text:
\section{Introduction}
The Ariki-Koike algebras have been introduced by Ariki and Koike in \cite{AK}.
They can be seen as natural generalizations of Iwahori-Hecke algebras of type
$A_{n-1}$ and $B_n$ and as  special cases of cyclotomic Hecke algebras introduced
by  Brou\'e and Malle in \cite{BM}.  Let  $R$ be a commutative ring, let $l\in{\mathbb{N}}$,
$n\in{\mathbb{N}}$ and let  $v,\ x_1,\ x_2,\ ...,\ x_{l}$ be  $l+1$ parameters in $R$.
The Ariki-Koike algebra $\mathcal{H}_{R,n}:=\mathcal{H}_{R,n}(v;x_1,...,x_{l})$
(or cyclotomic Hecke algebra  of type $G(l,1,n)$) over $R$  is the unital associative $R$-algebra presented by:
\begin{itemize}
\item generators: $T_0$, $T_1$,..., $T_{n-1}$,
\item relations: \begin{align*}
               & T_0 T_1 T_0 T_1=T_1 T_0 T_1 T_0,\\
               & T_iT_{i+1}T_i=T_{i+1}T_i T_{i+1}\ (i=1,...,n-2),\\
               & T_i T_j =T_j T_i\ (|j-i|>1),\\
                  &(T_0-x_1)(T_0-x_2)...(T_0-x_{l})  =  0,\\
                             &(T_i-v)(T_i+1)  =  0\ (i=1,...,n-1). \end{align*}
\end{itemize}
Assume that $R$ is a field of characteristic $0$. When $\mathcal{H}_{R,n}$ is a semisimple algebra,
 it is known that the  simple  $\mathcal{H}_{R,n}$-modules are given by the set of
  Specht modules $S^{ \underline{\lambda}}_{R}$ parametrized by the $l$-partitions of rank $n$.
   Using results of Dipper and Mathas \cite{DM}, the study of the
 representation theory of $\mathcal{H}_{R,n}$ in the non semisimple case can be reduced to
   the case where $R=\mathbb{C}$ and :
$$v=\eta_e,\qquad{x_i =\eta_e^{s_j},\ \ j=1,...,l,}$$
where $\displaystyle \eta_e:=\textrm{exp}{(\frac{2i\pi}{e})}\in\mathbb{C}$  and
where $s_j\in{\mathbb{Z}}$ for $j=1,2,...,l$.

In \cite{Ac}, Ariki has first provided a description of the simple modules in
this modular case. Proving and generalizing a conjecture by Lascoux, Leclerc
and Thibon, he has shown in \cite{Ad} that the associated decomposition
matrices can be identified with the Kashiwara-Lusztig canonical basis elements
of the irreducible $\mathcal{U}({\widehat{\mathfrak{sl}}_e})$-module of highest
weight $\Lambda_{s_1(\textrm{mod}\ e)}+\Lambda_{s_2 (\textrm{mod}\
e)}+...+\Lambda_{s_{l}(\textrm{mod}\ e)}$ (where the $\Lambda_i$ denote the
fundamental weights). It is known that these elements can be labeled by using
the ``crystal graph theory'' and there are several ways to do that. Each of
these ways corresponds to a certain realization of the Fock space as a module
over the quantum group $\mathcal{U}_q({\widehat{\mathfrak{sl}}_e})$.  In
particular, one of these realizations  leads to a labeling of the canonical
basis elements by the ``Kleshchev $l$-partitions'' and Ariki has given a
parametrization of the $\mathcal{H}_{\mathbb{C},n}$-simple modules by using
this class of $l$-partitions.

This paper is a continuation of the works done by Geck \cite{G}, Geck and Rouquier \cite{GR}
and  the author \cite{Jp}. The aim is to give natural parametrizations of the
simple modules for non semisimple Ariki-Koike algebras. Let ${\bf
s}_l=(s_1,s_2,...,s_l)\in{\mathbb{Z}^l}$ and let $y$ be an inderterminate. We
consider the Ariki-Koike algebra $\mathcal{H}_{A,n}$  over
$A:=\mathbb{C}[y,y^{-1}]$ with the following choice of parameters:
 \begin{eqnarray*}
&&u_j=y^{lm^{(j)}}\eta_l^{j-1} \textrm{ for}\ j=1,...,l,\\
&&v=y^l.
\end{eqnarray*}
where $\displaystyle \eta_l:=\textrm{exp}(\frac{2i\pi}{l})$ and where for
$j=1,...,l$, we have $\displaystyle
m^{(j)}=s_j-\frac{(j-1)e}{l}+\alpha e$ ($\alpha$ is a positive
integer such that $m^{(j)}\geq 0$ for $j=1,...,l$).

If we specialize $y$ to $\eta_{le}:=\textrm{exp}(\frac{2i\pi}{le})$ via
 a ring homomorphism $\theta$, we obtain the above Ariki-Koike algebra
 $\mathcal{H}_{\mathbb{C},n}:=\mathcal{H}_{\mathbb{C},n}(\eta_e;\eta_e^{s_1},...,\eta_e^{s_l})$. Let $\mathcal{H}_{\mathbb{C}(y),n}:=\mathbb{C}(y)\otimes_A \mathcal{H}_{A,n}$. Then
 $\mathcal{H}_{\mathbb{C}(y),n}$ is a split semisimple algebra and the simple
 $\mathcal{H}_{\mathbb{C}(y),n}$-modules are the Specht modules $S^{\underline{\lambda}}_{\mathbb{C}(y)}$
 defined over $\mathbb{C}(y)$.  We obtain a well-defined decomposition map $d$ between the
 Grothendieck groups of finitely generated $\mathcal{H}_{\mathbb{C}(y),n}$-modules
 and $\mathcal{H}_{\mathbb{C},n}$-modules. For $V\in{\Irr{\mathcal{H}_{\mathbb{C}(y),n}}}$, we have equations:
$$d([V])=\sum_{M\in{\Irr{\mathcal{H}_{\mathbb{C},n}}}}d_{V,M}[M].$$
Using the  symmetric algebra strucure of $\mathcal{H}_{A,n}$, we can attach an integer $a_{{\bf
s}_l}(V)$ to each simple $\mathcal{H}_{\mathbb{C}(y),n}$-module $V$. This is called the
$a$-value of $V$. Note that for Hecke algebras of type $A_{n-1}$ and $B_n$,
this value can also be defined using the Kazhdan-Lusztig theory but such a
theory  is not available
 for the wider case of Ariki-Koike algebras. The aim of this paper is to show
 the following theorem (see Theorem \ref{maing}).\\
\\
{\bf Main Theorem.} {\it For each $M\in{\Irr{\mathcal{H}_{\mathbb{C},n}}}$, there exists a
unique simple $\mathcal{H}_{\mathbb{C}(y),n}$-module $V_M$ such that the following two
conditions hold:
\begin{itemize}
\item $d_{V_M,M}=1$.
\item if there exists $W\in{\Irr{\mathcal{H}_{\mathbb{C}(y),n}}}$ such that $d_{W,M}\neq
  0$ then $a_{{\bf s}_l}(W)>a_{{\bf s}_l}(V_M)$.
\end{itemize}
 The function which sends $M$ to $V_M$ is injective. As a consequence the associated
 decomposition matrix is unitriangular
 and the following set is in natural bijection with $\Irr{\mathcal{H}_{\mathbb{C},n}}$:
$$\mathcal{B}=\{V_M\ |\ M\in{\Irr{\mathcal{H}_{\mathbb{C},n}}}\}.$$
Moreover $\mathcal{B}$ is parametrized by the crystal of the associated
$\mathcal{U}_q(\widehat{\mathfrak{sl}}_e)$-module in one of the realizations of
the Fock space mentioned above.} \vspace{0.3cm}

This result both generalizes and gives a new proof of the main result of
\cite{Jp} where the case $0\leq s_1 \leq ... \leq s_l<e$ was
considered. As the Ariki-Koike algebra $\mathcal{H}_{\mathbb{C},n}$
only depends on the classes modulo $e$ of the integers $s_j$
($j=1,...,l$), the theorem provides several natural
parametrizations for the simple modules of
$\mathcal{H}_{\mathbb{C},n}$. Note also that in the context of Hecke
algebras, the existence of such ``basic sets'' $\mathcal{B}$ is linked with the
existence of a Kazhdan-Lusztig theory (see \cite{Gs}). Hence, this result
gives indices for the existence of such a theory for Ariki-Koike algebras.

The paper is organized as follows. In section $2$ and $3$, we summarize known
results on the representation theory of Ariki-Koike algebras. We introduce
combinatoric objects and we recall works of Uglov on higher level $q$-deformed
Fock space which will be crucial for the proof of the main result. Section $4$
contains the main theorem of the paper, Theorem \ref{maing}.  The proof of this
Theorem  requires a combinatorial study of the deep results of Uglov
\cite{U}.\\
\\
{\it Acknowledgements.} I thank Bernard Leclerc and Xavier Yvonne for
precious remarks and discussions.

\section{Decomposition maps for Ariki-Koike algebras}\label{dec}

Let  $R$ be a commutative associative ring with unit and let  $v$, $x_1$,..., $x_{l}$ be $l+1$ invertible elements in $R$. Let $n\in{\mathbb{N}}$. Let  $\mathcal{H}_{R,n}:=\mathcal{H}_{R,n}{({v};{x}_1,...,{x}_{l})}$ be the associated Ariki-Koike algebra as it is defined in the introduction. For a complete survey of the representation theory  of $\mathcal{H}_{R,n}$, see \cite{Ma}.

It is known that this algebra is a ``cellular'' algebra in the sense of Graham and Lehrer \cite{GL} and thus has ``Specht modules''  which are parametrized by the $l$-partitions of rank $n$.  A $l$-partition  $\underline{\lambda}$ of rank $n$ is a sequence of $l$ partitions  $\underline{\lambda}=(\lambda^{(1)},...,\lambda^{(l)})$ such that  $\displaystyle{\sum_{k=1}^{l}{|\lambda^{(k)}|}}=n$. We denote by $\Pi^n_l$ the set of  $l$-partitions of rank $n$.

For each $l$-partition $\underline{\lambda}$ of rank $n$, we can associate a remarkable $\mathcal{H}_{R,n}$-module $S_R^{\underline{\lambda}}$ which is free over $R$. This is called  a Specht module (see the definition of ``dual'' Specht modules in \cite{DJM}). Assume that $R$ is a field. In general, the Specht modules are reducible and each of these modules can be endowed with a natural bilinear form. Let $\textrm{rad}(.)$ denotes the radical of this form. For $\underline{\lambda} \in{\Pi_l^n}$, we denote :
$$D^{\underline{\lambda}}_R:= S_R^{\underline{\lambda}}/\textrm{rad}(S_R^{\underline{\lambda}}).$$
Then the theory of cellular algebras gives the following result:
\begin{thm}(Graham-Lehrer \cite{GL}, Dipper-James-Mathas \cite{DJM}) Assume that $R$ is a field then:
\begin{enumerate}
\item Non zero $D_R^{\underline{\lambda}}$ form a complete set of non-isomorphic simple  $\mathcal{H}_{R,n}$-modules. Moreover, all of these modules are absolutely irreducible.
\item If  $\mathcal{H}_{R,n}$ is a semisimple algebra then it is split semisimple and   for all $\underline{\lambda} \in{\Pi_l^n}$ we have  $\textrm{rad}(S_R^{\underline{\lambda}})=0$. Thus, the  $S_R^{\underline{\lambda}}$ form a complete set of non isomorphic simple modules.
\end{enumerate}
\end{thm}

Using this theorem, when  $\mathcal{H}_{R,n}$    is semisimple, the simple
modules are explicitly known and are given by the Specht modules. The following
theorem gives a criterion of semisimplicty for the algebra
$\mathcal{H}_{R,n}$.

\begin{thm}\label{semi} (Ariki \cite{As})\label{semisimple}  $\mathcal{H}_{R,n}$ is split semisimple if and only if:
\begin{itemize}
\item for all $i\neq{j}$ and for all   $d\in{\mathbb{Z}}$ such that $|d|<{n}$, we have:
$$v^d x_i\neq{x_j},$$
\item $\displaystyle{\prod_{i=1}^{n}{(1+v+...+v^{i-1})}\neq{0}}.$
\end{itemize}
\end{thm}

Hence we are reduced to understand the representations of
$\mathcal{H}_{R,n}$ in the modular case.  Assume that $R$ is
the field of complex numbers $\mathbb{C}$. Then, using the results of Dipper and Mathas (\cite{DM}) and Mathas (\cite{Ms}), the non semisimple case  can be  reduced to the case  where all the $u_i$ are powers of the same root of unity $\eta_e:=\textrm{exp}(\frac{2i\pi}{e})$ with $e\geq 2$. Let  $\mathcal{H}_{\mathbb{C},n}$ be the Ariki-Koike algebra over $\mathbb{C}$ with the following choice of parameters:
\begin{eqnarray*}
&&x_j=\eta_e^{v_{j}}\ \textrm{for}\ j=1,...,l,\\
&&x=\eta_{e},
\end{eqnarray*}
 where $0\leq{v_l}\leq ...\leq v_{1}<e$. The problem of describing the simple modules of  $\mathcal{H}_{\mathbb{C},n}$  is linked with the problem of determining the decomposition map which we now define.

Let $\mathcal{H}_{A,n}$ be an Ariki-Koike defined over a commutative ring $A$ with unit. Let $K$ be the field of fractions of $A$. We assume that:
\begin{itemize}
\item[(1)] $A$ is integrally closed in $K$,
\item[(2)]  $\mathcal{H}_{K,n}:=K\otimes_A \mathcal{H}_{A,n}$ is split semisimple,
\item[(3)] we have a ring homomorphism $\theta: A\to \mathbb{C}$ such that $\mathbb{C}=\textrm{Frac}(\theta (A))$ and such that the specialized algebra $\mathbb{C}\otimes_A \mathcal{H}_{A,n}$ is the Ariki-Koike algebra  $\mathcal{H}_{\mathbb{C},n}$ with the above choice of parameters.
\end{itemize}
Then, by  \cite{Gm}, we have a well-defined decomposition map $d_{\theta}$ between the Grothendieck groups of finitely generated $\mathcal{H}_{K,n}$ and $\mathcal{H}_{\mathbb{C},n}$-modules.
In the context of cellular algebras, this function can be easily defined. Let:
$$\Phi^n_{\{e;v_1,...,v_l\}}:=\{\underline{\lambda} \in{\Pi_l^n}\ |\ D^{\underline{\lambda}}_{\mathbb{C}}\neq 0\}.$$

 Let  $R_0 (\mathcal{H}_{\mathbb{C},n})$  be  the Grothendieck group of finitely generated $\mathcal{H}_{\mathbb{C},n}$-modules. This is generated by the classes of simple $\mathcal{H}_{\mathbb{C},n}$-modules  $[D^{\underline{\lambda}}_{\mathbb{C}}]$ with $\underline{\lambda} \in{\Pi_l^n}$. Hence, for all $\underline{\lambda} \in{\Pi_l^n}$, there exist numbers $d_{\underline{\lambda},\underline{\mu}}$ with $\underline{\mu} \in{\Phi^n_{\{e;v_1,...,v_l\}}}$ such that:
$$[S^{\underline{\lambda}}_{\mathbb{C}}]=\sum_{\underline{\mu} \in{\Phi^n_{\{e;v_1,...,v_l\}}}}{d_{\underline{\lambda},\underline{\mu}}[D_{\mathbb{C}}^{\underline{\mu}}]}.$$
 The matrix $(d_{\underline{\lambda},\underline{\mu}})_{\underline{\lambda}\in{\Pi_l^n},\underline{\mu}\in{\Phi^n}}$ is called the decomposition matrix of $\mathcal{H}_{\mathbb{C},n}$ (where we denote $\Phi^n:=\Phi^n_{\{e;v_1,...,v_l\}}$). Let $\mathcal{F}^n$ be the $\mathbb{C}$-vector space which is generated by the symbols $\[S^{\underline{\lambda}}\]$ with $\underline{\lambda}\in{\Pi_l^n}$. We obtain a homomorphism:
$$\begin{array}{lll}
\mathcal{F}^n&\to&{ R_0 (\mathcal{H}_{\mathbb{C},n})}\\
 \[S^{\underline{\lambda}}\]&\mapsto&[S_{\mathbb{C}}^{\underline{\lambda}}]=\sum_{\underline{\mu}\in{\Phi^n}}{d_{\underline{\lambda},\underline{\mu}}[D_{\mathbb{C}}^{\underline{\mu}}]}.
\end{array}$$

  Now,  $\mathcal{F}^n$ can be naturally identified with the Grothendieck group of finitely generated modules over a semisimple algebra  $\mathcal{H}_{K,n}$ verifying $(1)-(3)$, by identifying the classes of simple  $\mathcal{H}_{K,n}$-modules $[S_K^{\underline{\lambda}}]$ with the symbols  $\[S^{\underline{\lambda}}\]$. Hence,  the decomposition map is defined as follows:
$$\begin{array}{llll}
d:& R_0 (\mathcal{H}_{K,n}) &\to&{ R_0 (\mathcal{H}_{\mathbb{C},n})}\\
 &[S^{\underline{\lambda}}_K]&\mapsto&[S_{\mathbb{C}}^{\underline{\lambda}}]=\sum_{\underline{\mu}\in{\Phi^n}}{d_{\underline{\lambda},\underline{\mu}}[D_{\mathbb{C}}^{\underline{\mu}}]}.
\end{array}$$

By results of Ariki and Uglov, the problem of determining the decomposition matrices for Ariki-Koike algebras can be translated to the problem of computing the canonical basis of $q$-deformed Fock spaces. In the next section, we recall these results.

\section{Canonical basis of higher level $q$-deformed Fock spaces}
The higher $q$-deformed Fock spaces have been introduced in
\cite{JM} (see also \cite{TU}). These spaces which are spanned by the set of
``multipartitions'' can be endowed with a structure of integrable
$\mathcal{U}_q (\widehat{sl}_e)$-module. In  \cite{U},  generalizing
works by  Leclerc and Thibon \cite{LT}, Uglov has given a
construction of  canonical bases for these spaces. In this part, we
review this construction following  \cite{U} (see also \cite{Us} and
\cite{Yt}). This will be used for the proof of the main result of this
paper. Then, we explain the links with the representation theory of Ariki-Koike algebras which are given by Ariki's theorem.

\subsection{$q$-wedge products and $q$-deformed Fock spaces}\label{bij}

 Let $q$ and  $z$ be indeterminates and let $l$ and $e$ be  positive integers. Let $V_e$ be an $e$-dimensional vector space over $\mathbb{Q}(q)$ with basis $v_1^{(e)}$, $v_2^{(e)}$,..., $v_e^{(e)}$.  We put $V_{e,l}:=(V_e\otimes V_l)[z,z^{-1}]$. For $a\in{\{1,...,e\}}$, $b\in{\{1,...,l\}}$ and $m\in{\mathbb{Z}}$, we put $k:=a+e(l-b)-elm$ and $u_k:=v^{(e)}_az^mv_b^{(l)}$. Then $V_{e,l}$ is a  $\mathbb{Q}(q)$ vector space with basis $\{u_k\ |\ k\in{\mathbb{Z}}\}$.

The $q$-wedge square $\bigwedge^2 V_{e,l}$ of  $V_{e,l}$ can be viewed as a $q$-deformation of the exterior square of   $V_{e,l}$. This is a  $\mathbb{Q}(q)$-vector space generated by the monomials $u_{k_1}\wedge u_{k_2}$ with $(k_1,k_2)\in{\mathbb{Z}^2}$. A basis of this space is given by the ``ordered'' monomials that is the monomials $u_{k_1}\wedge u_{k_2}$ such that $k_1> k_2$. Any monomial $u_{k_1}\wedge u_{k_2}$  can be expressed as a linear combination of ordered monomials using the following rules $(R1)$, $(R2)$, $(R3)$ and $(R4)$.

Let $k_1\leq k_2$ and for $i=1,2$, put $k_i=a_i+e(l-b_i)-elm_i$ where
 $a_i\in{\{1,...,e\}}$, $b_i\in{\{1,...,l\}}$ and $m_i\in{\mathbb{Z}}$.
 We define  $\alpha$ and $\beta$ to be the unique integers in $[0,el-1]$ such that $\alpha \equiv(a_2 -a_1 )(\textrm{mod }el) $ and $\beta\equiv(e(b_1-b_2))(\textrm{mod }el)$. Then the relations  $(R1)$, $(R2)$, $(R3)$ and $(R4)$ are given as follows:\\
\begin{itemize}
\item[(R1)] if $\alpha=0$ and  $\beta=0$: $$\displaystyle u_{k_1}\wedge u_{k_2}=-u_{k_2}\wedge u_{k_1};$$
\item[(R2)] if $\alpha\neq 0$ and $\beta=0$:
$$\begin{array}{ll}
u_{k_1}\wedge u_{k_2}=&-\displaystyle q^{-1}u_{k_2}\wedge u_{k_1}+\\
                     &\displaystyle +(q^{-2}-1) \sum_{m\geq 0}q^{-2m} u_{k_2-\alpha-elm}\wedge u_{k_1+\alpha+elm}- \\
                     & \displaystyle-(q^{-2}-1)\sum_{m\geq 1}q^{-2m+1} u_{k_2-elm}\wedge u_{k_1+elm};
\end{array}
$$
\item[(R3)] if $\alpha= 0$ and  $\beta\neq0$:
$$\begin{array}{ll}
u_{k_1}\wedge u_{k_2}=&\displaystyle q u_{k_2}\wedge u_{k_1}+\\
 &\displaystyle +(q^{2}-1)  \sum_{m\geq 0}q^{2m} u_{k_2-\beta-elm}\wedge u_{k_1+\beta+elm}+\\ &\displaystyle  +(q^2 -1)\sum_{m\geq 1}q^{2m-1} u_{k_2-elm}\wedge u_{k_1+elm};\end{array}$$

\item[(R4)]  if $\alpha\neq 0$ and  $\beta\neq 0$:
$$\begin{array}{ll}
  u_{k_1}\wedge u_{k_2}=&u_{k_2}\wedge u_{k_1}+\\
&+\displaystyle{(q-q^{-1})\sum_{m\geq{0}} \frac{q^{2m+1}+q^{-2m-1}}{q+q^{-1}}  u_{k_2-\beta-elm}\wedge u_{k_1+\beta+elm}}+\\
&+\displaystyle{(q-q^{-1})\sum_{m\geq{0}} \frac{q^{2m+1}+q^{-2m-1}}{q+q^{-1}}  u_{k_2-\alpha-elm}\wedge u_{k_1+\alpha+elm}}+\\
&+\displaystyle{(q-q^{-1})\sum_{m\geq{0}} \frac{q^{2m}-q^{-2m}}{q+q^{-1}}  u_{k_2-\beta-\alpha-elm}\wedge u_{k_1+\beta+\alpha+elm}+}\\
&+\displaystyle{(q-q^{-1})\sum_{m\geq{1}} \frac{q^{2m}-q^{-2m}}{q+q^{-1}}  u_{k_2-elm}\wedge u_{k_1+elm}};\end{array}$$
\end{itemize}
where the summations continue as long as the monomials appearing under
the sums are ordered.

For any integer $r\geq 2$, we can now define the $r$-fold $q$-wedge product
$\bigwedge^r V_{e,l}$. This is the  $\mathbb{Q}(q)$-vector space generated by
the elements $u_{k_1}\wedge u_{k_2}\wedge ... \wedge u_{k_r}$ with
$k_i\in{\mathbb{Z}}$. Again the ordered monomials, that is the monomials
$u_{k_1}\wedge u_{k_2}\wedge ... \wedge u_{k_r}$  with $k_1>k_2>...>k_r$, form
a basis of $\bigwedge^r V_{e,l}$. Moreover, an arbitrary monomial  can be
expressed  as a linear combination of ordered monomials using the relations
$(R1)$,  $(R2)$,  $(R3)$ and $(R4)$ in every adjacent pair of the factors.

Finally, for  $s\in{\mathbb{Z}}$, the semi-infinite $q$-wedge product $\bigwedge^{s+\frac{\infty}{2}} V_{e,l}$ of charge $s$ is the inductive limit of $\bigwedge^r V_{e,l}$ where the maps $\bigwedge^r V_{e,l}\to\bigwedge^{r+1} V_{e,l}$ are given by $v\mapsto v\wedge u_{s-r}$. Hence,  $\bigwedge^{s+\frac{\infty}{2}} V_{e,l}$ is the  $\mathbb{Q}(q)$-vector space generated by the semi-infinite monomials  $u_{k_1}\wedge u_{k_2}\wedge ...$ with  $k_i\in{\mathbb{Z}}$ and such that $k_i=s-i+1$ for $i>>0$. A basis is given by the ordered monomials  that is the monomials $u_{k_1}\wedge u_{k_2}\wedge ...$ with $k_1>k_2>....$ and the relations  $(R1)$-$(R4)$ provide a way to express an arbitrary monomial as a linear combinaison of ordered monomials.

Now, each semi-infinite ordered monomial can be labeled by a pair
$(\underline{\lambda},{\bf s}_l)$ where $\underline{\lambda}\in{\Pi_l^n}$ is a
$l$-partition of rank $n$ and ${\bf s}_l=(s_1,...,s_l)\in{\mathbb{Z}}^l$ is
such that $\sum_{i=1}^l s_i =s$. This labeling coincides with that of \cite{U}
up to transformation $(s_1,...,s_l)\mapsto (s_l,...,s_1)$ and
$(\lambda^{(1)},...,\lambda^{(l)})\mapsto (\lambda^{(l)},...,\lambda^{(1)})$.
Let  $u_{k_1}\wedge u_{k_2}\wedge ...\in{ \bigwedge^{s+\frac{\infty}{2}}
V_{e,l}}$ be a  semi-infinite ordered monomial. For $i=1,2,...$, we put
$k_i=a_i+e(l-b_i)-elm_i$ where   $a_i\in{\{1,...,e\}}$, $b_i\in{\{1,...,l\}}$
and $m_i\in{\mathbb{Z}}$. For $b=1,....,l$, let $k_1^{(b)}>k_2^{(b)}>...$ be
the semi-infinite sequence obtained by ordering the elements of the set
$\{a_i-em_i\ |\ b_i=b\}$ in strictly decreasing order. Then, there is a unique
$s_b\in{\mathbb{Z}}$ such that $k_i^{(b)}=s_b-i+1$ for $i>>1$. We put
$\underline{\lambda}=(\lambda^{(1)},...,\lambda^{(l)})$ where for $i>0$,
$\lambda^{(b)}_i=k_i^{(b)}-s_b+i-1$ and ${\bf s}_l=(s_1,...,s_l)$. This defines
a bijection between the set of semi-infinite ordered monomials and the set of
pairs  $(\underline{\lambda},{\bf s}_l)$ where
$\underline{\lambda}\in{\Pi_l^n}$ and ${\bf
s}_l=(s_1,...,s_l)\in{\mathbb{Z}}^l$ is such that $\sum_{i=1}^l s_i =s$.

 This bijection  can be described as follows (see \cite{Y} for much
 details).  First, the infinite decreasing sequence $\underline{k}=(k_1,k_2,...)$
 can be pictured as a set of black  beads on an infinite runner. For
 example, the following sequence :
 $$\underline{k}=(15,12,8,7,3,1,-2,-4,-5,-6,...)$$
  is represented by the following abacus:

\begin{center}
\leavevmode
\epsfxsize= 12cm
\epsffile{ess.eps}
\end{center}

On the other hand,  using the decomposition $k_i=a_i+e(l-b_i)-elm_i$, the same sequence can be represented by an $l$-abacus, that is, as a set of black  beads on $l$ infinite runners. For $e=4$ and $l=3$, we obtain:

\begin{center}
\leavevmode
\epsfxsize= 12cm
\epsffile{ess2.eps}
\end{center}

In the above representation, the beads are labeled by the integers $k_i$ (with $i=1,2,...$). We can alternatively labeled them by the integers $k_i^{(b)}$ (with $i=1,2,...$ and $b=1,2,...,l$) as follows:

\begin{center}
\leavevmode
\epsfxsize= 12cm
\epsffile{ess3.eps}
\end{center}

Let  $\underline{\lambda}$ be the $l$-partition  and ${\bf s}_l=(s_1,...,s_l)$ be the $l$-tuple of integers  associated to $\underline{k}$. For $i=1,2,...$ and $b=1,2,...,l$, we have $\lambda^{(b)}_i=k_i^{(b)}-s_b+i-1$. Thus, one can now easily determine  $\underline{\lambda}$   by counting the number of white beads at the left of each black bead on each runner. Continuing the above example, we obtain $\underline{\lambda}=((6,1),(2,2),(4,1))$ and ${\bf s}_l=(-2,2,3)$.

Now, the higher level $q$-deformed Fock space of charge ${\bf s}_l $ is defined to be the $\mathbb{C}(q)$-vector space generated by the symbols $|\underline{\lambda},{\bf s}_l \rangle$ with $\underline{\lambda}\in{\Pi_l^n}$ :
$$\mathcal{F}_{e,{\bf s}_l}:=\bigoplus_{\overset{\underline{\lambda}\in{\Pi_l^n}}{n\in{\mathbb{N}}}} \mathbb{C}(q)|\underline{\lambda},{\bf s}_l\rangle.$$
Thus, if we identify the semi-infinite ordered monomial $u_{\underline{k}}\in{\bigwedge^{s+\frac{\infty}{2}} V_{e,l}}$  with the pair  $|\underline{\lambda},{\bf s}_l\rangle$ defined by the above bijection, we get:
$$\bigwedge^{s+\frac{\infty}{2}} V_{e,l}=\bigoplus_{s_1+...+s_l=s}\mathcal{F}_{e,{\bf s}_l}.$$

\subsection{Canonical basis and Ariki's theorem}\label{ariki}

Let $\mathcal{U}_q({\widehat{\mathfrak{sl}}}_e)$ be the quantum group
 associated to  the Lie algebra ${\widehat{\mathfrak{sl}}}_e$. We
 denote by $e_i$, $f_i$ and $k_i$ with $i=0,...,e-1$, the Chevalley
 generators and by  $\mathcal{U}_q'({\widehat{\mathfrak{sl}}}_e)$ the
 subalgebra of $\mathcal{U}_q({\widehat{\mathfrak{sl}}}_e)$ generated
 by $e_i$, $f_i$, $k_i$, $k_i^{-1}$. We can construct a structure of
 $\mathcal{U}_q'({\widehat{\mathfrak{sl}}}_e)$-module on
 $\bigwedge^{s+\frac{\infty}{2}}V_{e,l}$ (see \cite[\S 2.1,\S 3.5, \S 4.2]{U})
such that each of the subspace $\mathcal{F}_{e,{\bf s}_l}$ is stable with respect to this action.
Before describing the action of the  $\mathcal{U}_q'({\widehat{\mathfrak{sl}}}_e)$-module $\mathcal{F}_{e,{\bf s}_l}$,  we need some combinatorial definitions.

  Let   $\underline\lambda={(\lambda^{(1)} ,...,\lambda^{(l)})}$ be a $l$-partition of rank  $n$. The diagram of  $\underline{\lambda}$ is the following set:
$$[\underline{\lambda}]=\left\{ (a,b,c)\ |\ 1\leq{c}\leq{l},\ 1\leq{b}\leq{\lambda_a^{(c)}}\right\}.$$
The elements of this diagram are called   the nodes of  $\underline{\lambda}$.
Let  $\gamma=(a,b,c)$ be a node of  $\underline{\lambda}$. The residue
of  $\gamma$  associated to the set  $\{e;{s_1},...,{s_{l}}\}$ is the
unique element of $[0,e[$ such that:
$$\textrm{res}{(\gamma)}\equiv (b-a+s_{c})(\textrm{mod}\ e).$$
If $\gamma $ is a node with residue $i$, we say that  $\gamma$ is an  $i$-node.
Let  $\underline{\lambda}$ and  $\underline{\mu}$ be two $l$-partitions of rank  $n$ and $n+1$ such that  $[\underline{\lambda}]\subset{[\underline{\mu}]}$. There exists a node $\gamma$ such that  $[\underline{\mu}]=[\underline{\lambda}]\cup{\{\gamma\}}$. Then, we denote $[\underline{\mu}]/[\underline{\lambda}]=\gamma$ and  if $\textrm{res}{(\gamma)}=i$, we say that  $\gamma$ is an addable $i$-node  for   $\underline{\lambda}$ and a removable  $i$-node for  $\underline{\mu}$. Now, we introduce an order on the set of nodes of a $l$-partition.
 We say that  $\gamma=(a,b,c)$ is {\it above}  $\gamma'=(a',b',c')$ if:
$$b-a+s_c<b'-a'+s_{c'}\ \textrm{or } \textrm{if}\ b-a+s_c=b'-a'+s_{c'}\textrm{ and }c'<c.$$
Let $\underline{\lambda}$ and $\underline{\mu}$ be two $l$-partitions of rank $n$ and $n+1$ such that  there exists an $i$-node $\gamma$ such that  $[\underline{\mu}]=[\underline{\lambda}]\cup{\{\gamma\}}$. We define the following numbers:

\begin{align*}
{N}_i^{a}{(\underline{\lambda},\underline{\mu})}=&   \sharp\{ \textrm{addable }\ i-\textrm{nodes of } \underline{\lambda}\ \textrm{ above } \gamma\} \\
 & -\sharp\{ \textrm{removable }\ i-\textrm{nodes of } \underline{\mu}\ \textrm{ above } \gamma\},\\
{N}_i^{b}{(\underline{\lambda},\underline{\mu})}= &   \sharp\{ \textrm{addable } i-\textrm{nodes of } \underline{\lambda}\ \textrm{ below } \gamma\}\\
&   -\sharp\{ \textrm{removable } i-\textrm{nodes of } \underline{\mu}\ \textrm{ below } \gamma\},\\
{N}_{i}{(\underline{\lambda})} =& \sharp\{ \textrm{addable } i-\textrm{nodes of } \underline{\lambda}\}\\
& -\sharp\{ \textrm{removable } i-\textrm{nodes of } \underline{\lambda}\}.
\end{align*}
Now, $\mathcal{F}_{e,{\bf s}_l}$ is an integrable $\mathcal{U}_q'(\widehat{\mathfrak{sl}_e})$-module with action:
\begin{align*}
e_{i}|\underline{\lambda},{\bf s}_l\rangle &=\sum_{\res{[\underline{\lambda}]/[\underline{\mu}]}\equiv i}{q^{-{N}_i^{a}{(\underline{\mu},\underline{\lambda})}}}|\underline{\mu},{\bf s}_l\rangle,\\
f_{i}|\underline{\lambda},{\bf s}_l\rangle &=\sum_{\res{[\underline{\mu}]/[\underline{\lambda}]}\equiv i}{q^{{N}_i^{b}{(\underline{\lambda},\underline{\mu})}}}|\underline{\mu},{\bf s}_l\rangle,\\
k_{i}|\underline{\lambda},{\bf s}_l\rangle&=q^{{N}_i{(\underline{\lambda})}}|\underline{\lambda},{\bf s}_l\rangle.
\end{align*}
where $0\leq{i}\leq{e-1}$.
Note that this action coincides with that of \cite{U} up to transformation $(s_1,...,s_l)\mapsto (s_l,...,s_1)$.

We now introduce another basis for $\mathcal{F}_{e,{\bf s}_l}$ namely  the Leclerc-Thibon canonical basis. This basis  is defined by using an involution on the wedge space  $\bigwedge^{s+\frac{\infty}{2}} V_{e,l}$ which has been introduced in \cite{LT} for $l=1$ and generalized to any $l$ in \cite{U}.
 Let $u_{\underline{k}}:=u_{k_1}\wedge u_{k_2}\wedge ...$ be a semi infinite monomial (ordered or not). For $i=1,2,...$, we put
 $k_i=a_i+e(l-b_i)-elm_i$ where $a_i\in{\{1,...,e\}}$,
 $b_i\in{\{1,...,l\}}$ and $m_i\in{\mathbb{Z}}$. For $r\in{\mathbb{N}}$, we put:
$$\omega(u_{\underline{k}})=\sharp \{i<j\ |\ a_i=a_j\},$$
$$\omega'(u_{\underline{k}})=\sharp \{i<j\ |\ b_i=b_j\}.$$
Then, for $r\geq \sum_{i=1}^{\infty} k_i-(s-i+1)$ we set:
$$\overline{u_{\underline{k}}}:=(-q)^{\omega'(u_{\underline{k}})}q^{-\omega(u_{\underline{k}})} u_{k_r}\wedge u_{k_{r-1}}\wedge .... \wedge u_{k_1}\wedge u_{k_{r+1}}\wedge u_{k_{r+2}}\wedge ...$$
One can prove that  this monomial is independent of $r$ and that
$u\mapsto{\overline{u}}$ defines an involution on
$\bigwedge^{s+\frac{\infty}{2}} V_{e,l}$. The canonical basis is now defined as
follows:
\begin{thm}(Leclerc-Thibon \cite[Theorem  4.1]{LT}, Uglov \cite[\S 4.4]{U})\label{cb} Let $s\in{\mathbb{Z}}$. There exists a unique basis:
 $$\{G(\underline{\lambda},{\bf s}_l)\ |\ \sum_{i=1}^l s_i=s,\ \underline{\lambda}\in{\Pi_l^n},\ n\in{\mathbb{N}}\}$$
of $\bigwedge^{s+\frac{\infty}{2}} V_{e,l}$ such that:
\begin{itemize}
\item $\overline{ G(\underline{\lambda},{\bf s}_l)}=G(\underline{\lambda},{\bf s}_l)$,
\item $G(\underline{\lambda},{\bf s}_l) - |\underline{\lambda},{\bf s}_l\rangle\in \bigoplus_{\underline{\mu}} q\mathbb{C}[q] |\underline{\mu},{\bf s}_l\rangle   $.
\end{itemize}
This is called the canonical basis of $\bigwedge^{s+\frac{\infty}{2}} V_{e,l}$.
 \end{thm}
In particular the set $\{G(\underline{\lambda},{\bf s}_l)\ |\ \underline{\lambda}\in{\Pi_l^n},\ n\in{\mathbb{N}}\}$ gives a basis
 of the Fock space $\mathcal{F}_{e,{\bf s}_l}$. Now, we consider the subspace $M_{{\bf s}_l}:=\mathcal{U}_q'(\widehat{\mathfrak{sl}_e}).|\underline{\emptyset},{\bf s}_l\rangle$.
It is well known that this is isomorphic to the irreducible $\mathcal{U}_q'(\widehat{\mathfrak{sl}_e})$-module $V(\Lambda)$ with highest weight
$\Lambda:=\Lambda_{s_{1}(\textrm{mod}\ e)}+\Lambda_{s_{2}(\textrm{mod}\ e)}+...+\Lambda_{s_{l}(\textrm{mod}\ e)}$.
Note that if ${\bf s}_l'=(s_1',...,s_l')\in{\mathbb{Z}^l}$ is such that $s_i\equiv s_i'(\textrm{mod }e)$, then the modules
 $M_{{\bf s}_l}$ and $M_{{\bf s}_l'}$ are isomorphic (but the action of  $\mathcal{U}_q'(\widehat{\mathfrak{sl}_e})$ on the elements
 of the standard basis $|\underline{\lambda},{\bf s}_l\rangle$ and  $|\underline{\lambda},{\bf s}_l'\rangle$ are different in general).

A basis of $M_{{\bf s}_l}$ can be given by using the canonical basis of $\mathcal{F}_{e,{\bf s}_l}$
 and by studying the associated crystal graph. This graph can be described combinatorially as follows.

Let   $\underline{\lambda}$ be a   $l$-partition and  let $\gamma$ be  an $i$-node of $\underline{\lambda}$,
 we say that  $\gamma$ is a normal $i$-node of  $\underline{\lambda}$ if,
whenever $\eta$ is an $i$-node of $\underline{\lambda}$   below  $\gamma$,
there are more removable $i$-nodes between $\eta$ and $\gamma$ than addable $i$-nodes between  $\eta$ and $\gamma$.
If $\gamma$ is the highest normal $i$-node of    $\underline{\lambda}$, we say that  $\gamma$ is a good $i$-node. Note that this notion depends on the choice of ${\bf s}_l$.

Then, the crystal graph of $\mathcal{F}_{e,{\bf s}_l}$ is given by:
\begin{itemize}
\item vertices: the $l$-partitions,
\item edges: $\displaystyle{{\underline{\lambda}\overset{i}{\rightarrow}{{\underline{\mu}}}}}$ if and only if $[\underline{\mu}]/[\underline{\lambda}]$ is a good  $i$-node.
\end{itemize}

By using properties of crystal bases, we can obtain the crystal graph of
$M_{{\bf s}_l}$: this is the connected components of that of
$\mathcal{F}_{e,{\bf s}_l}$ which contain the vacuum vector
$|\underline{\emptyset},{\bf s}_l\rangle$. The vertices of this graph are given
by the following class of $l$-partitions.

\begin{df} Let ${\bf s}_l\in{\mathbb{Z}^n}$.  The set of Uglov $l$-partitions $\Lambda^n_{e;{\bf s}_l}$ is defined   recursively  as follows.
\begin{itemize}
   \item We have $\underline{\emptyset}:=(\emptyset,\emptyset,...,\emptyset)\in{  \Lambda^n_{e;{\bf s}_l} }$.
    \item If $\underline{\lambda}\in\Lambda^n_{e;{\bf s}_l}$ , there exist $i\in{\{0,...,e-1\}}$ and a good $i$-node $\gamma$ such that if we remove $\gamma$ from  $\underline{\lambda}$, the resulting  $l$-partition is in $\Lambda^n_{e;{\bf s}_l}$.
\end{itemize}
\end{df}

\begin{re}
Assume that  ${\bf s}_l\in{\mathbb{Z}^n}$ is such that $0\leq s_1 \leq
s_2\leq ...\leq s_{l}<e$ then it is shown in \cite[Proposition 2.11]{FL} that the set of Uglov $l$-partitions are the  $l$-partitions  $\underline\lambda={(\lambda^{(1)} ,...,\lambda^{(l)})}$ such that:
\begin{enumerate}
\item for all $1\leq{j}\leq{l-1}$ and $i=1,2,...$, we have:
\begin{align*}
&\lambda_i^{(j)}\geq{\lambda^{(j+1)}_{i+s_{j+1}-s_j}},\\
&\lambda^{(l)}_i\geq{\lambda^{(1)}_{i+e+s_1-s_{l}}};
\end{align*}

\item  for all  $k>0$, among the residues appearing at the right ends of the length $k$ rows of   $\underline\lambda$, at least one element of  $\{0,1,...,e-1\}$ does not occur.
\end{enumerate}
 Such $l$-partitions are called FLOTW $l$-partitions in \cite{Jp}.

Assume that   ${\bf s}_l\in{\mathbb{Z}^n}$ is such that $s_1>> s_2>>
...>> s_{l}$, then the set  Uglov $l$-partitions $\Lambda^n_{e;{\bf
    s}_l}$ coincides with the set of  Kleshchev $l$-partitions by
\cite[\S 3]{Ac}.
\end{re}
Now, the canonical basis of $M_{{\bf s}_l}$ is the following set:
$$\{G(\underline{\lambda},{\bf s}_l)\ |\ \underline{\lambda}\in{\Lambda^{n}_{e;{\bf s}_l}},\ n\in{\mathbb{N}}\}.$$

The following theorem gives a link between the canonical basis elements of  $M_{{\bf s}_l}$ and the decomposition matrices of Ariki-Koike algebras.

\begin{thm}\label{Ar} (Ariki, see \cite[Theorem 4.49]{Ab})
  Let  $\mathcal{H}_{\mathbb{C},n}$ be the Ariki-Koike algebra over $\mathbb{C}$ with the following choice of parameters:
\begin{eqnarray*}
&&x_j=\eta_e^{v_{j}}\ \textrm{for}\ j=1,...,l,\\
&&v=\eta_{e},
\end{eqnarray*}
where $0\leq v_1\leq ...\leq v_l<e$. Let $\Phi^n:=\Phi^n_{\{e;v_l,...,v_l\}}$ and   $(d_{\underline{\lambda},\underline{\mu}})_{\underline{\lambda}\in{\Pi_l^n},\underline{\mu}\in{\Phi^n}}$ be the associated decomposition matrix.

Let  ${\bf s}_l=(s_1,...,s_l)\in{\mathbb{Z}^l}$ be such that $v_j\equiv s_j(\textrm{mod}\ e)$ for $j=1,...,l$. Then, for each $\underline{\nu}\in{\Lambda^n_{ e;{\bf s}_l   }}$, write
$$G(\underline{\nu},{\bf s}_l)=\sum_{\underline{\lambda}\in{\Pi^n_l}}d_{\underline{\lambda},\underline{\nu}}(q)|\underline{\lambda},{\bf s}_l\rangle,$$
 where $d_{\underline{\lambda},\underline{\nu}}(q)\in{\mathbb{C}[q]}$.
 Then there exists a  bijection $\kappa:{\Phi^n}\to \Lambda^n_{e;{\bf s}_l}$ such that for all $\underline{\lambda}\in{\Pi_l^n}$ and  $\underline{\mu}\in{\Phi^n}$, we have:
$$d_{\underline{\lambda},\underline{\mu}}=d_{\underline{\lambda},\kappa(\underline{\mu})}(1).$$
\end{thm}

In other words, Ariki's theorem asserts that the columns of the decomposition
matrix of  $\mathcal{H}_{\mathbb{C},n}$ with the above choice of parameters
coincides with the canonical basis elements of  $M_{{\bf s}_l}$ evaluated at
$q=1$ whenever $v_j\equiv s_j(\textrm{mod}\ e)$ for $j=1,...,l$. Note that this
decomposition matrix only depends on $\{e;v_1,...,v_l\}$ whereas  the canonical
basis elements of  $M_{{\bf s}_l}$ depends on $\{e;s_1,...,s_l\}$. In
particular, if $s_j\equiv s_j'(\textrm{mod}\ e)$ for  $j=1,...,l$,  the
labelings of the canonical basis elements  of  $M_{{\bf s}_l}$ and $M_{{\bf
s}_l'}$ by   $\Lambda^n_{e; {\bf s}_l}$ and $\Lambda^n_{e; {\bf s}_l'}$ are
different in general as we can see in the following example.
\begin{ex}\label{exem}
Assume that $l=2$, $e=4$, $v_1=0$, $v_2=1$. Then different values for ${\bf
s}_l$ lead to different labelings of the same crystal graph which are
given in Example \ref{exfin}.
\end{ex}

 In \cite{Ac}, Ariki has given an explicit description of $\Phi^n_{\{e;v_1,...,v_l\}}$ by showing that this set coincides with the set   $\Lambda^n_{ e;{\bf s}_l}$ with  $ s_1>> s_2>> ...>> s_{l}$ and  $v_j\equiv s_j(\textrm{mod}\ e)$ for $j=1,...,l$. In \cite{Jp}, another parametrization of the simple   $\mathcal{H}_{\mathbb{C},n}$-modules has been given by using the set  $\Lambda^n_{e; {\bf s}_l}$ with ${\bf s}_l=(v_1,...,v_l)$ (namely the set of FLOTW $l$-partitions) and an ordering of the rows of the decomposition matrices by Lusztig $a$-values. In the next section, we will show that each of the sets  $\Lambda^n_{ e;{\bf s}_l}$ with    $v_j\equiv s_j(\textrm{mod}\ e)$ has a natural interpretation in  the representation theory of  $\mathcal{H}_{\mathbb{C},n}$.

\section{Unitriangularity of the decomposition matrices of Ariki-Koike algebras}
 \subsection{Specialisations and Lusztig $a$-values}
Let $e$ be an integer and let ${\bf s}_l=(s_1,s_2,...,s_{l})\in{\mathbb{Z}^l}$ be a sequence of integers.  The aim of this part is to study the Ariki-Koike algebra $\mathcal{H}_{\mathbb{C},n}$ with the following choice of parameters:
\begin{eqnarray*}
&&u_j=\eta_e^{s_j(\textrm{mod }e)}\ \textrm{for}\ j=1,...,l,\\
&&v=\eta_e.
\end{eqnarray*}
For $j=1,...,l$, we define rational numbers:
$$m^{(j)}=s_j-\frac{(j-1)e}{l}+\alpha e,$$
where $\alpha$ is a positive  integer such that $m^{(j)}\geq{0}$  for $j=1,...,l$. Let $y$ be an indeterminate and put $A:=\mathbb{C}[y,y^{-1}]$. We consider the Ariki-Koike algebra $\mathcal{H}^{{\bf s}_l}_{A,n}$ over $A$   with the following parameters:
\begin{eqnarray*}
&&u_j=y^{lm^{(j)}}\eta_l^{j-1}\ \textrm{for}\ j=1,...,l,\\
&&v=y^l
\end{eqnarray*}
 (this algebra is  denoted by
$\mathcal{H}_{A,n}$ in the introduction but here, we use the
notation  $\mathcal{H}_{A,n}^{{\bf s}_l}$ to insist on the
dependance on ${\bf s}_l$).

By Theorem \ref{semi},   $\mathcal{H}_{\mathbb{C}(y),n}^{{\bf s}_l}:=\mathbb{C}(y)\otimes_A \mathcal{H}_{A,n}^{{\bf s}_l}$ is split semisimple.
Moreover, if we specialize the parameter $y$ to ${\eta_{le}:=\textrm{exp}(\frac{2i\pi}{le})}$,
we obtain the above Ariki-Koike algebra  $\mathcal{H}_{\mathbb{C},n}$. Hence, we have a well-defined
decomposition map between $R_0 (\mathcal{H}_{\mathbb{C}(y),n}^{{\bf s}_l})$  and  $R_0 (\mathcal{H}_{\mathbb{C},n})$.

We now associate to each simple $\mathcal{H}_{\mathbb{C}(y),n}^{{\bf s}_l}$-module $S^{\underline{\lambda}}_{\mathbb{C}(y)}$ an $a$-value following \cite{Jp}. Put ${\bf m}_l=(m^{(1)},...,m^{(l)})$ where the $m^{(j)}$ are defined above. We need to define the
notion of ``${\bf m}_l$-translated symbols'' associated to $l$-compositions.

Let $n\in{\mathbb{N}}$ and $l\in{\mathbb{N}}$. An $l$-composition $\underline{\lambda}$ of rank $n$  is an $l$-tuple $(\lambda^{(1)},....,\lambda^{(l)})$ where :
\begin{itemize}
\item for all $i=1,...,l$, we have $\lambda^{(i)}=(\lambda^{(i)}_1,...,\lambda^{(i)}_{h^{(i)}})$ for  $h^{(i)}\in{\mathbb{N}}$ and $\lambda^{(i)}_j\in{\mathbb{N}_{>0}}$ ($j=1,...,h^{(i)}$), $h^{(i)}$ is called the height  of $\lambda^{(i)}$,
\item $\displaystyle{\sum_{i=1}^{l}\sum_{j=1}^{h^{(i)}}\lambda^{(i)}_j=n}$.

\end{itemize}
 Let $\underline{\lambda}=(\lambda^{(1)},...,\lambda^{(l)})$ be a $l$-composition and let $h^{(i)}$ be the heights of the compositions $\lambda^{(i)}$. Then the height of $\underline{\lambda}$ is the following positive integer:
$$h_{\underline{\lambda}}:=\textrm{max}{\{h^{(1)},...,h^{(l)}\}}.$$
Let $k$ be a positive integer. The translated symbol $\textbf{B}[{\bf m}_l]'$ associated to ${\bf m}_l$, $k$ and $\underline{\lambda}$  is:
$$\textbf{B}[{\bf m}_l]':=(B'^{(1)},...,B'^{(l)}),$$
where $B'^{(i)}$, for $i=1,...,l$, is given by:
$$B'^{(i)}:=(B'^{(i)}_1,...,B'^{(i)}_{h_{\underline{\lambda}}+k}),$$
in which:
$$B'^{(i)}_{j}:=\lambda^{(i)}_j-j+h_{\underline{\lambda}}+k+m^{(i)}\  \  (1\leq j \leq h_{\underline{\lambda}}+k).$$
The integer  $h_{\underline{\lambda}}+k$ is called the height of $\textbf{B}[{\bf m}_l]'$.

Now, the $a$-value on the $\mathcal{H}_{\mathbb{C}(y),n}^{{\bf
    s}_l}$-module $S^{\underline{\lambda}}_{\mathbb{C}(y)}$ is defined
to be the lower degree of the associated the Schur element. These
    elements have been explicitely comptuted by
Geck, Iancu and Malle in \cite{GIM} (see \cite[Proposition 3.1]{Jp}).
We obtain the following definition:

\begin{df} Let  $\underline{\lambda}$ be a $l$-partition of rank $n$  and let  $S_{\mathbb{C}(y)}^{\underline{\lambda}}$ be the simple $\mathcal{H}^{{\bf s}_l}_{\mathbb{C}(y),n}$-module.
Let  $k$ be a positive integer and $\textbf{B}[{\bf m}_l]'$ be the
${\bf m}_l$-translated symbol associated to ${\bf m}_l$ and $k$. Then, if $h$ is the height of  $\textbf{B}[{\bf m}_l]'$, the $a$-value of  $S_{\mathbb{C}(y)}^{\underline{\lambda}}$ is the following rational number:
$$a_{{\bf s}_l}(\underline{\lambda})=f(n,h,{\bf m}_l)+\sum_{{1\leq{i}\leq{j}<l+1}\atop{{(a,b)\in{B'^{(i)}\times{B'^{(j)}}}}\atop{a>b\ \textrm{if}\ i=j}}}{\min{\{a,b\}}}   -\sum_{{1\leq{i,j}<l+1}\atop{{a\in{B'^{(i)}}}\atop{1\leq{k}\leq{a}}}}{\min{\{k,m^{(j)}\}}},$$
where $f(n,h,{\bf m}_l)$ is a rational number which only depends on
the parameters $\{e;s_1,...,s_{l}\}$, on $h$ and on $n$ (the
expression of $f$ is given in \cite[Proposition 3.2]{Jp}).
\end{df}
We need to introduce the following preorder on the set of $l$-compositions.\begin{df}
Let  $\underline{\mu}$ and  $\underline{\nu}$ be  $l$-compositions
of rank  $n$. Let $k$ and $k'$ be such that $h_{\underline{\mu}}+k=h_{\underline{\nu}}+k'$. Let     $\textbf{B}_{ \underline{\mu}}[{\bf m}_l]'$,
(resp.  $\textbf{B}_{ \underline{\nu}}[{\bf m}_l]'$) be the ${\bf
  m}_l$-translated symbol associated  to $\underline{\mu}$, ${\bf m}_l$ and $k$ (resp.   $\underline{\mu}$, ${\bf m}_l$ and $k'$). Then we write:
$$\underline{\mu} \prec{\underline{\nu}},$$
if:
$$\sum_{{1\leq{i}\leq{j}<l+1}\atop{{(a,b)\in{B_{\underline{\mu}}'^{(i)}
\times{B_{\underline{\mu}}'^{(j)}}}}\atop{a>b\ \textrm{if}\ i=j}}}
{\min{\{a,b\}}}   -\sum_{{1\leq{i,j}<l+1}
\atop{{a\in{B_{\underline{\mu}}'^{(i)}}}
\atop{1\leq{k}\leq{a}}}}{\min{\{k,m^{(j)}\}}}<$$
$$\sum_{{1\leq{i}\leq{j}<l+1}\atop{{(a,b)\in{B_{\underline{\nu}}'^{(i)}
\times{B_{\underline{\nu}}'^{(j)}}}}\atop{a>b\ \textrm{if}\ i=j}}}
{\min{\{a,b\}}}   -\sum_{{1\leq{i,j}<l+1}\atop{{a\in{B_{\underline{\nu}}
'^{(i)}}}\atop{1\leq{k}\leq{a}}}}{\min{\{k,m^{(j)}\}}}.$$
In particular, if  $\underline{\mu}$ and   $\underline{\nu}$ are
 $l$-partitions, as the translated symbol of $\underline{\mu}$ and
$\underline{\nu}$ have the same height $h$, we have:
$$\underline{\mu} \prec{\underline{\nu}}\iff{a_{{\bf s}_l}(\underline{\mu})
<a_{{\bf s}_l}(\underline{\nu})}.$$
\end{df}

 Before beginning the proof of the main result, we need to give  the
 following useful proposition which has been shown in \cite[Lemma 4.4]{Jp} in the case  $0\leq s_1 \leq ....\leq s_l <e$. One can easily check that the proof holds in the general case.

\begin{prop}\label{combi} Let  $\underline{\lambda}$ be a
 $l$-composition of rank $n$, let  $\textbf{B}:=(B^{(1)},...,B^{(l)})$
be an ordinary symbol of $\underline{\lambda}$,  let  $\beta_1$
and $\beta_2$ be two elements of   $\textbf{B}[{\bf m}_l]'$, we assume
 that:
$$\beta_1 < \beta_2.$$
 Let  $r\in{\mathbb{N}}$. We add  $r$ nodes  to $\underline{\lambda}$
 on the part associated to    $\beta_1$. Let  $\underline{\mu}$
 be the resulting $l$-composition of rank $n+r$.  We add  $r$ nodes
 to $\underline{\lambda}$  on the part associated to    $\beta_2$.
 Let  $\underline{\nu}$ be the resulting $l$-composition of rank
$n+r$.   Then, we have:
$$\underline{\nu} \prec{\underline{\mu}}.$$
In particular, if   $\underline{\mu}$ and   $\underline{\nu}$
are some  $l$-partitions, as the translated symbols of
$\underline{\mu}$ and   $\underline{\nu}$ have the same height, we have:
$$a_{{\bf s}_l}(\underline{\mu})>a_{{\bf s}_l}(\underline{\nu}).$$
\end{prop}
Assume that  $\underline{\lambda}$ is a  $l$-composition of rank $n$.

\begin{center}
\leavevmode
\epsfxsize= 12cm
\epsffile{parti5.eps}
\end{center}

Assume  that   $\underline{\mu}$ and   $\underline{\mu}'$ are two
$l$-compositions which are obtained from  $\underline{\lambda}$ by
adding $r$ nodes on the parts  $\lambda^{(N)}_i$ and
$\lambda^{(N')}_j$ respectively where $1\leq N, N'\leq l$. Assume in addition that we have:
$$\lambda^{(N)}_i-i+s_N-\frac{(N-1)e}{l}>\lambda^{(N')}_j-j+s_{N'}-\frac{(N'-1)e}{l}.$$
Keeping the above representation of $\underline{\lambda}$, $\underline{\mu}$ and   $\underline{\mu}'$ are respectively given as follows:

\begin{center}
\leavevmode
\epsfxsize= 12cm
\epsffile{parti4.eps}
\end{center}

\begin{center}
\leavevmode
\epsfxsize= 12cm
\epsffile{parti6.eps}
\end{center}

Then Proposition \ref{combi} asserts that we have:
$$\underline{\mu} \prec{\underline{\mu}'}.$$

The aim of the following part is to show that the matrix associated to the  canonical basis elements of the irreducible modules $M_{{\bf s}_l}$ is unitriangular with respect to $a$-values.

\subsection{Study of the involution on the semi-infinite $q$-wedge product}

Let $s\in{\mathbb{Z}}$. We keep the notations of the previous sections. We will work with the  semi-infinite $q$-wedge product $\bigwedge^{s+\frac{\infty}{2}} V_{e,l}$. First, we attach to each semi-infinite monomial an $a$-value.  Let $u_{\underline{k}}\in \bigwedge^{s+\frac{\infty}{2}} V_{e,l}$ be a   semi-infinite $q$-wedge product and let  $u_{\underline{\tilde{k}}}$ be the monomial obtained by reordering the $k_i$ in strictly decreasing order. Then, the bijection described in section \ref{bij} shows that we can associate  to  $u_{\underline{\tilde{k}}}$ a symbol $|\underline{\lambda},{\bf s}_l\rangle$ with $\underline{\lambda}\in{\Pi_l^n}$ and ${\bf s}_l=(s_1,...,s_l)\in{\mathbb{Z}^l}$ verifying $\sum_{i=1}^l s_i =s$. We put:
\begin{align*}
\pi (u_{\underline{k}})&:=|\underline{\lambda},{\bf s}_l\rangle,\\
a(u_{\underline{k}})&:=a_{{\bf s}_l}(\underline{\lambda}).
\end{align*}
In this part, we show the following proposition:
\begin{prop}\label{main2} Let  $u_{\underline{k}}\in \bigwedge^{s+\frac{\infty}{2}} V_{e,l}$ be a   semi-infinite ordered $q$-wedge product. Then, we have:
$$\overline{u_{\underline{k}}}= u_{\underline{k}}+\textrm{ sum of ordered monomials }u_{\underline{r}}\textrm{ with }a(u_{\underline{r}})>a(u_{\underline{k}}), $$
where the involution $\overline{ }$ is defined in section \ref{ariki}.
\end{prop}
We note that a similar property is shown in \cite{U} but for a  (partial) order which is not compatible with the preorder induced by the $a$-values.

Let  $u_{\underline{k}}$ be an arbitrary semi-infinite monomial such that there exists  $i\in{\mathbb{N}}$ such that $k_i<k_{i+1}$. Then the relations $(R1)-(R4)$ in section \ref{bij} show how to express   $u_{\underline{k}}$ in terms of semi-infinite monomials   $u_{\underline{k}'}$ such that $k_i'>k_{i+1}'$. Let  $u_{\underline{k}'}$ be such a   semi-infinite monomial such that $k_i'\neq k_{i+1}$ and $k_{i+1}'\neq k_i$. Then the  $l$-partitions associated to $u_{\underline{k}}$  and  $u_{\underline{k}'}$  have the same charge ${\bf s}_l$. We put :
\begin{align*}
|\underline{\lambda},{\bf s}_l\rangle&:=\pi (u_{\underline{k}}),\\
|\underline{\lambda}',{\bf s}_l\rangle&:=\pi (u_{\underline{k}'}).
\end{align*}
From the relations $(R1)-(R4)$, using the representations of   $u_{\underline{k}}$  and  $u_{\underline{k}'}$ by $l$-abacus, we can see that  $u_{\underline{k}'}$ is obtained from  $u_{\underline{k}}$ by moving two black beads representing $k_i$ and $k_{i+1}$  to two black beads representing  $k_i'$ and $k_{i+1}'$ and lying in the same runners as $k_i$ and $k_{i+1}$. As a consequence, the $l$-partition $\underline{\lambda}'$ is obtained from $\underline{\lambda}$ by removing a ribbon $R$ of size $x$ from a component $b$ and adding a ribbon $R'$ of size $x$ to a component $b'$ as in the following figure.

\begin{center}
\leavevmode
\epsfxsize= 12cm
\epsffile{parti.eps}
\end{center}

 Hence, there exist positive integers  $r$, $s$, $p$ and $t$ such that:
$$\lambda'^{(b)}=(\lambda^{(b)}_1,....,\lambda^{(b)}_{r-1},\lambda^{(b)}_{r+1}-1,...,,\lambda^{(b)}_{s}-1,\lambda^{(b)}_r-(x-(r+s)),\lambda^{(b)}_{s+1},...,\lambda_{h^{(b)}}^{(s)}),$$
$$\lambda'^{(b')}=(\lambda^{(b')}_1,...,\lambda_{p-1}^{(b')},\lambda_t^{(b')}+(x-(p-k)),\lambda^{(b')}_{p}+1,...,\lambda^{(b')}_{t-1}+1,\lambda^{(b')}_{t+1},...,\lambda^{(b')}_{h^{(b')}}).   $$
In fact, the two $l$-partitions $\underline{\lambda}'$ and
$\underline{\lambda}$ are both obtained by adding $R$ and $R'$ to the
components $b$ and $b'$ of  the same $l$-partition
$\underline{\nu}=\pi(u_{\underline{r}})$ where
$$\underline{r}=(k_1,...,k_{i-1},\hat{k}_i,\hat{k}_{i+1},k_{i+2},...).$$
Here,  $\hat{k}_i$ and $\hat{k}_{i+1}$ are the minimal integers of
$\{k_i,k_{i+1},k_i ',k_{i+1} '\}$ such that the black beads associated to  $\hat{k}_i$ and $\hat{k}_{i+1}$ lie at the components $b$ and $b'$ of the $l$-abacus of  $u_{\underline{r}}$ and   $\hat{k}_{i+1}<\hat{k}_{i}$.

\begin{lem}\label{pcomb} We keep the above notations.
Assume that $\underline{\lambda}\neq \underline{\lambda}'$. Put
$\hat{k}_i=a+e(l-b)-el m$ and  $\hat{k}_{i+1}=a'+e(l-b')-el m'$ where
$a,a'\in{\{1,...,e\}}$, $b,b'\in{\{1,...,l\}}$ and,
$m,m'\in{\mathbb{Z}}$.    Then we have:
$$a-em-\frac{(b-1)e}{l}>a'-em'-\frac{(b'-1)e}{l}.$$
\end{lem}
\emph{Proof}. For $j=i,\ i+1$, put $k_j=a_j+e(l-b_j)-elm_j$ where  $a_j\in{\{1,...,e\}}$,
$b_j\in{\{1,...,l\}}$ and $m_j\in{\mathbb{Z}}$. As in section
\ref{bij}, we define  $\alpha$
and $\beta$ to be the unique integers in $[0,el-1]$ such that $\alpha
  \equiv(a_{i+1} -a_i )(\textrm{mod }el) $ and
  $\beta\equiv e(b_i-b_{i+1})(\textrm{mod }el)$. As
  $\underline{\lambda}\neq \underline{\lambda}'$, we can assume that
  $\alpha\neq 0$ or $\beta\neq 0$.

Now, regarding the relations $(R1)-(R4)$ in section \ref{bij}, we have
$5$ cases to consider.

\begin{itemize}
\item  Assume that $k_{i}'=k_{i+1}-elm$ and
  $k_{i+1}'=k_{i}+elm$ with $m> 0$. If $\beta=0$, we have  $\hat{k}_i=k_{i+1}'$
  and $\hat{k}_{i+1}=k_{i }$ and since $k_{i+1}'>k_i$, the result is
  clear. If $\beta\neq 0$,  we have  $\hat{k}_i=k_{i}'$
  and $\hat{k}_{i+1}=k_{i }$.

As $k_{i}'>k_{i+1}'$, we have:
$$a_{i+1}+e(l-b_{i+1})-el(m_{i+1}+m)>a_{i}+e(l-b_{i})-el(m_{i}-m).$$
Now, as $m>0$ and $\displaystyle a_{i+1}-a_i>\frac{a_{i+1}-a_i}{l}-e$, we conclude
that:
$$a_{i+1}-e(m_{i+1}+m)-(b_{i+1}-1)\frac{e}{l}>a_{i}-em_{i}-(b_{i}-1)\frac{e}{l}.$$

\item Assume that $k_{i}'=k_{i+1}-\beta-elm$ and
  $k_{i+1}'=k_{i}+\beta+elm$ with $m\geq 0$ and $\beta\neq 0$. Then, we have
  $\hat{k}_i=k_{i+1}'$    and $\hat{k}_{i+1}=k_{i } $.

\begin{itemize}
 \item If $b_{i+1}<b_{i}$ then $\beta=e(b_i-b_{i+1})$ and we have:
\begin{align*}
\hat{k}_i&=a_{i}+e(l-b_{i})-elm_i+e(b_i-b_{i+1})+elm\\
&=a_i+e(l-b_{i+1})-el(m_i-m).\end{align*}
We conclude that:
$$a_i-e(m_i-m)-(b_{i+1}-1)\frac{e}{l}>a_i-em_i-(b_i-1)\frac{e}{l}.$$
 \item If $b_i<b_{i+1}$  then $\beta=e(b_i-b_{i+1})+el$ and we have:
\begin{align*}
\hat{k}_i&=a_{i}+e(l-b_{i})-elm_i+e(b_i-b_{i+1})+elm+el\\
&=a_i+e(l-b_{i+1})-el(m_i-m-1).\end{align*}
 We conclude that:
$$a_i-e(m_i-m-1)-(b_{i+1}-1)\frac{e}{l}>a_i-em_i-(b_i-1)\frac{e}{l}.$$
\end{itemize}

\item Assume that $k_{i+1}'=k_{i}+\alpha+elm$ and
  $k_{i}'=k_{i+1}-\alpha-elm$ with $m\geq 0$ and $\alpha\neq 0$. If
  $\beta=0$, we have $\hat{k}_i=k_{i+1}'$    and $\hat{k}_{i+1}=k_{i
  }$ and since $k_{i+1}'>k_i$, the result is clear. If $\beta\neq 0$, we have
  $\hat{k}_i=k_{i}'$    and $\hat{k}_{i+1}=k_{i }$.

\begin{itemize}
 \item If $a_{i+1}-a_i>0$ then $\alpha=a_{i+1}-a_i$ and we have:
\begin{align*}
\hat{k}_i&=a_{i+1}+e(l-b_{i+1})-elm_{i+1}-a_{i+1}+a_{i}+elm\\
&=a_i+e(l-b_{i+1})-el(m_{i+1}-m).\end{align*}
We conclude that:
$$a_i-e(m_{i+1}-m)-(b_{i+1}-1)\frac{e}{l}>a_i-em_{i}-(b_i-1)\frac{e}{l}.$$
Indeed, if $m_{i+1}-m<m_i$, the result is clear. If otherwise, as
$m_{i+1}\leq m_i$, we have $m=0$ and $m_{i+1}=m_i$. Hence, as
$k_{i}'>k_{i+1}'$, we have $b_{i+1}<b_i$.
\item  If $a_{i+1}-a_i<0$ then $\alpha=a_{i+1}-a_i+el$ and we have:
\begin{align*}
\hat{k}_i&=a_{i+1}+e(l-b_{i+1})-elm_{i+1}-a_{i+1}+a_{i}+el(m+1)\\
&=a_i+e(l-b_{i+1})-el(m_{i+1}-m-1).\end{align*}
We conclude that:
$$a_i-e(m_{i+1}-m-1)-(b_{i+1}-1)\frac{e}{l}>a_i-em_{i}-(b_i-1) \frac{e}{l}.$$
\end{itemize}
\item Assume that $k_{i+1}'=k_{i}+\alpha+\beta+elm$ and
  $k_{i}'=k_{i+1}-\alpha-\beta-elm$ with $m\geq 0$, $\beta\neq 0$ and
  $\alpha\neq 0$. Then, we have
  $\hat{k}_i=k_{i+1}'$    and $\hat{k}_{i+1}=k_{i }$.
\begin{itemize}
\item if $a_{i+1}>a_i$ and $b_{i}>b_{i+1}$ then $\alpha=a_{i+1}-a_i$
  and $\beta=e(b_{i}-b_{i+1})$. We have:
\begin{align*}
\hat{k}_i&=a_{i}+e(l-b_{i})-elm_i+e(b_i-b_{i+1})+(a_{i+1}-a_i)+elm\\
&=a_{i+1}+e(l-b_{i+1})-el(m_i-m).\end{align*}
We conclude that:
$$a_{i+1}-e(m_i-m)-(b_{i+1}-1)\frac{e}{l}>a_i-em_i-(b_i-1)\frac{e}{l}.$$
\item if $a_{i+1}<a_i$ and $b_{i}>b_{i+1}$ then $\alpha=a_{i+1}-a_i+el$
  and $\beta=e(b_{i}-b_{i+1})$. We have:
\begin{align*}
\hat{k}_i&=a_{i}+e(l-b_{i})-elm_i+e(b_i-b_{i+1})+(a_{i+1}-a_i)+el(m+1)\\
&=a_{i+1}+e(l-b_{i+1})-el(m_i-m-1).\end{align*}
We conclude that:
$$a_{i+1}-e(m_i-m-1)-(b_{i+1}-1)\frac{e}{l}>a_i-em_i-(b_i-1)\frac{e}{l}.$$
\item if $a_{i+1}>a_i$ and $b_{i}<b_{i+1}$ then $\alpha=a_{i+1}-a_i$
  and $\beta=e(b_{i}-b_{i+1})+el$. We have:
\begin{align*}
\hat{k}_i&=a_{i}+e(l-b_{i})-elm_i+e(b_i-b_{i+1})+(a_{i+1}-a_i)+elm+el\\
&=a_{i+1}+e(l-b_{i+1})-el(m_i-m-1).\end{align*}
We conclude that:
$$a_{i+1}-e(m_i-m-1)-(b_{i+1}-1)\frac{e}{l}>a_i-em_i-(b_i-1)\frac{e}{l}.$$
\item if $a_{i+1}>a_i$ and $b_{i}<b_{i+1}$ then $\alpha=a_{i+1}-a_i+el$
  and $\beta=e(b_{i}-b_{i+1})+el$. We have:
\begin{align*}
\hat{k}_i&=a_{i}+e(l-b_{i})-elm_i+e(b_i-b_{i+1})+(a_{i+1}-a_i)+elm+2el\\
&=a_{i+1}+e(l-b_{i+1})-el(m_i-m-2).\end{align*}
We conclude that:
$$a_{i+1}-e(m_i-m-2)-(b_{i+1}-1)\frac{e}{l}>a_i-em_i-(b_i-1)\frac{e}{l}.$$
\end{itemize}
  \end{itemize}
\rightline{$\Box$}

We are now able to prove Proposition \ref{main2}. Keeping the notations
of the beginning of this section,  Lemma \ref{pcomb} shows that we have:
$$\nu_{s}^{(b)}-s+s_{b}-\frac{(b-1)e}{l}>\nu_{t}^{(b')}-t+s_{b'}-\frac{(b'-1)e}{l}.\ \ \ \ \ \ (*)$$

Assume that we have $\lambda_i'^{(b')}=\nu_i^{(b')}+y_i$ for integers $y_i$ with $i=p,p+1,...,t$ and assume that we have   $\lambda_j^{(b)}=\nu_j^{(b)}+x_j$ for integers $x_j$ with $j=r,r+1,...s$.
We define a $l$-composition $\underline{\lambda}[1]$ as follows.
\begin{itemize}
\item If $y_t\leq x_s$, then we put $j_1:=s$ and we define:
$$\underline{\lambda}[1]_i^{(k)}=\left\{\begin{array}{ll}
{\lambda}_i'^{(k)}&\textrm {if }(i,k)\neq (s,b) \textrm{ and }(i,k)\neq (t,b'),\\
{\lambda}_i'^{(k)}+y_t&\textrm {if }(i,k)= (s,b),\\
{\lambda}_i'^{(k)}-y_t&\textrm {if }(i,k)= (t,b').\\
\end{array}\right.$$
In this case, $\underline{\lambda}[1]$ and $\underline{\lambda}'$ are both obtained by adding the same number of nodes on a $l$-composition. Then, by $(*)$ and by Proposition \ref{combi}, we get:
$$\underline{\lambda}[1]\prec\underline{\lambda}'.$$

\item If  $y_t> x_s$, there exists $j_1\in{\{r,r+1,...,s-1\}}$ such that:
$$x_{j_1}+x_{j_1+1}+....+x_s\geq y_t \geq x_{j_1+1}+...+x_{s-1}+x_s.$$
Then, we define:
$$ \underline{\lambda}[1]_i^{(k)}=\left\{  \begin{array}{ll}
{\lambda}_i'^{(k)}+x_i&\textrm {if }k=b\textrm{ and }\\
                      & \textrm{   }s\geq i\geq j_1+1,\\
{\lambda}_i'^{(k)}+y_t-(x_{j_1+1}+...+x_{s-1}+x_s)&\textrm {if }(i,k)=(j_1,b),\\
{\lambda}_i'^{(k)}-y_t&\textrm {if }(i,k)= (t,b'),\\
{\lambda}_i'^{(k)}&\textrm {if otherwise}.\\
\end{array}\right.$$
For all $i$ such that $r\leq i\leq s$, observe that
$\nu^{(b)}_s+x_{i+1}+...+x_{s-1}+x_s=\nu^{(b)}_{i}+(s-i)$. Then, by $(*)$, for all $i$ such that $j_1\leq i\leq s$, we have:
$$\nu_{i}^{(b)}-i+s_{b}-\frac{(b-1)e}{l}>\nu_{t}^{(b')}-t+x_{i+1}+...+x_{s}+s_{b'}-\frac{(b'-1)e}{l}.$$
Thus,  for all $i$ such that $j_1\leq i\leq s$, we obtain:
$${\lambda'}_{i}^{(b)}-i+s_{b}-\frac{(b-1)e}{l}>{\lambda'}_{t}^{(b')}-t+x_{i+1}+...+x_{s-1}+x_s-y_t+s_{b'}-\frac{(b'-1)e}{l}.$$

Thus, using Proposition \ref{combi}, we get:
$$\underline{\lambda}[1]\prec\underline{\lambda}'.$$
\end{itemize}
Keeping the above figure, $\underline{\lambda}[1]$ is  as follows.
\begin{center}
\leavevmode
\epsfxsize= 12cm
\epsffile{parti1.eps}
\end{center}
 As we have $\nu_{t}^{(b')}-1+y_t=\nu_{t-1}^{(b')}$, note that:
$$\lambda[1]_{j_1}^{(b)}-j_1+s_{b}-\frac{(b-1)e}{l}>\nu_{t-1}^{(b')}-(t-1)+s_{b'}-\frac{(b'-1)e}{l}.\ \ \ \ \ \ (**)$$
We now define a $l$-composition $\underline{\lambda}[2]$ as follows. Put :
$$x_{j_1}':=x_{j_1}+x_{j_1+1}+...+x_s-y_t.$$
\begin{itemize}
\item If $y_{t-1}\leq x_{j_1}'$,  we put $j_2:=j_1$ and we define:
$$\underline{\lambda}[2]_i^{(k)}=\left\{ \begin{array}{ll}
{\lambda}[1]_i^{(k)}&\textrm {if }(i,k)\neq (j_1,b) \textrm{ and }(i,k)\neq (t-1,b'),\\
{\lambda}[1]_i^{(k)}+y_{t-1}&\textrm {if }(i,k)= (j_1,b),\\
{\lambda}[1]_i^{(k)}-y_{t-1}&\textrm {if }(i,k)= (t-1,b').\\
\end{array}\right.$$
By $(**)$, we have:
$${\lambda}[1]_{j_1}^{(b)}-j_1+s_b-\frac{(b-1)e}{l}>{\lambda}[1]_{t-1}^{(b')}-y_{t-1}-(t-1)+s_b-\frac{(b'-1)e}{l}.$$
By Proposition \ref{combi}, we get:
$$\underline{\lambda}[2]\prec\underline{\lambda}[1].$$

\item If  $y_{t-1}> x_{j_1}'$, there exists $j_2\in{\{r,r+1,...,j_1-1\}}$ such that:
$$x_{j_2}+x_{j_2+1}+...+x_{j_1-1}+x_{j_1}'\geq y_{t-1}>x_{j_2+1}+...+x_{j_1-1}+x_{j_1}'.$$
Then, we define:

$$\underline{\lambda}[2]_i^{(k)}=\left\{\begin{array}{ll}
{\lambda}[1]_i^{(k)}+x_{j_1}'&\textrm {if }(i,k)= (j_1,b),\\
{\lambda}[1]_i^{(k)}+x_{i}&\textrm {if }k=b\textrm{ and }\\
 & \textrm{   } j_1-1\geq i>j_2,\\
{\lambda}[1]_i^{(k)}-y_{t-1}&\textrm {if }(i,k)= (t-1,b'),\\
{\lambda}[1]_i^{(k)}+y_{t-1}-(x_{j_2+1}+...+x_{j_1-1}+x_{j_1}')&\textrm {if }(i,k)= (j_2,b),\\
{\lambda}[1]_i^{(k)}& \textrm {if otherwise.}
\end{array}\right.$$
By $(**)$, we have:
$${\lambda}[1]_{j_1}^{(b)}-i+s_{b}-\frac{(b-1)e}{l}>{\lambda}[1]_{t-1}^{(b')}-(t-1)-y_{t-1}+s_{b'}-\frac{(b'-1)e}{l}.$$
Moreover, we have  $\lambda[1]_{j_1}^{(b)}+x_{j_1}'=\lambda[1]_{j_1-1}^{(b)}+1$ and $\lambda[1]^{(b)}_i+x_i=\lambda[1]^{(b)}_{i-1}+1$ for all $i$ such that $j_1-1\geq i\geq r$. Thus, for all $i$ such that $j_1-1\geq i\geq j_2$, by $(**)$, we obtain:
$${\lambda}[1]_{i}^{(b)}-i+s_{b}-\frac{(b-1)e}{l}>{\lambda}[1]_{t-1}^{(b')}-(t-1)+x_{i+1}+...+x_{j_1-1}+x_{j_1}'-y_{t-1}+s_{b'}-\frac{(b'-1)e}{l}.$$
Hence, by Proposition \ref{combi}, we conclude that:
$$\underline{\lambda}[2]\prec\underline{\lambda}[1].$$
\end{itemize}
Keeping the above figure, $\underline{\lambda}[2]$ is  as follows.
\begin{center}
\leavevmode
\epsfxsize= 12cm
\epsffile{parti2.eps}
\end{center}

Continuing in this way, we obtain the $l$-composition  $\underline{\lambda}[p-t+1]=\underline{\lambda}$:
\begin{center}
\leavevmode
\epsfxsize= 12cm
\epsffile{parti3.eps}
\end{center}
We have :
$$\underline{\lambda}\prec\underline{\lambda}'.$$
Thus, we conclude that:
$$a_{{\bf s}_l}(\underline{\lambda})< a_{{\bf s}_l}(\underline{\lambda}').$$
Now  all the ordered monomials $u_{\underline{p}}$ which appear in $\overline{u_{\underline{k}}}$ are obtained recursively by using the relations $(R1)-(R4)$. Hence, by induction,  there exists $\beta$ such that:
$$\overline{u_{\underline{k}}}= \beta u_{\underline{k}}+\textrm{ sum of ordered monomials }u_{\underline{r}}\textrm{ with }a(u_{\underline{r}})>a(u_{\underline{k}}), $$
By \cite[Remark 3.24]{U}, we have $\beta=1$. This concludes the proof of the proposition.

\subsection{Consequences}

The result of the previous section leads to the following theorem:
\begin{thm}\label{mainm}
Let  ${\bf s}_l\in{\mathbb{Z}^n}$ and let $M_{{\bf s}_l}:=U_q' (\widehat{\mathfrak{sl}_e}).|\underline{\emptyset},{\bf s}_l\rangle\subset \mathcal{F}_{e,{\bf s}_l}$. Let
$$\{G(\underline{\mu},{\bf s}_l)\ |\ \ \underline{\mu}\in{\Lambda^{n}_{e;{\bf s}_l}   },\ n\in{\mathbb{N}}\}$$
be the canonical basis elements. Then, for all  $n\in{\mathbb{N}}$ and $\underline{\mu}\in{\Lambda^{n}_{e;{\bf s}_l}}$, we have:
$$G(\underline{\mu},{\bf s}_l)=|\underline{\mu},{\bf s}_l\rangle+\sum_{\underline{\lambda}\in{\Pi_l^n}, a_{{\bf s}_l}(\underline{\lambda})>a_{{\bf s}_l}(\underline{\mu})} d_{\underline{\lambda},\underline{\mu}}(q)|\underline{\lambda},{\bf s}_l\rangle,$$
where $d_{\underline{\lambda},\underline{\mu}}(q)\in q\mathbb{C}[q]$.
\end{thm}
\emph{Proof}.
 By Proposition \ref{main2}, we have for all $\underline{\lambda}\in{\Pi_l^n}$:
$$\overline{|\underline{\lambda},{\bf s}_l\rangle}=|\underline{\lambda},{\bf s}_l\rangle+\textrm{ sum of  } |\underline{\mu},{\bf s}_l\rangle \textrm{ with }
 a_{{\bf s}_l}(\underline{\mu})>a_{{\bf s}_l}(\underline{\lambda}).$$
Assume that  $\underline{\lambda}\in{\Lambda^{n}_{e;{\bf s}_l}}$ and that  $\underline{\mu}$ is one of the minimal $l$-partitions with respect to $a_{{\bf s}_l}$ which appears in $G(\underline{\lambda},{\bf s}_l)$. Then the characterization of the canonical basis in Theorem \ref{cb} immediately implies that   $\underline{\lambda}=\underline{\mu}$.\\
\rightline{$\Box$}

Now, we give the consequences  on the decomposition matrices of Ariki-Koike algebras. By Ariki's theorem, we obtain the following result. Note that this generalizes and gives a new proof of the main result of \cite{Jp} where the case $0\leq s_1\leq ...\leq s_l<e$ was considered (but the proof of \cite{Jp} gave an explicit construction of the canonical basis elements). Note also that it provides an interpretation of the parametrization by the Kleshchev $l$-partitions by using $a$-values.

Let ${\bf s}_l=(s_1,...,s_l)\in{\mathbb{Z}^l}$. Let $\mathcal{H}_{\mathbb{C},n}$ be the Ariki-Koike algebra over $\mathbb{C}$ with the following choice of parameters:
\begin{eqnarray*}
&&x_j=\eta_e^{s_{j}}\ \textrm{for}\ j=1,...,l,\\
&&x=\eta_{e},
\end{eqnarray*}

Let  $\mathcal{H}_{\mathbb{C}(y),n}^{{\bf s}_l}$ be the Ariki-Koike algebra over $\mathbb{C}(y)$   with the following parameters:
\begin{eqnarray*}
&&u_j=y^{lm^{(j)}}\eta_l^{j-1}\ \textrm{for}\ j=1,...,l,\\
&&v=y^l,
\end{eqnarray*}
where $\displaystyle m^{(j)}=s_j-\frac{(j-1)e}{l}+\alpha e$ for
$j=1,...,l$ and where $\alpha$ is a positive integer such that
$m^{(j)}>0$ for  $j=1,...,l$. Then the specialisation $\theta:\mathbb{C}[y,y^{-1}]\to \mathbb{C}$ which sends $y$ to $\displaystyle \textrm{exp}(\frac{2i\pi}{le})$ induces a decomposition map:
$$d:R_0 (\mathcal{H}^{{\bf s}_l}_{\mathbb{C}(y),n})\to R_0 (\mathcal{H}_{\mathbb{C},n}).$$
\begin{thm}\label{maing}  For each $M\in{\Irr{\mathcal{H}_{\mathbb{C},n}}}$, there exists a   unique simple $\mathcal{H}_{\mathbb{C}(y),n}^{{\bf s}_l}$-module $V_M$  such that the following two conditions hold:
\begin{itemize}
\item $d_{V_M,M}= 1$,
\item if there exists $W\in{\mathcal{H}_{\mathbb{C}(y),n}^{{\bf s}_l}}$ such that $d_{W,M}\neq 0$ then $a(W)>a(V_M)$.
\end{itemize}
  Moreover the function which sends $M$ to $V_M$ is injective. As a consequence the associated decomposition matrix is unitriangular and the following set is in natural bijection with $\Irr{\mathcal{H}_{\mathbb{C},n}}$:
$$\mathcal{B}_{{\bf s}_l}=\{V_M\ |\ M\in{\Irr{\mathcal{H}_{\mathbb{C},n}}}\}.$$
Finally, we have:
$$\mathcal{B}_{{\bf s}_l}=\{S^{\underline{\lambda}}_{\mathbb{C}(y)}\ |\ \underline{\lambda}\in{\Lambda^{n}_{e;{\bf s}_l}} \}.$$
\end{thm}
\begin{ex}\label{exfin}
The decomposition matrices of Ariki-Koike
algebras can be computed using the algorithm described in
\cite{Ja}. We put $l=2$,
$x=\eta_4$, $x_1=\eta_4^0$ and $x_2=\eta_4^1$. Then Theorem \ref{maing} shows
how we can order the rows of the associated decomposition matrix to
have a unitriangular shape with $1$ along the diagonal. To do this, we
must choose $(s_1,s_2)$ such that $s_1\equiv 0 (\textrm{mod }4)$ and
$s_1\equiv 1 (\textrm{mod }4)$ and order the rows with respect to the
$a$-values.  These rows are indexed by the $2$-partitions of rank $4$,
we also give the associated $a$-values. We obtain the explicit
bijection between  $\Lambda^{4}_{4,(s_1,s_2)}$ and $\Irr{\mathcal{H}_{\mathbb{C},n}}$.
\begin{itemize}
\item When $s_1=0$ and $s_2=1$, the simple $\mathcal{H}_{\mathbb{C},n}$-modules may be labeled by $\Lambda^{4}_{4,(0,1)}$.

$$\Lambda^4_{4,(0,1)}=\{(\emptyset,(4));((1),(2,1));((1,1),(1,1));((1),(3));((1,1),(2));$$
$$((2),(1,1));((2),(2));((2,1),(1));((2,1,1),\emptyset);((2,2),\emptyset);((3),(1));$$
$$((3,1),\emptyset);((4),\emptyset)\};$$
$$
\begin{array}{cc}
      ((4),\emptyset) & 0 \\
     ((3),(1))  & 1\\
     (\emptyset,(4)) & 1\\
     ((3,1),\emptyset)  &1\\
     ((2),(2))  &2\\
     ((2,2),\emptyset)   &2\\
     ((1),(3))  &2\\
   ((2,1),(1))  &3\\
    ((2,1,1),\emptyset)  &4\\
    ((2),(1,1))  &4\\
    ((1,1),(2))  &4\\
    ((1),(2,1))  &5\\
   ((1,1),(1,1)) &6 \\
     (\emptyset,(3,1))  &4\\
   ((1,1,1),(1))  &6\\
     (\emptyset,(2,2))   &6\\
   ((1,1,1,1),\emptyset)  &9\\
    (\emptyset,(2,1,1))  &9\\
   ((1),(1,1,1))  &9\\
  (\emptyset,(1,1,1,1))  &16
\end{array}
\left(
\begin{array}{cccccccccccccc}
  &1&.&.&.&.&.&.&.&.&.&.&.&.\\
  &.&1&.&.&.&.&.&.&.&.&.&.&.\\
  &.&.&1&.&.&.&.&.&.&.&.&.&.\\
  &1&.&.&1&.&.&.&.&.&.&.&.&.\\
  &.&1&.&.&1&.&.&.&.&.&.&.&.\\
  &.&.&.&.&.&1&.&.&.&.&.&.&.\\
 &1&.&1&.&.&.&1&.&.&.&.&.&.\\
 &.&.&.&.&.&.&.&1&.&.&.&.&.\\
  &.&.&.&1&.&.&.&.&1&.&.&.&.\\
  &.&.&.&.&.&.&.&.&.&1&.&.&.\\
  &1&.&.&1&.&.&1&.&.&.&1&.&.\\
  &.&.&.&.&.&.&.&.&.&.&.&1&.\\
  &.&.&.&.&.&1&.&.&.&.&.&.&1\\
  &.&.&1&.&.&.&1&.&.&.&.&.&.\\
  &.&.&.&1&.&.&.&.&1&.&1&.&.\\
  &.&.&.&.&1&.&.&.&.&.&.&.&.\\
  &.&.&.&.&.&.&.&.&1&.&.&.&.\\
  &.&.&.&.&.&.&1&.&.&.&1&.&.\\
  &.&.&.&.&.&.&.&.&.&.&.&.&1\\
  &.&.&.&.&.&.&.&.&.&.&1&.&.\\
\end{array}\right)$$

\item When $s_1=4$ and $s_2=1$, the simple $\mathcal{H}_{\mathbb{C},n}$-modules may be labeled by $\Lambda^{4}_{4,(4,1)}$.

$$\Lambda^4_{4,(4,1)}=\{((1,1,1),(1));((1),(2,1);((1,1),(1,1));((1),(3));((1,1),(2));$$
$$((2),(1,1));((2),(2));((2,1),(1));((2,1,1),\emptyset);((2,2),\emptyset);((3),(1));$$ $$((3,1),\emptyset);((4),\emptyset)\};$$

$$
\begin{array}{cc}
((4),\emptyset)& 0\\
((3,1),\emptyset)& 1\\
((2,2),\emptyset)& 2\\
((2,1,1),\emptyset)& 3\\
((3),(1))& 5\\
((2,1),(1))& 6\\
((1,1,1),(1))& 8\\
((2),(2))& 9\\
((1,1),(2))& 10\\
((2),(1,1))& 12\\
((1),(3))& 12\\
((1,1),(1,1))& 13\\
((1),(2,1))& 16\\
((1,1,1,1),\emptyset)& 6\\
(\emptyset,(4))& 14\\
(\emptyset,(3,1))& 19\\
(\emptyset,(2,2))& 22\\
(\emptyset,(2,1,1))& 25\\
((1),(1,1,1))& 21\\
(\emptyset,(1,1,1,1))& 32\\
\end{array}
\left(\begin{array}{cccccccccccccc}
1 & . & .&.&.&.&.&.&.&.&.&.&.\\
1&1&.&.&.&.&.&.&.&.&.&.&.\\
.&.&1&.&.&.&.&.&.&.&.&.&.\\
.&1&.&1&&.&.&.&.&.&.&.&.\\
.&.&.&.&1&.&.&.&.&.&.&.&.\\
.&.&.&.&.&1&.&.&.&.&.&.&.\\
.&1&.&1&.&.&1&.&.&.&.&.&.\\
.&.&.&.&1&.&.&1&.&.&.&.&.\\
1&1&.&.&.&.&1&.&1&.&.&.&.\\
.&.&.&.&.&.&.&.&.&1&.&.&.\\
1&.&.&.&.&.&.&.&1&.&1&.&.\\
.&.&1&.&.&.&.&.&.&.&.&1&.\\
.&.&.&.&.&.&.&.&.&.&.&.&1\\
.&.&.&1&.&.&.&.&.&.&.&.&.\\
.&.&.&.&.&.&.&.&.&.&1&.&.\\
.&.&.&.&.&.&.&.&1&.&1&.&.\\
.&.&.&.&.&.&.&1&.&.&.&.&.\\
.&.&.&.&.&.&1&.&1&.&.&.&.\\
.&.&.&.&.&.&.&.&.&.&.&1&.\\
.&.&.&.&.&.&1&.&.&.&.&.&.
\end{array}\right)
$$
\item When $s_1=0$ and $s_2=5$, the simple $\mathcal{H}_{\mathbb{C},n}$-modules may be labeled by $\Lambda^{4}_{4,(0,5)}$.

$$\Lambda^4_{4,(0,5)}=\{(\emptyset,(2,1,1));(\emptyset,(2,2)),((1),(1,1,1));((1,1),(1,1));((1,1),(2));$$ $$((2),(1,1));((1),(2,1));((1,1,1),(1));((1,1),(2));((1),(3));(\emptyset,(3,1));$$ $$(\emptyset,(4));((2,1),(1)),((2),(2))\}.$$

$$
\begin{array}{cc}
(\emptyset,(4))& 0\\
(\emptyset,(3,1))& 1\\
(\emptyset,(2,2))& 2\\
((1),(3))& 3\\
(\emptyset,(2,1,1))& 3\\
((1),(2,1))& 4\\
((2),(2))& 5\\
((1),(1,1,1))& 6\\
((2),(1,1))& 6\\
((1,1),(2))& 8\\
((1,1),(1,1))& 9\\
((2,1),(1))& 10\\
((1,1,1),(1))& 15\\
(\emptyset,(1,1,1,1))& 6\\
((4),\emptyset)& 6\\
((3),(1))& 6\\
((3,1),\emptyset)& 11\\
((2,2),\emptyset)& 14\\
((2,1,1),\emptyset)& 17\\
((1,1,1,1),\emptyset)& 24\\
\end{array}
\left(\begin{array}{cccccccccccccc}

 1&. &. &.&. &. &. &. &. &. &. &. &.\\
 1&1&.&.&.&.&.&.&.&.&.&.&.\\
 .&.&1&.&.&.&.&.&.&.&.&.&.\\
1&1&.&1&.&.&.&.&.&.&.&.&.\\
.&1&.&.&1&.&.&.&.&.&.&.&.\\
.&.&.&.&.&1&.&.&.&.&.&.&.\\
.&.&1&.&.&.&1&.&.&.&.&.&.\\
.&.&.&.&.&.&.&1&.&.&.&.&.\\
.&.&.&.&.&.&.&.&1&.&.&.&.\\
.&1&.&1&1&.&.&.&.&1&.&.&.\\
.&.&.&.&.&.&.&1&.&.&1&.&.\\
.&.&.&.&.&.&.&.&.&.&.&1&.\\
.&.&.&.&1&.&.&.&.&1&.&.&1\\
.&.&.&1&.&.&.&.&.&.&.&.&.\\
.&.&.&.&.&.&1&.&.&.&.&.&.\\
.&.&.&.&1&.&.&.&.&.&.&.&.\\
.&.&.&1&.&.&.&.&.&1&.&.&.\\
.&.&.&.&.&.&.&.&.&.&1&.&.\\
.&.&.&.&.&.&.&.&.&1&.&.&1\\
.&.&.&.&.&.&.&.&.&.&.&.&1
\end{array}\right)
$$

\end{itemize}
\end{ex}

\bibliographystyle{amsalpha}

redpajama_set_name: RedPajamaArXiv


text:
Integration pioneers to be honored with state historical marker
A marker to commemorate U.S. Navy Band B-1 will be installed at 10 a.m. Saturday, May 27.
Special to the Outlook
CHAPEL HILL-The first African Americans to serve in the modern Navy at general rating will be honored with the dedication of a permanent historical marker on the 75th anniversary of their enlisting.
The marker, which will be installed at 10 a.m. Saturday, May 27 at the intersection of West Franklin and South Roberson streets, commemorates U.S. Navy Band B-1, which was attached to the Navy's PreFlight School on the UNC campus from July 1942 to April 1944.
Attending the installation ceremony will be two of the original members of B-1, Simeon Holloway of Las Vegas and Calvin Morrow of Greensboro, as well as many of the veterans' family members. Only four of the original 44 members of the band are still living.
After the installation, a reception in B-1's honor will be held at the Hargraves Center at 216 N. Roberson Street. The marker dedication program and reception are both free and open to the public.
U.S. Navy B-1 was comprised primarily of young men who were enrolled at North Carolina A and T and Dudley High School. Enlistees also came from North Carolina Central, Hampton Institute, South Carolina State, Johnson C. Smith, and Hillside High School in Durham.
B-1 bandsmen were formally inducted into the Navy on May 27, 1942 at the Raleigh recruiting station. After training at Norfolk, they were transferred to Chapel Hill. Because of North Carolina's segregationist laws, the bandsmen could not live or be fed on campus. They were housed in a newly constructed community building on Roberson Street that is now the Hargraves Recreation Center.
Their service included daily marches from Hargraves to the UNC campus, where B-1 played for the raising of colors for the cadets every morning. These marches were big events for Chapel Hill's Black community. Rebecca Clark, longtime Chapel Hill resident and Civil Rights advocate, vividly recalled for B-1's official history how wonderful it felt to see "all those handsome young men in their uniforms" marching to work. "The kids were out at every corner watching them parade. They were the best thing that could have happened to our community."
Rebecca Clark's son, John Clark, said, "Doug and I and all the kids in the neighborhood would run out to Roberson Street when we heard the band coming, and we followed them as far as we could."
The band's marching route would have taken them by the spot where the state's historical marker will be installed.
During their service at Chapel Hill, B-1 bandsmen often played music for occasions that they would not have been allowed to attend because of their race. UNC-CH Chancellor James Moeser apologized to them in 2007 for how they were treated by the university during World War II, and at Kenan Stadium the bandsmen were made honorary members of the Marching Tar Heels.
Prior to B-1's service, Blacks were limited to work in the Navy as cooks and porters. B-1 was also the first Navy band to serve without having trained at the Navy's School of Music, which did not admit Blacks. B-1 bandsmen were also the first African Americans to work on the UNC campus in jobs that did not involve cooking, cleaning and laundry work.
North Carolina's historical marker program is administered by the state's Department of Cultural Resources and the Department of Transportation. Since 1935, the program has erected over 1,500 state highway historical markers. The B-1 marker is the fourth to be installed in Chapel Hill, where markers also commemorate the founding of the University of North Carolina, the training of U.S. astronauts, and the 1947 Journey of Reconciliation. A Carrboro marker commemorating Elizabeth Cotten is about five blocks from where the B-1 marker will be installed.
Category National News
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Black jobless rate was 6.60 percent in April, the lowest since 1972, but…
Young Black man honors his deceased mother's wish and admitted to Savannah State
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Lawsuit filed to block Obama Presidential Center in Chicago

redpajama_set_name: RedPajamaCommonCrawl


text:
Articles - 'a', 'an' and 'the'
Choose the missing articles (a, an or the) in the spaces.
a a a a an the the the the the The the the the the the The the
There is public outcry in Australia over controversial plans to use aanthe Sydney Opera House as aanthe "billboard" to advertise aanthe horse race. Tens of thousands of Australians have signed aanthe petition to block aanthe plans to project aanthe advertisement for aanthe Everest Cup onto aanthe iconic white sails of the Opera House. The race is the richest horse race in the world. The decision to use aanthe Opera House as a platform for the ad even has the backing of Australia's Prime Minister Scott Morrison. He said: "It's not like they're painting it on there. I mean, it's lights flashing up there for aanthe brief moment of time....I don't understand why we tie ourselves up in knots about these things."
aanthe former chief executive of the Opera House described aanthe decision to promote the race on the world-famous landmark as "crass, inappropriate and offensive". He said: "I find it extraordinary that aanthe state politicians on both sides have somehow decided that this is in aanthe interests of Sydney, New South Wales or Australia to corrupt aanthe way the Opera House works, to corrupt aanthe art integrity of the building." aanthe son of Peter Hall, the architect who helped complete the Opera House, said: "My father would have been sickened by it....He would not have condoned advertising on aanthe building in any way. Lucky he's not around to see the desecration of our beautiful, iconic masterpiece."


Ready for a fun challenge?
Destruction Brook Woods offers wide trails and spectacular scenery along with providing a fun and challenging course that is 100% trail race. Enjoy beautiful autumn views of South Dartmouth at DNRT's 5th Annual Trail Race. This unique 3.3 mile course will ramble through one of DNRT's most popular properties, Destruction Brook Woods.
Walkers are welcome! We are also offering a 2.4-mile walk through Destruction Brook Woods for our walking friends.
Spectators are also welcome – both start & finish lines will be easily visible for spectators. So bring along a friend or family member to cheer you on as you run or walk through DNRT's beautiful property. Your registration fee helps support land preservation, open space, and DNRT's 35+ mile trail network, which remains open for public access free of charge, thanks to you!
All who register before October 22 will receive a long-sleeve t-shirt. We hope to have enough shirts for everyone, but can only guarantee t-shirts for those signed up before this date.
Need another reason to register? Free post-race pizza and beer!
Registration will be open closer to the race date.


I have a piece out today in Inside Higher Ed on why it matters that poetry reading in the US has almost doubled in five years.
My tenure case was approved by the department and university in February, and the new contract officially began this month.
I feel fortunate, certainly. My colleagues are friendly and generous; my university is materially committed to the humanities; my students are bright and driven to do good in the world. In 2014, my current position was my only offer of employment, a continual reminder that it could have turned out otherwise. It is still turning out otherwise for many brilliant friends and acquaintances who have been seeking full-time academic jobs every year since then.
I feel responsible for helping to improve the conditions under which my colleagues labor and my students learn. I am part of a faculty group engaged in holding our university to its own commitment to social justice. This summer I began attending organizing meetings of the Boston chapter of March For Our Lives, a non-partisan group that opposes gun violence, racism, economic inequality, and education inequity. I have been learning a lot.
On July 1, my job security improved. I am now part of the ludicrously small proportion of the academic workface with tenure: 21% in the U.S. in 2015, according to this AAUP chart based on IPEDS data, as against 57% part-time and full-time non-tenure-track faculty (the other 22% are grad students and junior faculty on the tenure track). I first applied for full-time academic jobs in the fall of 2008—that glorious season! Though private and public university management used the financial crisis to justify the austerity they imposed, austerity did not end with the official end of the Great Recession, at least when it comes to hiring faculty into good jobs. The MLA's most recent report on its Job Information list begins by remarking that four years of decline in both English and foreign-language job ads brings numbers to "another new low," having sunk beneath the level of the "troughs" of earlier decades. "How's the market this year?" the senior academics ask Ph.D.-job-seekers, as though they were asking about the weather. But as with the weather, short-term fluctuations do not disguise the menacing trend.
I have spent the entirety of my academic career so far watching the intensified hollowing-out of my profession. The destruction is not limited to those friends and grad-school colleagues whose "job hunt" turned up nothing—or turned up academic jobs which make the same demands as the tenure track without the same job security. The harm can be counted, too, in the numberless person-hours every academic I know has spent tailoring job application materials, drafting custom syllabuses, and performing all the other rituals of applicant abjection. If you care about the work scholars do, the atmosphere is demoralizing. It is, to be sure, worse in worse jobs: when I was a part-time adjunct, I found the isolation particularly depressing, and I liked my "individualized" health insurance plan even less. But even in a good job with outstanding colleagues and students all around, something eats away at the ordinary routines of my academic life: all the day-to-day work of simply doing the job (teaching the students, carrying on the research, going to the meetings, the meetings, the meetings) takes on more than a tinge of denial, something for the few of us who have good academic jobs to do while we wait for the last curtain to fall on professional scholarship. Nor is it encouraging to witness the parade of more active forms of denial: bad-faith solutions, illusory comforts, and intellectualized excuses for selfishness.
But mostly I regret the good work that could have been done by all of us in a better, more just system. It is hard to be optimistic about the possibility of bringing such a system into existence, but faculty with tenure and a conscience ought to put to good use the protections, such as they are, that they have from being sanctioned for openly addressing and seeking to change the conditions of the university. Change, and not defense alone, must be the aim. The latest headlines draw our attention to attacks on faculty coming from reactionary forces outside the university. Protecting our colleagues and our students from these attacks is an urgent task. But we should recognize that the very same corporatizing process that has eaten away at tenure has increased the vulnerability of faculty to political attacks, since those attacks work by pressuring an administrative class that has fewer and fewer common interests with the people who do the research and the teaching. Management protects the brand, not the faculty. Organized action by faculty and students is the best hope for guarding what is worthy in the university and remaking the rest in the name of the public good.
Tenure can hardly be said to confer any special perspective or (Cthulhu forbid) leadership role in such a movement for the public good. Discouragingly, experience suggests rather the opposite about the capacity of the tenured faculty for leading radical change. Still, I recognize a debt. What I owe to everyone else who should have been "coming up" at the same time as I did is this: pursuing the collective interest in more just academic institutions, even if this sometimes conflicts with the narrow interests of the ever-shrinking group who share my academic rank.


Calling All Financial Regulatory Fortune Tellers: What Would Jay Clayton's Policy Be?
President- elect Trump announced Jan. 4 that he will nominate Jay Clayton, currently a law partner at Sullivan & Cromwell LLP, to head the Securities and Exchange Commission. He will replace Mary Jo White who announced her intention to step down in January.
"Jay Clayton is a highly talented expert on many aspects of financial and regulatory law, and he will ensure our financial institutions can thrive and create jobs while playing by the rules at the same time," President-elect Trump's transition team said.
"We need to undo many regulations which have stifled investment in American businesses, and restore oversight of the financial industry in a way that does not harm American workers," President- elect Trump said.
Larry Shover—chief investment officer at Solution Funds Group and a member of the Public Company Accounting Oversight Board's Investor Advisory Group—responded to an e-mail from Bloomberg BNA Jan. 4.
Shover wrote that he was familiar with and admired Clayton's work as one of the lawyers representing Ally Financial. The case involved a $25 billion settlement in 2012 between banks and 49 states and the federal government over fraudulent mortgage servicing.
Kurt Schacht, Managing Director of Standards & Advocacy at CFA Institute and Chair of the Investor Advisory Committee of the SEC, wrote "The investment management profession looks forward to having someone in the SEC Chair role who will be a strong advocate for investor interests while also supporting capital formation. We look forward to working with the incoming Chair and any new Commissioners on achieving these goals," in an e-mail response to Bloomberg BNA.
Questions for the Delphic Oracle.
This is where the need for a modern day Delphic oracle comes in to answer whether Clayton will favor the Wall Street giants that he used to represent and assist in dismantling the Dodd–Frank Wall Street Reform and Consumer Protection Act—signed into federal law by President Barack Obama in 2010—to address problems underlying the 2008-2009 financial crisis.
"I am aware some have suggested amending Dodd-Frank, but no one has specifically proposed repealing the provisions that authorize the PCAOB to oversee auditors of broker-dealers," Public Company Accounting Oversight Board Chairman James Doty, wrote in a Dec. 2016 e-mail response to Bloomberg BNA in Dec. 2016 about threats to dismantle the Dodd–Frank .
Will Clayton Strike a Balance Between Oversight and Job Creation?
Some fear that Clayton will embrace the guidance of The Financial Choice Act, a bill sponsored by House Financial Services Committee Chairman Jeb Hensarling (R-Texas)—that advanced in September after a 30-26 vote.
Hensarling is a vocal proponent of free-market economic principles and an equally vocal proponent of discarding many of the restrictions of Dodd- Frank.
Supporters of the Financial Choice Act argue that it gives the SEC greater authority to penalize big banks while changing regulations to allow community banks to foster economic growth.
Future of the Financial Choice Act Fluid.
At this point the Financial Choice Act's future is still fluid and it is safe to predict much more scrutiny of its provisions before a new era of financial regulation is unleashed.
Jeffrey Mahoney, General Counsel at the Council of Institutional Investors told Bloomberg BNA in an e-mail that CII had serious concerns about some of the provisions in the Act and hoped that those provisions would be removed from the next version.
Mahoney included a copy of the CII's letter to Hersarling in his e-mail response.
CII objects to relaxing requirements for public companies to have an external, independent auditor attest to and report on management's assessment of internal controls over financial reporting)—an auditing high risk area where Public Company Accounting Board Inspections staff continually find "deficiencies' or failures.
Narrowing the scope of clawbacks—recovery of erroneous bonus and incentive awards to current or former executive that may have been caused by company fraud—is another provision of the Act that CII opposes.
Clayton's biography is available on the Sullivan & Cromwell LLP website. His practice is described as including public and private mergers and acquisitions transactions, capital markets offerings, regulatory and enforcement proceedings. His clients have included Goldman Sachs, Bear Stearns, Barclays Capital, British Airways, and an ownership group for the Atlanta Hawks NBA franchise, according to the biography.
Clayton has a B.S. in engineering from the University of Pennsylvania, a B.A. in Economics from the University of Cambridge, and a J.D. from the University of Pennsylvania Law School. Until 2015 he was an adjunct professor of law at the University of Pennsylvania.
Continue the discussion at Bloomberg BNA Accounting LinkedIn.


shares dropped today following the company's third quarter report. INKT stock over the past week reached its lowest point since early December.
Folks worry about growth over the next few quarters, especially as Inktomi absorbs FastForward Networks. Let the nervous nellies have their day; the reality is that Inktomi the company looks as solid as ever.
Forward guidance from Inktomi executives remained bullish. Looking at the larger industry picture, outlook from JDS Uniphase (Nasdaq: JDSU) indicates high-speed optical network demand will continue to be strong. As long as that network buildout continues, demand for Inktomi's caching technology will be there.
had a decent quarter, but that doesn't matter. Again, the outlook is what counts, and this technology distributor provides more reason for optimism, in the form of a forecast calling for fourth quarter revenue growth in the low double digits on a percentage basis.
That's not bad considering Ingram generated revenue of $7.83 billion in the fourth quarte of last year. Ingram also continues to revive its gross margin.
This morning's gains have brought IM stock nearly in-line with competitors such as Tech Data (Nasdaq: TECD). At its current price around 16, Ingram is valued at roughly 11.5 times estimated earnings for the next 12 months, compared to a forward multiple of 12.2 for TECD shares, which also gained ground today.
Tech distribution generally has been a lousy sector for investing this year. But parts prices are falling now, margins are rising and demand looks reasonably healthy for 2001. Other industries would love to be in that kind of situation.
offers evidence that the market is trading purely on valuation today. The online media company posted a quarter during which sales fell short of some expectations as the company rejiggered its strategy.
Yet investors decided to lift the stock by 21 percent, taking NBCi out of the penny stock realm. Considering NBCi still has an awful lot to prove, you'd have to conclude that people simply decided shares had been beaten up enough, regardless of what the future may hold.
shareholders approved their company's merger with Terra Networks (Nasdaq: TRRA). Talk about changing attitudes: last year CMGi (Nasdaq: CMGI) bailed on the USA Networks (Nasdaq: USAI) deal, mainly because its value plummeted as the market reacted.
Now? Not even a 63 percent plunge is enough to dissuade folks from a merger. In other words, they ended up agreeing with Barry Diller's contention that Internet stocks deserved a far lower valuation than previously thought. Diller just happened to say it a year too early.
-- USA Networks' most visible Web property -- now is thinking about moving back in with its parents. Along the same lines, Barnesandnoble.com (Nasdaq: BNBN) just got a little closer to mother.