%\begin{lemma}
%\label{sg-c}Denote by $\mathring{\mathcal{C}_{c}}$ the cone generated by
%$\mathring{\mathcal C}$. For all compact set $K\subset\mathring{\mathcal
%{C}_{c}}$ and all large enough $t>0$, we have $tK\subset\mathring{\mathcal C}$.
%\end{lemma}
%\begin{proof}
%We have $K\subset\mathring{\mathcal{C}_{c}}$. For all $x\in K$, there exists
%$t>0$ such that $tx\in\mathring{\mathcal C}$, and there is a closed ball
%centered at $tx$ and contained in $\mathring{\mathcal C}$. Coming back to $x$
%we obtain the following: for all $x\in K$, there exist $r>0$ and $t>0$ such
%that $t\overline{B}(x,r)\subset\mathring{\mathcal C}$. We can cover $K$ by
%finitely many such balls $B(x,r)$, and it is sufficient to prove the expected
%property for each of these balls.
%We start from the fact that, for a particular $t=t_{0}>0$ we have
%$t\overline{B}(x,r)\subset\mathring{\mathcal C}$, and we want to prove that it
%is true for all large enough $t$.
%We fix $\varepsilon>0$ such that $t_{0}\overline{B}(x,r+\varepsilon
%)\subset\mathring{\mathcal C}$. Let $\vec{u}$ denote a unitary vector in
%$\mathbb{R}^{n}$. For each $\vec{u}$ such that the half-line $\mathbb{R}%
%^{+}\vec{u}$ meets the ball $\overline{B}(x,r)$, we have
%\begin{equation}
%\mathbb{R}^{+}\vec{u}\cap t_{0}\overline{B}(x,r+\varepsilon)=[a(\vec
%{u}),b(\vec{u})]\vec{u} \label{an1}%
%\end{equation}
%where $a$ and $b$ are two continuous functions with $0<a<b$. By compacity, the
%quotient $b/a$ stays greater than a number $\rho>1$ when the vector $\vec{u}$
%varies as above. By the semigroup property of $\mathring{\mathcal C}$, for all
%positive integer $n$,
%\begin{equation}
%\lbrack na(\vec{u}),nb(\vec{u})]\vec{u}\subset\mathring{\mathcal C}.
%\label{an2}%
%\end{equation}
%Fix an integer $n_{0}>{\frac{1}{\rho-1}}.$ For all $n\geq n_{0}$, we have
%$nb\geq(n+1)a$, hence
%\begin{equation}
%\cup_{n\geq n_{0}}[na,nb]=[n_{0}a,+\infty). \label{an3}%
%\end{equation}
%From (\ref{an2}) and (\ref{an3}), we deduce that for all real $s\geq n_{0}$,
%\[
%s[a(\vec{u}),b(\vec{u})]\vec{u}\subset\mathring{\mathcal C}.
%\]
%With the help of (\ref{an1}), we conclude that for all real $s\geq n_{0}$,
%\[
%st_{0}\overline{B}(x,r)\subset\mathring{\mathcal C}.
%\]
%This concludes the proof.
%\end{proof}
%We now prove the following result, which is crucial in this work:
%\begin{lemma}
%\label{psc} One gets $\qquad\mathbb{P}\left[  \ S_{\ell}\in\mathcal{C}%
%\ \forall{\ell}\geq0\right]  >0. $
%\end{lemma}
%\begin{proof}
%By hypothesis (h3), one may fix $a$ in $\mathring{\mathcal C}$ such that, for
%all $\varepsilon>0$,
%\[
%\mathbb{P}\left[  S_{1}\in\mathcal{C},\ldots,S_{\ell_{0}-1}\in\mathcal{C}%
%,S_{\ell_{0}}\in B(a,\varepsilon)\right]  >0.
%\]
%By the strong law of large numbers, the sequence $\left(  \frac{1}{\ell
%}S_{\ell}\right)  _{\ell\geq1}$ converges almost surely to $\mathbf{m}$.
%Therefore, thanks to Hypothesis (h3) and Lemma~\ref{sg-c}, almost surely, one
%gets $S_{\ell}\in\mathring{\mathcal C}$ for any large enough ${\ell}$, that
%is
%\[
%\lim_{L\rightarrow+\infty}\mathbb{P}\left[  \forall{\ell}\geq L,\ S_{\ell}
%\in\mathring{\mathcal C}\right]  =1.
%\]
%By Lemma~\ref{sg-c}, we know that, for any $x\in\mathbb{R}^{n}$, for all large
%enough $k\in\mathbb{N}$ one gets $x+ka\in\mathring{\mathcal C}$. Thus
%\[
%\left[  \forall{\ell}\geq L,\ S_{\ell}\in\mathring{\mathcal C}\right]
%=\bigcup_{k\geq0}\left[  (\forall{\ell}\geq L,\ S_{\ell}\in\mathcal{C}^{\circ
%})\text{ and }(\forall{\ell}<L,\ S_{\ell}+ka\in\mathring{\mathcal C})\right]
%\]
%and therefore $\displaystyle\left[  \forall{\ell}\geq L,\ S_{\ell}
%\in\mathring{\mathcal C}\right]  \subset\bigcup_{k\geq0}\left[  \forall{\ell
%},\ S_{\ell}+ka\in\mathring{\mathcal C}\right]  $ since $\mathring{\mathcal
%C}$ is a semigroup. Hence, there exists $k\geq0$ such that $\mathbb{P}\left[
%\forall{\ell},\ S_{\ell}+ka\in\mathring{\mathcal C}\right]  >0.$ Fix such a
%$k$ and $\delta>0$ such that $B(a,\delta)\subset\mathring{\mathcal C}$.
%We consider now $k$ independent repetitions of the event
%\[
%A_{1}:=\left[  S_{1}\in\mathcal{C},\ldots,S_{\ell_{0}-1}\in\mathcal{C}
%,\ S_{\ell_{0}}\in B(a,\frac{\delta}{k+1})\right]  ,
%\]
%that is we consider the events $A_{j}$, $1\leq j\leq k+1$, defined by
%\[
%A_{j}:=\left[  S_{(j-1)\ell_{0}+1}-S_{(j-1)\ell_{0}}\in\mathcal{C}
%,\ldots,S_{(j-1)\ell_{0}+\ell_{0}-1}-S_{(j-1)\ell_{0}}\in\mathcal{C}
%,\ S_{j\ell_{0}}-S_{(j-1)\ell_{0}}\in B(a,\frac{\delta}{k+1})\right]  .
%\]
%In the following claim, we use the semigroup property: if the events
%$A_{1},\ldots,A_{k+1}$ are simultaneously realized, then
%\[
%S_{1}\in\mathcal{C},\ldots,S_{(k+1)\ell_{0}}\in\mathcal{C}\text{ and }\Vert
%S_{(k+1)\ell_{0}}-ka-a\Vert\leq\sum_{j=1}^{k+1}\Vert S_{j\ell_{0}
%}-S_{(j-1)\ell_{0}}-a\Vert\leq\delta,
%\]
%so $S_{(k+1)\ell_{0}}-ka\in\mathring{\mathcal C}$. We thus have%
%\begin{multline*}
%\mathbb{P}\left[  \forall{\ell}\geq1,\ S_{\ell}\in\mathcal{C}\right]
%\geq\mathbb{P}\left[  \left(  \cap_{j=1}^{k+1}A_{j}\right)  \text{ and
%}\left(  \forall{\ell}>(k+1)\ell_{0},\ S_{\ell}-S_{(k+1)\ell_{0}}%
%+ka\in\mathring{\mathcal C}\right)  \right] \\
%=\prod_{j=1}^{k+1}\mathbb{P}[A_{j}]\ \times\ \mathbb{P}\left[  \forall{\ell
%}>0,\ S_{\ell}+ka\in\mathring{\mathcal C}\right]  =\left(  \mathbb{P}%
%[A_{1}]\right)  ^{k+1}\ \times\ \mathbb{P}\left[  \forall{\ell}>0,\ S_{\ell
%}+ka\in\mathring{\mathcal C}\right]  >0.
%\end{multline*}
%\end{proof}
%We will also need a quite sharp estimation from below of the probability that
%the random walk stays inside the semigroup $\mathcal{C}$ up to time $n$;
%indeed, we have the
%\begin{lemma}
%\label{lem-marc} Assume the random variables $X_{\ell}$ are almost surely
%bounded. Let $\alpha\in]1/2,2/3[$. If the sequence $({\ell}^{-\alpha}\Vert
%g_{\ell}-{\ell}\mathbf{m}\Vert)_{\ell}$ is bounded, then there exists $c>0$
%such that, for all large enough $\ell$,
%\begin{equation}
%\mathbb{P}\left[  S_{1}\in\mathcal{C},\ldots,S_{\ell}\in\mathcal{C},S_{\ell
%}=g_{\ell}\right]  \geq\exp\left(  -c\ell^{\alpha}\right)  . \label{garbit}%
%\end{equation}
%\end{lemma}
%\begin{proof}
%Thanks to hypothesis (h3) we can fix $\delta>0$ such that the closure
%$\overline{B}(\mathbf{m},\delta)$ of the open ball $B(\mathbf{m},\delta)$ is
%contained in the cone $\mathring{\mathcal{C}_{c}}$ generated by $\mathring
%{\mathcal C}$. Thanks to Lemma~\ref{sg-c}, we know that $B(t\mathbf{m}%
%,t\delta)\subset\mathring{\mathcal C}$ for all large enough $t$.
%On the other hand, the law of large numbers tells us that $\displaystyle\lim
%_{\ell\rightarrow+\infty}\mathbb{P}\left[  \left\Vert \frac{1}{\ell}S_{\ell
%}-\mathbf{m}\right\Vert <\delta\right]  =1$. Combined with Lemma~\ref{psc},
%this implies that, for $\ell_{1}$ large enough,
%\[
%\mathbb{P}\left[  S_{1}\in\mathcal{C},\ldots,S_{\ell_{1}}\in\mathcal{C}%
%,S_{\ell_{1}}\in B(\ell_{1}\mathbf{m},\ell_{1}\delta)\right]  >0.
%\]
%From the two preceding claims, we deduce the existence of an integer $\ell
%_{1}>0$ and a point $y_{1}\in B(\ell_{1}\mathbf{m},\ell_{1}\delta
%)\cap\mathbb{Z}^{n}$ such that
%\[
%\mathbb{P}\left[  S_{1}\in\mathcal{C},\ldots,S_{\ell_{1}}\in\mathcal{C}%
%,S_{\ell_{1}}=y_{1}\right]  >0\quad\mathrm{and}\quad B(t\mathbf{m}%
%,t\delta)\subset\mathring{\mathcal C}\quad\mathrm{forall}\quad t\geq\ell_{1}.
%\]
%We fix such a pair $(\ell_{1},y_{1})$. Since $y_{1}\in\mathring{\mathcal C}$
%we have $y_{1}+\mathcal{C}_{c}\subset\mathcal{C}$, by hypothesis (h1). For
%$\ell\geq\ell_{1}$, we have
%\begin{multline*}
%\mathbb{P}\left[  S_{1}\in\mathcal{C},\ldots,S_{\ell}\in\mathcal{C},S_{\ell
%}=g_{\ell}\right] \\
%\geq\mathbb{P}\big[S_{1}\in\mathcal{C},\ldots,S_{\ell_{1}}\in\mathcal{C}%
%,S_{\ell_{1}}=y_{1},S_{\ell_{1}+1}\in y_{1}+\mathcal{C}_{c},S_{\ell_{1}+2}\in
%y_{1}+\mathcal{C}_{c},\ldots\\
%\ldots,S_{\ell}\in y_{1}+\mathcal{C}_{c},S_{\ell}-S_{\ell_{1}}=g_{\ell}%
%-y_{1}\big]\\
%=\mathbb{P}\left[  S_{1}\in\mathcal{C},\ldots,S_{\ell_{1}}\in\mathcal{C}%
%,S_{\ell_{1}}=y_{1}\right]  \times\mathbb{P}\left[  S_{1}\in\mathcal{C}%
%_{c},S_{2}\in\mathcal{C}_{c},\ldots,S_{\ell-\ell_{1}}\in\mathcal{C}%
%_{c},S_{\ell-\ell_{1}}=g_{\ell}-y_{1}\right]
%\end{multline*}
%Now, since the sequence $\left(  \ell^{-\alpha}\Vert g_{\ell+\ell_{1}}%
%-y_{1}-\ell\mathbf{m}\Vert\right)  $ is bounded, one may apply Lemma~4.4 of
%\cite{LLP}, with the cone $\mathcal{C}_{c}$, which satisfies the required
%properties with respect to the random walk; one gets
%%\[
%%\liminf_{\ell\to\infty}\left(  \mathbb{P}\left(  S_{1}\in\mathcal{C}%
%%_{c},\ldots,S_{\ell}\in\mathcal{C}_{c},S_{\ell}=g_{\ell+\ell_{1}}%
%%-y_{1}\right)  \right)  ^{{\ell}^{-\alpha}}>0,
%%\]
%%hence%
%\[
%\liminf_{\ell\rightarrow\infty}\left(  \mathbb{P}\left[  S_{1}\in
%\mathcal{C}_{c},\ldots,S_{\ell-\ell_{1}}\in\mathcal{C}_{c},S_{\ell-\ell_{1}%
%}=g_{\ell}-y_{1}\right]  \right)  ^{{\ell}^{-\alpha}}>0,
%\]
%so that
%\[
%\liminf_{\ell\rightarrow\infty}\left(  \mathbb{P}\left[  S_{1}\in
%\mathcal{C},\ldots,S_{\ell}\in\mathcal{C},S_{\ell}=g_{\ell}\right]  \right)
%^{{\ell}^{-\alpha}}>0.
%\]
%\end{proof}
%\bigskip
%\begin{proof}
%[Proof of Theorem \ref{RQT}]We follow the lines of the proof of Theorem 4.3 in
%\cite{LLP}. The claim \textquotedblleft For all $\ell\geq1$, $B(\ell
%\mathbf{m},\ell\delta)\subset\mathcal{C}$\textquotedblright\ has to be
%replaced by \textquotedblleft For all large enough $\ell$, $B(\ell
%\mathbf{m},\ell\delta)\subset\mathcal{C}$\textquotedblright\ but this does not
%disrupt the proof. In the proof of Theorem 4.3 in \cite{LLP}, we use Lemma
%\ref{lem-marc} and the following claim: there exists $\varepsilon>0$ such that
%when the walk goes out of $\mathcal{C}$, its distance to the point
%$k\mathbf{m}$ is greater than $k\varepsilon$. As it is stated, this claim is
%not necessary true in our present situation; fortunately this is certainly
%true for all large enough $k$ and it is applied only to values of $k$ greater
%than $\ell^{\alpha}$. For all the rest, the proof in \cite{LLP} can be
%followed line by line.
%\end{proof}
%\bigskip


We now turn to the proof of Proposition~\ref{Cor_RSK}. We first need to
interpret the RSK correspondence in terms of crystal basis theory. We refer
the reader to \cite{Kashi} ($\diamondsuit=\emptyset$), \cite{BKK}
($\diamondsuit=h)$, and \cite{GHSKM} ($\diamondsuit=s$) for detailed
expositions. With each irreducible representation $V^{\diamondsuit}(\lambda)$,
its crystal graph $B^{\diamondsuit}(\lambda)$ is associated, which is an
oriented graph with arrows $\overset{i}{\rightarrow}$ colored by symbols
$i\in\{1,\ldots,n-1\}$ for $\mathfrak{g}=\mathfrak{gl}(n),$ $i\in
\{\overline{m-1},\ldots,\overline{1},0,1,\ldots,n-1\}$ for $\mathfrak{g}%
=\mathfrak{gl}(m,n)$ and $i\in\{1,\ldots,n-1,\overline{1}\}$ for
$\mathfrak{g}=\mathfrak{q}(n)$.

Below we provide the crystal $B^{\diamondsuit}$ of the defining representation
$V^{\diamondsuit}$:
\[%
\begin{array}
[c]{lll}%
\text{for\ }\mathfrak{g}=\mathfrak{gl}(n) & \ B^{\emptyset}\ \text{ is} &
1\overset{1}{\rightarrow}2\overset{2}{\rightarrow}\cdots\overset
{n-1}{\rightarrow}n,\\
\text{for\ }\mathfrak{g}=\mathfrak{gl}(m,n) & \ B^{h}\ \text{ is} &
\overline{m}\overset{\overline{m-1}}{\rightarrow}\overline{m-1}\overset
{\overline{m-2}}{\rightarrow}\cdots\overset{\overline{1}}{\rightarrow
}\overline{1}\overset{0}{\rightarrow}1\overset{1}{\rightarrow}2\overset
{2}{\rightarrow}\cdots\overset{n-1}{\rightarrow}n,\\
\text{for\ }\mathfrak{g}=\mathfrak{q}(n) & \ B^{s} \ \text{ is} &
%TCIMACRO{\QATOPD{.}{.}{\overset{1}{\rightarrow}}{\underset{\overline{1}%
%}{\dashrightarrow}}}%
%BeginExpansion
\genfrac{.}{.}{0pt}{}{\overset{1}{\rightarrow}}{\underset{\overline{1}%
}{\dashrightarrow}}%
%EndExpansion
2\overset{2}{\rightarrow}\cdots\overset{n-1}{\rightarrow}n.
\end{array}
\]
Observe that the vertices of $B^{\diamondsuit}$ coincide with the letters of
$\mathcal{A}^{\diamondsuit}$. The vertices of $(B^{\diamondsuit})^{\otimes
\ell}$ are thus labeled by the words of $(\mathcal{A}^{\diamondsuit})^{\ell}$
by identifying each vertex $b=x_{1}\otimes\cdots\otimes x_{\ell}$ of
$(B^{\diamondsuit})^{\otimes\ell}$ with the word $b=x_{1}\cdots x_{\ell}$.

More generally, any representation $M^{\diamondsuit}$ (irreducible or not)
appearing in a tensor product $(V^{\diamondsuit})^{\otimes\ell}$ admits a
crystal $B^{\diamondsuit}(M)$. In this paper, each crystal $B(M)$ is realized
as a subcrystal of a crystal $(B^{\diamondsuit})^{\otimes\ell}$. The crystal
$B^{\diamondsuit}(M)$ is graded by the weights of $\mathfrak{g}$. There is a
map $\mathrm{wt}:B^{\diamondsuit}(M)\rightarrow P$. To obtain the
decomposition of $M^{\diamondsuit}$ in its irreducible components, it then
suffices to obtain the decomposition of $B^{\diamondsuit}(M)$ into its
connected components. The crystal $B^{\diamondsuit}(M\otimes
N)=B^{\diamondsuit}(M)\otimes B^{\diamondsuit}(N)$ associated with the tensor
product $M\otimes N$ of two representations can be constructed from the
crystal of $M$ and $N$ by simple combinatorial rules. A highest weight vertex
in $B^{\diamondsuit}(M)$ is a vertex $b$ for which no arrow $b^{\prime
}\overset{i}{\rightarrow}b$ exists in $B^{\diamondsuit}(M)$; a lowest weight
vertex in $B^{\diamondsuit}(M)$ is a vertex $b$ for which no arrow
$b\overset{i}{\rightarrow}b^{\prime}$ exists in $B^{\diamondsuit}(M)$.

Let $\sigma_{0}$ be the element of the Weyl group of $\mathfrak{g}$ defined
by
\[
\left\{
\begin{array}
[c]{rlll}%
\sigma_{0}(\beta_{1},\beta_{2},\ldots,\beta_{n})\kern-7pt & =(\beta_{n}%
,\beta_{n-1},\ldots,\beta_{1}) & \text{ for all }\beta\in\mathbb{Z}^{n}, &
\text{if }\diamondsuit=\emptyset,s,\\
\kern-3pt
\sigma_{0}(\beta_{\overline{m}},\ldots,\beta_{\overline{1}},\beta_{1}%
,\ldots,\beta_{n})\kern-7pt & =(\beta_{\overline{1}},\ldots\beta_{\overline
{m}},\beta_{n},\ldots,\beta_{1}) & \text{ for all }\beta\in\mathbb{Z}^{m+n}, &
\text{if }\diamondsuit=h.
\end{array}
\right.
\]
For $\diamondsuit=\emptyset\mathfrak{\ }$and $\lambda\in\mathcal{P}%
^{\emptyset}$, the crystal $B^{\emptyset}(\lambda)$ contains a unique highest
weight vertex $b^{\emptyset,\lambda}$ and a unique lowest weight vertex
$b_{\lambda}^{\emptyset}$. They have respective weights $\lambda$ and
$\sigma_{0}(\lambda)$. For $\diamondsuit=s$ and $\lambda\in\mathcal{P}^{s}$,
the crystal $B^{s}(\lambda)$ contains a unique highest weight vertex
$b^{s,\lambda}$ but may admit several lowest weight vertices. Nevertheless it
admits a unique lowest weight vertex $b_{\lambda}^{s}$ with weight $\sigma
_{0}(\lambda)$. For $\diamondsuit=h\mathfrak{\ }$and $\lambda\in
\mathcal{P}^{h}$, the crystal $B^{s}(\lambda)$ may admit several highest or
lowest weight vertices, and the situation is more complicated.

By crystal basis theory, for $\diamondsuit\in\{\emptyset,h, s\}$ the
multiplicity of $V^{\diamondsuit}(\lambda)$ in $M^{\diamondsuit}$ is given by
the number of highest weight vertices of weight $\lambda$ in $B^{\diamondsuit
}(M)$ or equivalently by the number of its lowest weight vertices of weight
$\sigma_{0}(\lambda)$.

\begin{lemma}
\label{Lem_split}Assume that $\diamondsuit\in\{\emptyset, s\}$, and consider
$u\otimes v\in B^{\diamondsuit}(\lambda)\otimes B^{\diamondsuit}(\mu)$, where
$\lambda,\mu\in\mathcal{P}^{\diamondsuit}$. Then

\begin{enumerate}
\item for $\diamondsuit=\emptyset$, the vertex $u\otimes v$ is a highest
weight vertex only if $u=b^{\emptyset,\lambda}$,

\item for $\diamondsuit=s$, the vertex $u\otimes v$ is a lowest weight vertex
only if $v=b_{\mu}^{s}$.
\end{enumerate}
\end{lemma}

\begin{remark}
Assertion~(1) does not hold in general for $\diamondsuit\in\{h, s\}$.
Moreover, Assertion~(2) also fails when $\diamondsuit=h$, which causes some
complications. The lack of Assertion~(1) for $\diamondsuit\in\{h,s\}$ explains
also why in these cases the paths which remain in $\mathcal{P}^{\diamondsuit}$
are not fixed by the Pitman transforms we have defined.
\end{remark}

Two crystals $B$ and $B^{\prime}$ are isomorphic when there exists a bijection
$\phi:B\rightarrow B^{\prime}$ which respects the graph structure, i.e., which
has the property that $\phi(a)\overset{i}{\rightarrow}\phi(b)$ in $B^{\prime}$
if and only if $a\overset{i}{\rightarrow}b$ in $B$. When $B$ and $B^{\prime}$
are crystals associated with irreducible representations, such a crystal
isomorphism exists if and only if these representations are isomorphic, and in
that case it is unique. For any $w\in(B^{\diamondsuit})^{\otimes\ell}$, write
$B^{\diamondsuit}(w)$ for the connected component of $(B^{\diamondsuit
})^{\otimes\ell}$ containing $w$. We can now interpret Theorem~\ref{Th_RSK} in
terms of crystal basis theory.

\begin{theorem}
\label{Th_RSKcrystal}Consider two vertices $w_{1}$ and $w_{2}$ of
$(B^{\diamondsuit})^{\otimes\ell}$. Then

\begin{enumerate}
\item $P^{\diamondsuit}(w_{1})=P^{\diamondsuit}(w_{2})$ if and only if
$B^{\diamondsuit}(w_{1})$ is isomorphic to $B^{\diamondsuit}(w_{2})$ and the
unique associated isomorphism sends $w_{1}$ on $w_{2}.$

\item $Q^{\diamondsuit}(w_{1})=Q^{\diamondsuit}(w_{2})$ if and only if
$B^{\diamondsuit}(w_{1})=B^{\diamondsuit}(w_{2})$.

\item For any standard $\diamondsuit$-tableau $T,$ the set of words
$B^{\diamondsuit}(T)$ defined in \S ~\ref{subsec_RSK} has the structure of a
crystal graph isomorphic to the abstract crystal $B^{\diamondsuit}(\lambda)$,
where $\lambda$ is the shape of $T$.

\item If we denote by $\phi:B^{\diamondsuit}(\lambda)\rightarrow
B^{\diamondsuit}(T)$ this isomorphism, we have $\mathrm{wt}(b)=\mathrm{wt}
(\phi(b))$ for all $b\in B^{\diamondsuit}(T),$ that is, the weight graduation
defined on the abstract crystal $B^{\diamondsuit}(T)$ is compatible with the
weight graduation defined on words.
\end{enumerate}
\end{theorem}

\subsection{Proof of {Proposition~\ref{Cor_RSK}}}

We finish this section with the proof of a crucial consequence of the
Robinson--Schensted--Knuth correspondence.

Consider $\mu\in\mathcal{P}^{\diamondsuit}$ and $T$ a standard tableau of
shape $\mu$. Let $\ell$ be a nonnegative integer and $\mathcal{U}_{\ell
,T}^{\diamondsuit}$ be the set of pairs $(P,Q)$, where $Q$ is a $\diamondsuit
$-standard tableau with $\ell+\left\vert \mu\right\vert $ boxes containing $T$
as a subtableau (that is, $T$ is the subtableau obtained by considering only
the entries $1,\ldots,\left\vert \mu\right\vert $ of $Q$) and $P$ a
$\diamondsuit$-tableau with the same shape as $Q$. By Theorem~\ref{Th_RSK},
the restriction of $\theta_{\ell+\left\vert \mu\right\vert }^{\diamondsuit}$
%(see Theorem~\ref{Cor_RSK})
to the subset $B^{\diamondsuit}(T)\times(\mathcal{A}^{\diamondsuit})^{\ell
}\subset\mathcal{A}_{\ell+\left\vert \mu\right\vert }^{\diamondsuit}$ yields
the one to one correspondence
\[
\theta_{\ell,T}^{\diamondsuit}:\left\{
\begin{array}
[c]{clc}%
B^{\diamondsuit}(T)\otimes(B^{\diamondsuit})^{\otimes\ell} & \rightarrow &
\mathcal{U}_{\ell,T}^{\diamondsuit}\\
w_{T}\otimes w & \mapsto & (P^{\diamondsuit}(w_{T}w),Q^{\diamondsuit}%
(w_{T}w)).
\end{array}
\right.
\]
Indeed, for any $u\in(B^{\diamondsuit})^{\otimes\ell+\left\vert \mu\right\vert
}$, $T$ is a subtableau of $Q^{\diamondsuit}(u)$ if and only if $u$ can be
written in the form $u=w_{T}\otimes w$, with $w_{T}\in B^{\diamondsuit}(T)$
and $w\in(B^{\diamondsuit})^{\otimes\ell}$. By crystal basis theory, for
$(B^{\diamondsuit})^{\otimes\ell+\left\vert \mu\right\vert }$, the number of
connected components of $B^{\diamondsuit}(T)\otimes(B^{\diamondsuit}%
)^{\otimes\ell}$ isomorphic to some $B^{\diamondsuit}(\lambda)$ is equal to
$f_{\lambda/\mu}^{\diamondsuit}$; using Theorem~\ref{Th_RSKcrystal}, we see
that it coincides with the number of standard $\diamondsuit$-tableaux of shape
$\lambda$ containing $T$ as a subtableau. There is a natural bijection between
the set of such tableaux and the set of skew standard $\diamondsuit$-tableaux
of shape $\lambda/\mu$: to any standard tableau $Q$ containing $T$, it
associates the skew tableau obtained by deleting the boxes of $T$ in $Q$ and
subtracting $\left\vert \mu\right\vert $ from the entries of the remaining
boxes. This proves Assertion~(1) of Proposition~\ref{Cor_RSK}.

\bigskip

Our method to prove Assertion~(2) of Proposition~\ref{Cor_RSK} depends on
$\diamondsuit$.

For $\diamondsuit=\emptyset,$ let $b=b_{1}\otimes b_{2}$ be a highest weight
vertex in $B^{\emptyset}(\kappa)\otimes B^{\emptyset}(\mu)$ of weight
$\lambda$. By Lemma~\ref{Lem_split}, we must have $b_{1}=b^{\emptyset,\kappa}%
$. Since $\mathrm{wt}(b)=\mathrm{wt}(b_{1})+\mathrm{wt}(b_{2})$, this implies
that $b_{2}\in B^{\diamondsuit}(\mu)$ has weight $\lambda-\kappa$. Therefore
$m_{\kappa,\mu}^{\emptyset,\lambda}\leq K_{\mu,\lambda-\kappa}^{\emptyset}$
since we can define an injective map from the set of highest weight vertices
of weight $\lambda$ in $B^{\emptyset}(\kappa)\otimes B^{\emptyset}(\mu)$ to
the set of vertices of weight $\lambda-\kappa$ in $B^{\emptyset}(\mu)$.

For $\diamondsuit=s,$ let $b=b_{1}\otimes b_{2}$ be a lowest weight vertex in
$B^{s}(\kappa)\otimes B^{s}(\mu)$ of weight $\lambda$. We must have
$b_{2}=b_{\mu}^{s}$, so $\mathrm{w}(b_{1})=\sigma_{0}(\lambda)-\sigma_{0}%
(\mu)=\sigma_{0}(\lambda-\mu)$. Similarly, we deduce that $m_{\kappa,\mu
}^{s,\lambda}=m_{\mu,\kappa}^{s,\lambda}\leq K_{\kappa,\sigma_{0}(\lambda
-\mu)}^{s}$. But $K_{\kappa,\sigma_{0}(\lambda-\mu)}^{s}=K_{\kappa,\lambda
-\mu}^{s}$, since $\sigma_{0}$ is an element of the Weyl group of
$\mathfrak{q}(n)$. Thus $m_{\mu,\kappa}^{s,\lambda}\leq K_{\kappa,\lambda-\mu
}^{s}$, as desired.

It remains to consider the case where $\diamondsuit=h$, i.e., the case of
$\mathfrak{gl}(m,n)$. Since Lemma~\ref{Lem_split} is no longer true in this
case, we shall need a different strategy. We use the Littlewood--Richardson
rule established in \cite{KK}. We say that a word $w=x_{1}\cdots x_{\ell}$
with letters in $\mathbb{Z}_{>0}$ is a
\textit{lattice permutation} if, for $k=1,\ldots,\ell$ and all positive
integers $i$, the number of letters $i$ in the prefix $w^{(k)}=x_{1}\cdots
x_{k}$ is greater than or equal to the number of letters $i+1$.

Consider $\lambda,\mu,\kappa$ in $\mathcal{P}^{h}$ such that $\left\vert
\mu\right\vert =\left\vert \lambda\right\vert -\left\vert \kappa\right\vert $.
Decompose the diagram of $\lambda$ into $\lambda^{(1)}$ (obtained by
considering its first $m$ rows) and $\lambda^{(2)}$, as in \S ~\ref{subsec_YD}%
. We denote by $LR_{\mu,\kappa}^{\lambda}$ the set of tableaux $T$ obtained by
filling the Young diagram $\lambda/\kappa$ with positive integers (see
Example~\ref{exam_teta}) such that:

\begin{enumerate}
\item the rows of $T$ weakly increase from left to right,

\item the columns of $T$ strictly increase from top to bottom,

\item the word $w$ obtained by reading first the $m$ rows of $\left(
\lambda/\kappa\right)  ^{(1)}$ from right to left and top to bottom, next the
$n$ columns of $\left(  \lambda/\kappa\right)  ^{(2)}$ from right to left and
top to bottom, is a lattice permutation such that, for all integers $k\geq1$,
the number of letters $k$ in $w$ is equal to the length of the $k$-th row of
the Young diagram of $\mu$, that is the $k$-th coordinate of $(\mu^{(1)}%
,(\mu^{(2)})^{\prime})$, where $(\mu^{(2)})^{\prime}$ is the conjugate
partition of $\mu^{(2)}$.
\end{enumerate}

We obtain the following proposition.

\begin{proposition}
[\cite{KK}]With the previous notations, we have $m_{\mu,\kappa}^{h,\lambda
}=\mathrm{card}(LR_{\mu,\kappa}^{\lambda})$.
\end{proposition}

By the previous proposition, in order to prove that $m_{\kappa,\mu}%
^{h,\lambda}\leq K_{\mu,\lambda-\kappa}^{h}$ it suffices to construct an
embedding $\theta$ from $LR_{\mu,\kappa}^{\lambda}$ into the set of
$h$-tableaux of shape $\mu$ and weight $\lambda-\kappa$. We proceed as
follows. Consider $T\in LR_{\mu,\kappa}^{\lambda}$ and write $R_{\overline{m}%
},\ldots,R_{\overline{1}}$ for the rows with boxes in $\lambda^{(1)}$ and
$C_{1},\ldots,C_{n}$ for the columns with boxes in $\lambda^{(2)}$. In
particular, for $i=1,\ldots,m$ and $j=1,\ldots,n$, the row $R_{\overline{i}}$
contains $\lambda_{\overline{i}}^{(1)}-\kappa_{\overline{i}}^{(1)}$ letters,
and the column $C_{j}$ contains $\lambda_{j}^{(2)}-\kappa_{j}^{(2)}$ letters.

We want to define the $h$-tableau $\theta(T)$. We first construct recursively
a tableau $T^{(1)}$ with $\lambda_{k}^{(1)}-\kappa_{k}^{(1)}$ letters
$\overline{k}$ for $k=1,\ldots,m$. We begin by defining $T_{\overline{m}}$ as
the row with $\lambda_{\overline{m}}^{(1)}-\kappa_{\overline{m}}^{(1)}$
letters $\overline{m}$. Assume that $T_{\overline{k+1}}$, $k\in\{2,\ldots
,m\}$, is already constructed. Then $T_{\overline{k}}$ is obtained by adding a
letter $\overline{k}$ in the $i$-th row of $T_{\overline{k+1}}$ for each
occurrence of the integer $i$ in $R_{\overline{k}}$.

Set $T^{(1)}=T_{\overline{1}}$. We construct $\theta(T)$ by adding
successively letters to $T^{(1)}$. First define $T_{1}$ by adding a letter $1$
in the $i$-th row of $T_{\overline{1}}$ for each integer $i$ in $C_{1}$. Now,
if $T_{k}$ is given, for some $k\in\{1,\ldots,m-1\}$, we construct $T_{k+1}$
by adding a letter $k$ in the $i$-th row of $T_{k}$ for each integer $i$
appearing in $C_{k}$. Finally, we set $\theta(T)=T_{n}$. Observe that, by
construction, the $k$-th row of $\theta(T)$ contains as many boxes as the
number of letters $k$ in the lattice permutation $w$ associated with $T$ by
Assertion~(3). Moreover, $\theta(T)$ is an $h$-tableau since $w$ is a lattice
permutation. It has shape $\mu$ and weight $\lambda-\kappa$, as desired.
Moreover, the map $\theta$ is injective since $\theta(T)$ records both the
number and the positions of the letters $k$ in the skew shape $\lambda-\kappa
$. This proves that $m_{\kappa,\mu}^{h,\lambda}\leq K_{\mu,\lambda-\kappa}%
^{h}$, as desired.

\begin{example}
\label{exam_teta}Take $\lambda=(3,3,3\mid3,3)$ with $m=n=3,$ $\kappa
=(2,0,0\mid0,0)$, and $\mu=(3,3,2\mid3,2)=(3,3,2,2,2,1)$. For
\[
T=%
\begin{tabular}
[c]{ll|l}\cline{3-3}
&  & \multicolumn{1}{|l|}{$1$}\\\hline
\multicolumn{1}{|l}{$1$} & \multicolumn{1}{|l|}{$1$} &
\multicolumn{1}{|l|}{$2$}\\\hline
\multicolumn{1}{|l}{$2$} & \multicolumn{1}{|l|}{$2$} &
\multicolumn{1}{|l|}{$3$}\\\hline
\multicolumn{1}{|l}{$\mathbf{3}$} & \multicolumn{1}{|l|}{$\mathbf{4}$} &
\\\cline{1-2}%
\multicolumn{1}{|l}{$\mathbf{4}$} & \multicolumn{1}{|l|}{$\mathbf{5}$} &
\\\cline{1-2}%
\multicolumn{1}{|l}{$\mathbf{5}$} & \multicolumn{1}{|l|}{$\mathbf{6}$} &
\\\cline{1-2}%
\end{tabular}
\ ,
\]
we get
\[
\theta(T)=%
\begin{tabular}
[c]{|l|ll}\hline
$\bar{3}$ & $\bar{2}$ & \multicolumn{1}{|l|}{$\bar{2}$}\\\hline
$\bar{2}$ & $\bar{1}$ & \multicolumn{1}{|l|}{$\bar{1}$}\\\hline
$\bar{1}$ & $\mathbf{1}$ & \multicolumn{1}{|l}{}\\\cline{1-2}\cline{2-2}%
$\mathbf{1}$ & $\mathbf{2}$ & \multicolumn{1}{|l}{}\\\cline{1-2}\cline{2-2}%
$\mathbf{1}$ & $\mathbf{2}$ & \multicolumn{1}{|l}{}\\\cline{1-2}%
$\mathbf{2}$ &  & \\\cline{1-1}%
\end{tabular}
\ ,
\]
where $w=1211322456345$ is a lattice permutation.
\end{example}

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\end{document}