text:
\section{Introduction}

Representations of affine Hecke algebras ${\mathcal H_\vtg(r)}={\mathcal H_\vtg(r)}_{\mathbb C}$ of type $A$ with a parameter $q\in\mathbb C$ link, on the one hand, representations of Ariki-Koike algebra \cite{AK} or cyclotomic Hecke algebra \cite{BM} $\mathcal{H}_{\underline{u}}(r)$ with
extra parameters $\underline{u}=(u_1,\ldots,u_m)$ and link, on the other hand, representations
of quantum loop algebras ${\rm U}_q(\widehat{\frak{gl}}_n)$ of $\mathfrak{gl}_n$, when $q$ is not a root of unity. In the latter case, irreducible ${\mathcal H_\vtg(r)}$-modules $V_{\mathfrak s}$ have been classified by Rogawski \cite{Ro1}, built on Zelevinsky \cite{Ze}, in terms of multisegments ${\mathfrak s}$. Moreover, by the affine quantum Schur--Weyl duality developed in \cite{DDF}, this classification determines in turn the isomorphism types of irreducible modules of the affine $q$-Schur algebras ${\mathcal S}_{\vtg}(\cycn,r)$,
which are inflated to irreducible ${\rm U}_q(\widehat{\frak{gl}}_n)$-modules. Thus, by \cite{FM}, we may index these irreducible modules by Drinfeld multipolynomials.

However, irreducible ${\mathcal H_\vtg(r)}$-modules are also inflations of irreducible $\mc H_{\und{u}}(r)$-modules which are labelled by Kleshchev multipartitions \cite{Ar}. Thus, a Kleshchev multipartition defines an irreducible ${\mathcal H_\vtg(r)}$-modules and, hence, an irreducible ${\rm U}_q(\widehat{\frak{gl}}_n)$-module and then a Drinfeld multipolynomial. So it is natural to ask how a Kleshchev multipartition of $r$ determines an $n$-tuple of Drinfeld polynomials. Since this determination relation is a many-to-one relation, a second natural question is how to single out certain Kleshchev multipartitions to obtain a one-to-one relation. The purpose of this paper is to answer these two questions.

Our approach was motivated by the work of Vazirani \cite{Va} and Fu and the first author \cite{DF}. First, it suffices to consider the Ariki--Koike algebras whose extra parameters $u_i$ are powers of $q^2$. These algebras are called Ariki--Koike algebras of integral type.  When $q$ is not a root of unity, Kleshchev multipartitions relative to such an Ariki--Koike algebra have an easy characterization following \cite{Va,BK}.
Second, it is enough to consider the irreducible ${\mathcal H_\vtg(r)}$-modules indexed by integral multisegments.
Following \cite{Ro1}, we order an integral multisegment in a unique way so that it defines a certain so-called standard words. This motivated us to introduce {\it standard} Kleshchev multipartitions. In this way, we immediately solved the one-to-one relation problem between standard Kleshchev multipartitions and integral multisegments.

We further extend the construction for the one-to-one relation to a many-to-one relation. Thus, we start with an arbitrary multipartition $\underline{\lambda}$ which is dual to a Kleshchev multipartition and
defines an irreducible representation $D^{\underline{\lambda}}$ of an Ariki-Koike algebra of integral type.
We form an integral multisegment ${\mathfrak s}$ by reading the column residual segments and establish an ${\mathcal H_\vtg(r)}$-module isomorphism $V_{\mathfrak s}\cong D^{\underline{\lambda}}$
by taking advantage of the connection between Kleshchev multipartitions and multisegments given in \cite{Va}.
Our key observation is that, for an arbitrary integral segment ${\mathfrak s}$, twisting the irreducible ${\mathcal H_\vtg(r)}$-module $M_{\mathfrak s}$ considered in \cite{Va} by Zelevinsky's involution results in, up to isomorphism, the irreducible module $V_{{\mathfrak s}}$ constructed in \cite{Ro1}; see Lemma~\ref{lem:VsMs}.

Next, we extend the construction, by the affine quantum Schur--Weyl duality developed in \cite{DDF}, to
irreducible representations of the quantum loop algebra ${\rm U}_q(\widehat{\frak{gl}}_n)$.  By introducing Drinfeld polynomials of integral type, we prove that tensoring $D^{\underline{\lambda}}$ with the tensor space gives an irreducible ${\rm U}_q(\widehat{\frak{gl}}_n)$-module whose Drinfeld polynomials having roots associated with the row residual segments of $\underline{\lambda}$. Like the Hecke algebra case, we show that irreducible polynomial ${\rm U}_q(\widehat{\frak{gl}}_n)$-modules are completely determined by those associated with Drinfeld polynomials of integral type. This is another interesting application of affine $q$-Schur algebras and the Schur--Weyl duality. Moreover,
the symmetry between column-reading for multisegments and row-reading for Drinfeld polynomials reveals  a key role played by Kleshchev multipartitions
in the affine quantum  Schur--Weyl duality.

We organise the paper as follows. We first briefly review in \S2 affine Hecke algebras and the classification of their irreducible modules, following Rogawski. We then introduce integral multisegments and their associated standard words. In \S3, we introduce Specht modules of Ariki--Koike algebras and explicitly work out the action of Jucys--Murphy operators on Specht modules. We further interpret the inflation of a Specht module as  an induced ${\mathcal H_\vtg(r)}$-module.
Then, we introduce Ariki--Koike algebras of integral type and their associated Kleshchev multipartitions.   With these preparations, we are ready to get the main results of the paper in the next three sections. Standard Kleshchev multipartitions are introduced in \S4 and are proved to be in one-to-one correspondence with integral multisegments. In \S5, we construct, for every irreducible module of an Ariki--Koike algebra of integral type, the corresponding multisegment when regarded as an irreducible ${\mathcal H_\vtg(r)}$-module. Drinfeld polynomials of integral type are introduced in \S6 where we will extend the one-to-one relation to irreducible representations of affine $q$-Schur algebras. In particular, we prove that the irreducible polynomial representations with Drinfeld polynomials of integral type determine all irreducible polynomial representations. Finally, in the last section, we provide an application to the skew shape representations.

{\it Throughout the paper, all algebras and modules considered are defined over $\mathbb C$.
Let $\mathbb C^*=\mathbb C\setminus\{0\}$ and assume that
 $q\in\mathbb C^*$ is not a root of unity.}

\section{Irreducible representations of affine Hecke algebras}
In this section, we shall give a review on
 the classification of irreducible representations
of affine Hecke algebras,  following
Rogawski~\cite{Ro1} and Zelevinsky~\cite{Ze}. We will focus on the subcategory ${\mathcal H_\vtg(r)}$-{\sf mod}$^{\mathbb Z}$ of ${\mathcal H_\vtg(r)}$-modules with integral weights.

\subsection{The affine Hecke algebra ${\mathcal H_\vtg(r)}$.}
For $r\geq 1$, the affine Hecke algebra ${\mathcal H_\vtg(r)}$ is the associative
algebra over $\mathbb C$ generated by $T_1,\ldots, T_{r-1}$, $X_1,\ldots, X_r$ and $X_1^{-1},\ldots, X_r^{-1}$ subject to the
following relations:
\begin{align*}
(T_i-q^2)(T_i+1)=1,\quad 1\leq i\leq r-1,\\
T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1},\quad T_iT_j=T_jT_i, \quad |i-j|> 1,\\
X_iX_i^{-1}=1=X_i^{-1}X_i,\quad X_iX_j=X_jX_i,\quad 1\leq i,j\leq r,\\
T_iX_iT_i=q^2X_{i+1}, X_jT_i=T_iX_j,\quad j\neq i,i+1.
\end{align*}

Let $\mathfrak S_r$ be the symmetric group on $\{1,2,\ldots,r\}$
which is generated by the simple transpositions $s_i=(i,i+1)$
with $1\leq i\leq r-1$.
Let $\mathcal{H}(r)$ be the subalgebra of ${\mathcal H_\vtg(r)}$ generated by $T_1,\ldots, T_{r-1}$.
Then $\mathcal{H}(r)=\mathcal{H}(\mathfrak{S}_r)$ is the (Iwahori-)Hecke algebra associated to the symmetric group $\mathfrak{S}_r$.
For $w\in\mathfrak S_r$ with a reduced expression $w=s_{i_1}s_{i_2}\cdots s_{i_k}$,
let $T_w=T_{i_1}T_{i_2}\cdots T_{i_k}$.
It is known that $T_w$ is independent of the choice of the reduced expression
of $w$ and moreover $\{T_w|w\in\mathfrak S_r\}$ is a basis of the Hecke algebra $\mathcal H(r)$.
Generally, given a composition $\mu=(\mu_1,\mu_2,\ldots,\mu_{\ell})$ of $r$, that is,
a finite sequence of non-negative integers
$\mu=(\mu_1,\ldots,\mu_\ell)$ such that $|\mu|=\mu_1+\cdots+\mu_\ell=r$,
denote by $\mathfrak S_{\mu}$
the associated Young subgroup and let $\mathcal H(\mu)$ be the subalgebra of $\mathcal H(r)$
corresponding to $\mathfrak S_{\mu}$. Furthermore, let $\mathcal H_{{\!\vartriangle\!}}(\mu)$
be the corresponding subalgebra of  $\mathcal H_{{\!\vartriangle\!}}(r)$, that is, the
subalgebra generated by  $X^{\pm1}_1,\ldots, X^{\pm1}_r$ and $T_i$ for $1\leq i\leq r-1$ such that $s_i\in\mathfrak S_\mu$.
In other words, $\mathcal{H}_{{\!\vartriangle\!}}(\mu)=\langle\mathcal{H}(\mu),X^{\pm1}_1,\ldots, X^{\pm1}_r\rangle\cong
\mathcal{H}_{\!\vartriangle\!}(\mu_1)\otimes\cdots\otimes\mathcal{H}_{\!\vartriangle\!}(\mu_\ell)$.

\subsection{\bf Classification of irreducible ${\mathcal H_\vtg(r)}$-modules.}
Take $\underline{a}=(a_1,a_2,\ldots,a_r)\in(\mathbb C^*)^r$ and set
$$
M(\underline{a})={\mathcal H_\vtg(r)}/J(\underline{a}),
$$
the quotient of ${\mathcal H_\vtg(r)}$ by the left ideal $J(\underline{a})$
generated by $X_1-a_1,\ldots,X_r-a_r$.
Suppose $M$ is a finite dimensional left ${\mathcal H_\vtg(r)}$-module
and $\underline{a}=(a_1,a_2,\ldots,a_r)\in(\mathbb C^*)^r$,
define the $\underline{a}$-weight space
\begin{align*}
M_{\underline{a}}&=\{m\in M\mid(X_k-a_k)m=0, 1\leq k\leq r\}.
\end{align*}

\begin{lem}[{\cite{Ro1}}]\label{lem:Ro1}
The following holds:
\begin{enumerate}
\item Every (finite dimensional) irreducible ${\mathcal H_\vtg(r)}$-module is isomorphic to
a quotient of a certain $M(\underline{a})$.

\item For $\underline{a}=(a_1,a_2,\ldots,a_r)\in(\mathbb C^*)^r$,
the restriction $M(\underline{a})|_{\mathcal H(r)}$ is isomorphic to the left
regular representation of $\mathcal H(r)$
via the linear map
\begin{align*}
\psi:\mathcal{H}(r)&\longrightarrow M(\underline{a})\\
T_w&\longmapsto \overline{T}_w, \quad \text{ for } w\in {\mathfrak S}_r,
\end{align*}
where $\overline{T}_w$ denotes the image of $T_w$ in the quotient $M(\underline{a})$.
\end{enumerate}
\end{lem}

Let $l:\mathfrak S_r\to\mathbb N$ be the length function relative to the $s_i$.
Let $\leq$ be the Bruhat order on $\mathfrak S_r$, and for $w'\leq w$, let
$P_{w',w}(q)$ be the Kazhdan-Lusztig polynomial \cite{KL}.
Define the element $C_w\in\mathcal H(r)$ by
$$
C_w=q^{l(w)}\sum_{w'\leq w}
(-1)^{l(w)-l(w')}q^{-2l(w')}P_{w',w}(q^{-2})T_{w'}.
$$
It is known from \cite{KL} that $\{C_w|w\in\mathfrak S_r\}$ is a basis of $\mathcal H(r)$,
and moreover
\begin{equation} \label{sign}
T_iC_w=-C_w \text{ if }  s_iw<w.
\end{equation}

For a composition $\mu$ of $r$, let $w^0_{\mu}$ be the longest element in the Young subgroup
$\mathfrak S_{\mu}$. Then by \eqref{sign} we obtain
\begin{equation}\label{cell}
T_{\sigma} C_{w^0_{\mu}}=(-1)^{l(\sigma)}C_{w^0_{\mu}}
\end{equation}
for any $\sigma\in \mathfrak S_{\mu}$.
Let
$$
I(\mu)=\mathcal{H}(r) C_{w^0_{\mu}}
$$
be the left ideal of $\mathcal{H}(r)$ generated by the element $C_{w^0_{\mu}}$.
It is known that $I(\mu)$ has dimension given by
\begin{equation}\label{dimImu}
{\rm dim}I(\mu)=\displaystyle\frac{r!}{\prod_i\mu_i!}.
\end{equation}

The cyclic subgroup
$\langle q^2\rangle$ of $\mathbb C^*$ is isomorphic to $\mathbb Z$. We will call the elements
$L_a:=a\langle q^2\rangle$ of the quotient group $\mathbb C^*/\langle q^2\rangle$ ``lines''.
A finite {\it consecutive} subset of a line is called a segment. Thus, a
{\em segment} ${\mathsf s}$ that lies on line $L_a$ is an ordered
sequence of the form ${\mathsf s}=(aq^{2i},aq^{2(i+1)},\ldots,aq^{2j})$ for some
$i,j\in\mathbb Z$ and $i\leq j$.
Given a segment ${\mathsf s}=(aq^{2i},\ldots,aq^{2j})$,
set
$$\widetilde{{\mathsf s}}=(aq^{2j},aq^{2j-2},\ldots,aq^{2i})$$ and $|{\mathsf s}|=j-i+1$ called the length of ${\mathsf s}$.
Denote by $\mathscr C$ the set of segments.
Given a sequence of segments ${\bf s}=({\mathsf s}_1, {\mathsf s}_2,\ldots, {\mathsf s}_t)\in\mathscr C^t$ for $t\geq 1$,
let
\begin{equation}\label{tilde s}
{\bf s}^\vee={\mathsf s}_1\vee\cdots\vee{\mathsf s}_t\;\quad(\text{resp.},
\widetilde{\bf s}^\vee=\widetilde{{\mathsf s}}_1\vee\cdots\vee\widetilde{{\mathsf s}}_t)
\end{equation}
 be the $r$-tuple obtained by juxtaposing the sequences\footnote{Here we changed the notation $\chi({\bf s})$ in \cite{Ro1} to ${\bf s}^\vee$ to be consistent with the notation $\lambda^\vee$ in \S3.1.}
${\mathsf s}_1,\ldots,{\mathsf s}_t$ (resp., $\widetilde{{\mathsf s}}_1,\ldots,\widetilde{{\mathsf s}}_t$)
and set
$$\mu({\bf s})=(|{\mathsf s}_1|, |{\mathsf s}_2|,\ldots, |{\mathsf s}_t|).$$

As an $\mathcal{H}(r)$-module,
$M({\bf s}^\vee)$ can be identified with the left regular representation
by Lemma~\ref{lem:Ro1}(2),
via which any subspace of $\mathcal{H}(r)$ and hence $I(\mu({\bf s}))$
can be identified with a subspace of $M({\bf s}^\vee)$.
To simplify notations, we write
$$I_{{\bf s}}=I(\mu({\bf s})).$$
Let $E_{\mu({\bf s})}$ be the left cell module of $\mathcal{H}(r)$ containing $w^0_{\mu({\bf s})}$.
It is known that as an $\mathcal{H}(r)$-module,  $E_{\mu({\bf s})}$
occurs with multiplicity one in $I_{{\bf s}}$.

Given
${\bf s}=({\mathsf s}_1,\ldots,{\mathsf s}_t),{\bf s}'=({\mathsf s}'_1,\ldots,{\mathsf s}'_t)\in\mathscr C^t$ for some $t\geq 1$,
we define an equivalence relation ${\bf s}\thicksim{\bf s}'$
if ${\bf s}$ and ${\bf s}'$ are equal up to a rearrangement, that is, there exists
$\sigma\in\mathfrak S_t$ such that ${\bf s}'_k={\bf s}_{\sigma(k)}$ for $1\leq k\leq t$.
The equivalent class of ${\bf s}=({\mathsf s}_1,\ldots,{\mathsf s}_t)$, denoted by $\bar{{\bf s}}=\{{\mathsf s}_1,\ldots,{\mathsf s}_t\}$, is  called a {\em multisegment}.
For $t\geq 1$, set
$$
\mathscr C^t_r=\{{\bf s}=({\mathsf s}_1,\ldots,{\mathsf s}_t)\in\mathscr C^t\mid\sum_i|{\mathsf s}_i|=r\}
$$
and let
$$
\mathscr S_r=\bigcup_{t\geq 1}( \mathscr C^t_r/\thicksim)
$$
be the set of multisegments of total length $r$.
Then we have the following results due to Rogawski \cite{Ro1} (cf. Zelevinsky \cite{Ze}).

\begin{prop}[{\cite[Proposition 4.4, Theorem 5.2]{Ro1}}]
\label{prop:Ro2}
Suppose ${\bf s},{\bf s}' \in \mathscr C^t_r$ for $t\geq 1$.
Then
\begin{enumerate}
\item $I_{\bf s}$ is an ${\mathcal H_\vtg(r)}$-submodule of $M({\bf s}^\vee)$.

\item $C_{w^0_{\mu({\bf s})}}\in (I_{\bf s})_{\widetilde{{\bf s}}^\vee}$.

\item As an ${\mathcal H_\vtg(r)}$-module, $I_{\bf s}$ has a unique composition factor $V_{{\bf s}}$
such that as an $\mathcal{H}(r)$-module, $E_{\mu({\bf s})}$ appears as a constituent
of $V_{\bf s}$.

\item $V_{\bf s}\cong V_{{\bf s}'}$ if and only if ${\bf s}\thicksim{\bf s}'$.

\item Every (finite dimensional) irreducible ${\mathcal H_\vtg(r)}$-module is isomorphic to
$V_{\bf s}$ for some ${\bf s}\in\mathscr C^t_r$ with $t\geq 1$. Hence the set $\mathscr S_r$
parametrizes all irreducible ${\mathcal H_\vtg(r)}$-modules.
\end{enumerate}
\end{prop}

By Proposition~\ref{prop:Ro2}(4), isomorphism types of irreducible ${\mathcal H_\vtg(r)}$-modules are indexed by the set $\mathscr S_r$.
Thus, we will use the notation $V_{\mathfrak s}$ to denote a member in the isomorphism class labelled by
${\mathfrak s}\in\mathscr S_r$.
We also note that, for ${\bf s}\sim{\bf s}'$, $I_{\bf s}$ is not necessarily isomorphic to $I_{{\bf s}'}$.
\subsection{\bf The structure of $I_{\bf s}$.}
It is well known that, for any $u\in\mathbb C^*$, there is an evaluation homomorphism
\begin{align}
{\rm ev}_u: {\mathcal H_\vtg(r)}&\twoheadrightarrow\mathcal{H}(r)\notag\\
X_1&\mapsto u,\quad T_i\mapsto T_i \label{ev}
\end{align}
for $1\leq i\leq r-1$.
Given an $\mathcal{H}(r)$-module $N$, let ${\rm ev}^*_u(N)$ denote the inflation of $N$
to an ${\mathcal H_\vtg(r)}$-module via this homomorphism ${\rm ev}_u$. In particular,  the ``trivial'' and ``sign'' modules $\bf 1$ and $-\bf1$ of $\mathcal{H}(r)$ give two 1-dimensional ${\mathcal H_\vtg(r)}$-modules ${\rm ev}^*_u({\bf1})$ and
${\rm ev}^*_u({\bf -1})$.

For a segment ${\mathsf s}=(aq^{2i},\ldots, aq^{2(i+r-1)})$, introduce the notation
 \begin{equation}\label{trivial-sign}
 \mathbb C_{{\mathsf s}}:={\rm ev}^*_{aq^{2i}} ({\bf1})\quad\text{ and }\quad \mathbb C_{\widetilde{{\mathsf s}}}:={\rm ev}^*_{aq^{2(i+r-1)}} ({\bf -1}).\end{equation}
Then, for $1\leq p\leq r-1$
and $1\leq k\leq r$,  $T_p$ acts as $q^2$ and $X_k$ acts as $aq^{2(i+k-1)}$ on $\mathbb C_{\mathsf s}$,
while on $\mathbb C_{\widetilde{\mathsf s}}$ the elements $T_p$ and $X_k$ act as $-1$ and $aq^{2(i+r-k)}$, respectively.
\begin{lem}\label{lem:induced}
Suppose ${\bf s}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t)\in\mathscr C_r^t$ for some $t\geq 1$.
Then
\begin{enumerate}
\item
The one-dimensional space $\mathbb C C_{w^0_{\mu({\bf s})}}$ affords an $\mathcal H_{{\!\vartriangle\!}}(\mu({\bf s}))$-submodule of $I_{\bf s}$ and moreover
$$
\mathbb C C_{w^0_{\mu({\bf s})}}\cong \mathbb C_{\widetilde{{\mathsf s}}_1}\otimes\cdots\otimes\mathbb C_{\widetilde{{\mathsf s}}_t}.
$$

\item As ${\mathcal H_\vtg(r)}$-modules, we have
$$
{\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu({\bf s}))}
(\mathbb C_{\widetilde{{\mathsf s}}_1}\otimes\cdots\otimes\mathbb C_{\widetilde{{\mathsf s}}_t})
\cong I_{\bf s}.
$$
\end{enumerate}
\end{lem}
\begin{proof}
The first part follows from \eqref{cell}, \eqref{trivial-sign} and Proposition~\ref{prop:Ro2}(2).
Then by Frobenius reciprocity,
there exists an ${\mathcal H_\vtg(r)}$-homomorphism
$$\aligned
\psi:
{\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu({\bf s}))}\mathbb C C_{w^0_{\mu({\bf s})}}&\longrightarrow
I_{\bf s}\\
\quad h\otimes C_{w^0_{\mu({\bf s})}}&\longmapsto hC_{w^0_{\mu({\bf s})}},\quad \forall h\in{\mathcal H_\vtg(r)}.\endaligned
$$
On the other hand, $I_{\bf s}$ is generated by $C_{w^0_{\mu({\bf s})}}$
and this implies that $\psi$ is surjective.
Observe that  ${\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu({\bf s}))}\mathbb C C_{w^0_{\mu({\bf s})}}$
and $I_{\bf s}$ have the same dimension given by~\eqref{dimImu},
therefore the second part of the lemma is proved.
\end{proof}

It is known from \cite{Ze} that if $\bar{\bf s}=\overline{{\bf s}'}={\mathfrak s}$ then $I_{\bf s}$ and $I_{{\bf s}'}$ have the same composition factors including the unique $V_{\mathfrak s}$. We now look at a sufficient condition on ${\bf s}$ for $V_{\mathfrak s}$ appearing in the head of $I_{\bf s}$.

\begin{defn}\label{defn:order}
\begin{enumerate}
\item
Let ${\mathsf s}_1$ and ${\mathsf s}_2$ be on the same line $L_a$, say
$
{\mathsf s}_1=(aq^{2i_1},\ldots,aq^{2j_1}),$ and $
{\mathsf s}_2=(aq^{2i_2},\ldots,aq^{2j_2}).
$
We say that ${\mathsf s}_1$ {\em precedes} ${\mathsf s}_2$ and write ${\mathsf s}_1\preccurlyeq{\mathsf s}_2$
if either $j_1<j_2$ or  $j_1=j_2$ and $i_1\geq i_2$ (cf. \cite[Theorem 5.2]{Ro1}).

\item Given ${\bf s}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t)\in\mathscr C^t$ with $t\geq 1$,
we say that ${\bf s}$ is {\em standard} (relative to $\preccurlyeq$) if
there exist $0=k_0< k_1<\ldots< k_p= t$ and disjoint lines $L_{a_1}, L_{a_2},\ldots,L_{a_p}$
with $a_1,\ldots,a_p\in\mathbb C^*$
such that, for each $1\leq i\leq p$, the segments
${\mathsf s}_j$  for $k_{i-1}+1\leq j\leq k_i$ lie
 on the line $L_{a_i}$
and ${\mathsf s}_{k_{i-1}+1}\preccurlyeq{\mathsf s}_{k_{i-1}+2}\preccurlyeq\ldots\preccurlyeq{\mathsf s}_{k_i}$.
In this case, we set ${\bf s}^{(i)}=({\mathsf s}_{k_{i-1}+1},\ldots,{\mathsf s}_{k_i})$
for $1\leq i\leq p$ and write
$$
{\bf s}={\bf s}^{(1)}\cup{\bf s}^{(2)}\cup\cdots\cup{\bf s}^{(p)}.
$$
\item For a segment ${\mathsf s}=(aq^{2i},\ldots,aq^{2j})$ lying on line $L_a$, define
 $${\mathsf s}^{-1}=(a^{-1}q^{-2j},\ldots,a^{-1}q^{-2i})$$
 a segment lying on line $L_{a^{-1}}$.
A sequence ${\bf s}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t)$ of segments is said to be {\em anti-standard}
if ${\bf s}^{-1}:=({\mathsf s}_1^{-1},{\mathsf s}_2^{-1},\ldots,{\mathsf s}_t^{-1})$ is standard.
\end{enumerate}
\end{defn}

Following~\cite[Definition 3.1]{Ro2},
we can define another order ${\mathsf s}_1\preccurlyeq'{\mathsf s}_2$
if either $j_1<j_2$ or  $j_1=j_2$ and $i_1\leq i_2$ for two segments
$
{\mathsf s}_1=(aq^{2i_1},\ldots,aq^{2j_1})$ and $
{\mathsf s}_2=(aq^{2i_2},\ldots,aq^{2j_2})
$
on the same line $L_a$.
Analogous to Definition~\ref{defn:order}(2), one can define the standard sequence of segments with respect to the order $\preccurlyeq'$.

\begin{prop}[{\cite[Theorem 3.3]{Ro2}}]\label{standardprime}
For ${\mathfrak s}\in\mathscr S_r$, if ${\bf s}\in{\mathfrak s}$ is standard with respect to the order $\preccurlyeq'$,
then $I_{\bf s}$ has simple head ${\rm hd}(I_{\bf s})$ and moreover
$$
V_{{\mathfrak s}}\cong{\rm hd}(I_{{\bf s}}).
$$
\end{prop}

\begin{lem}[{\cite[Proposition 5.2]{Ro2}}]\label{notlinked}
Suppose that ${\mathsf s}_1$ and ${\mathsf s}_2$ are on the same line $L_a$ with
${\mathsf s}_1=(aq^{2i_1},\ldots,aq^{2j_1})$, ${\mathsf s}_2=(aq^{2i_2},\ldots,aq^{2j_2}).$
If $j_1=j_2$, then
$$
{\rm ind}^{\mathcal H_{{\!\vartriangle\!}}(r)}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}(\mathbb C_{\widetilde{{\mathsf s}}_1}\otimes\mathbb C_{\widetilde{{\mathsf s}}_2})
\cong {\rm ind}^{\mathcal H_{{\!\vartriangle\!}}(r)}_{\mathcal H_{{\!\vartriangle\!}}(\mu^o)}(\mathbb C_{\widetilde{{\mathsf s}}_2}\otimes\mathbb C_{\widetilde{{\mathsf s}}_1}),
$$
that is, $I_{({\mathsf s}_1,{\mathsf s}_2)}\cong I_{({\mathsf s}_2,{\mathsf s}_1)}$,
where $\mu=(|{\mathsf s}_1|, |{\mathsf s}_2|)$, $\mu^o=(|{\mathsf s}_2|,|{\mathsf s}_1|)$ and $r=|{\mathsf s}_1|+|{\mathsf s}_2|$.

\end{lem}

\begin{lem}\label{lem:twoorder}
Let ${\bf s}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t)\in\mathscr C_r^t$ be standard relative to $\preccurlyeq$
with ${\mathsf s}_1,\ldots,{\mathsf s}_t$ being on the same line.
Reorder the segments in ${\bf s}$ such that ${\bf s}'=({\mathsf s}_{i_1},{\mathsf s}_{i_2},\ldots,{\mathsf s}_{i_t})$ is standard with respect to $\preccurlyeq'$.
Then
$$
I_{{\bf s}}\cong I_{{\bf s}'}.
$$
\end{lem}
\begin{proof}
Suppose ${\mathsf s}_1,\ldots,{\mathsf s}_t$ are on the line $L_a$ and ${\mathsf s}_k=(aq^{2i_k},\ldots,aq^{2j_k})$ for $1\leq k\leq t$.
Observe that there exist unique $0=k_0<k_1<k_2<\cdots<k_d=t$ such that
$j_{k_{b-1}+1}=j_{k_{b-1}+2}=\cdots=j_{k_b}$
and $i_{k_{b-1}+1}\geq i_{k_{b-1}+2}\geq\cdots\geq i_{k_b}$ for each $1\leq b\leq d$
and $j_{k_{e}}<j_{k_{e}+1}$ for $1\leq e\leq d-1$ .
Then ${\bf s}'$ has the form given by
$$
{\bf s}'=({\mathsf s}_{k_1},{\mathsf s}_{k_1-1},\ldots,{\mathsf s}_1,{\mathsf s}_{k_2},{\mathsf s}_{k_2-1}\cdots,{\mathsf s}_{k_1+1},\ldots,{\mathsf s}_{k_d},{\mathsf s}_{k_d-1},\ldots,{\mathsf s}_{k_{d-1}+1}).
$$
For $1\leq b\leq d$, set
\begin{align*}
I_{b}&={\rm ind}^{\mathcal H_{{\!\vartriangle\!}}(r_b)}_{\mathcal H_{{\!\vartriangle\!}}(\mu^{(b)})}
\mathbb C_{\widetilde{{\mathsf s}}_{k_{b-1}+1}}\otimes\cdots\otimes\mathbb C_{\widetilde{{\mathsf s}}_{k_b-1}}\otimes \mathbb C_{\widetilde{{\mathsf s}}_{k_b}}, \\
I'_{b}&={\rm ind}^{\mathcal H_{{\!\vartriangle\!}}(r_b)}_{\mathcal H_{{\!\vartriangle\!}}((\mu^{(b)})^o)}
\mathbb C_{\widetilde{{\mathsf s}}_{k_b}}\otimes\mathbb C_{\widetilde{{\mathsf s}}_{k_b-1}}\otimes\cdots\otimes \mathbb C_{\widetilde{{\mathsf s}}_{k_{b-1}+1}},
\end{align*}
where $\mu^{(b)}=(|{\mathsf s}_{k_{b-1}+1}|,\ldots,|{\mathsf s}_{k_b}|)$, $(\mu^{(b)})^o=(|{\mathsf s}_{k_b}|,\ldots,|{\mathsf s}_{k_{b-1}+1}|)$ and $r_b=|\mu^{(b)}|$.
By Lemma~\ref{notlinked}, we obtain
$$
I_b\cong I'_b
$$
for $1\leq b\leq d$.
Hence, by Lemma~\ref{lem:induced}(2) and associativity of induction, one can deduce that
$$
I_{{\bf s}}\cong {\rm ind}^{\mathcal H_{{\!\vartriangle\!}}(r)}_{\mathcal H_{{\!\vartriangle\!}}(\mu)} I_1\otimes\cdots\otimes I_d
\cong {\rm ind}^{\mathcal H_{{\!\vartriangle\!}}(r)}_{\mathcal H_{{\!\vartriangle\!}}(\mu)} I'_1\otimes\cdots\otimes I'_d\cong I_{{\bf s}'},
$$
where $\mu=(|\mu^{(1)}|,\ldots,|\mu^{(d)}|)$.
\end{proof}

\begin{prop}\label{prop:Ro3}
For ${\mathfrak s}\in\mathscr S_r$, if ${\bf s}\in{\mathfrak s}$ is standard (with respect to $\preccurlyeq$),
then $I_{\bf s}$ has simple head ${\rm hd}(I_{\bf s})$ and
$$
V_{{\mathfrak s}}\cong{\rm hd}(I_{{\bf s}}).
$$
\end{prop}
\begin{proof}
Suppose ${\bf s}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t)$ is standard with the decomposition ${\bf s}={\bf s}^{(1)}\cup{\bf s}^{(2)}\cup\cdots\cup{\bf s}^{(p)}$
in the sense of Definition~\ref{defn:order}(2).
Reorder the segments in ${\bf s}^{(i)}$ such that $({\bf s}^{(i)})'$ is standard with respect to the order $\preccurlyeq'$ for $1\leq i\leq p$.
Then ${\bf s}':=({\bf s}^{(1)})'\cup({\bf s}^{(2)})'\cup\cdots\cup({\bf s}^{(p)})'$ is standard with respect to the order $\preccurlyeq'$.
Meanwhile by Lemma~\ref{lem:induced}(2) we have
$$
I_{{\bf s}}\cong {\rm ind}^{\mathcal H_{{\!\vartriangle\!}}(r)}_{\mathcal H_{{\!\vartriangle\!}}(\mu)} I_{{\bf s}^{(1)}}\otimes\cdots\otimes I_{{\bf s}^{(p)}}\quad\text{and}\quad
I_{{\bf s}'}\cong {\rm ind}^{\mathcal H_{{\!\vartriangle\!}}(r)}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}I_{({\bf s}^{(1)})'}\otimes\cdots\otimes I_{({\bf s}^{(p)})'}.
$$
Thus, Lemma~\ref{lem:twoorder} implies $I_{{\bf s}}\cong I_{{\bf s}'}$ and so the result follows from  Proposition~\ref{standardprime}.
\end{proof}

Note that this result is not true in general for non-standard ${\bf s}$.
From now on, by standard sequence of segments we mean those satisfying the property given in
Definition~\ref{defn:order}(2) where the order $\preccurlyeq$ is used.

\subsection{The category ${\mathcal H_\vtg(r)}$-${\sf mod}^{\mathbb Z}$,
integral multisegments and standard words.}
 Let ${\mathcal H_\vtg(r)}$-${\sf mod}^{\mathbb Z}$ denote the full subcategory of
finite dimensional left ${\mathcal H_\vtg(r)}$-modules on which
the eigenvalues of $X_1,\ldots,X_r$ are powers of $q^2$ (i.e., the weights are integral).
It is known (cf. \cite{Kl, Gr}) that to understand irreducible ${\mathcal H_\vtg(r)}$-modules,
it suffices to study those irreducible modules in ${\mathcal H_\vtg(r)}$-${\sf mod}^{\mathbb Z}$.
More precisely, given standard ${\bf s}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t)\in\mathscr C^t_r$ with
$t\geq 1$, and assume
$
{\bf s}={\bf s}^{(1)}\cup{\mathsf s}^{(2)}\cup\cdots\cup{\mathsf s}^{(p)}
$
in the sense of Definition~\ref{defn:order}(2).
Then it is known \cite[Lemma 6.1.2]{Kl} that ${\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}(V_{{\bf s}^{(1)}}\otimes\cdots\otimes V_{{\bf s}^{(p)}})$
is irreducible and hence by Proposition~\ref{prop:Ro3} the following holds
\begin{equation}\label{gen&int}
V_{\bf s}\cong{\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}(V_{{\bf s}^{(1)}}\otimes\cdots\otimes V_{{\bf s}^{(p)}}),
\end{equation}
where $\mu=(\mu_1,\ldots,\mu_p)$ with $\mu_i=\sum_j|{\mathsf s}^{(i)}_j|$.
Therefore, it is reduced to study the irreducible modules $V_{{\bf s}}$ for
${\bf s}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t)$
with ${\mathsf s}_1,\ldots,{\mathsf s}_t$ being on the same line.

 For $a\in\mathbb C^*$, let $\sigma_a$ be the automorphism of ${\mathcal H_\vtg(r)}$ given by
\begin{equation}\label{sigma}
\sigma_a(T_i)=T_i,\quad \sigma_a(X_k)=aX_k
\end{equation}
for $1\leq i\leq r-1$ and $1\leq k\leq r$.
For an ${\mathcal H_\vtg(r)}$-module $V$, we can twist the action with $\sigma_a$ to get a
new module denoted by $V^{\sigma_a}$.
Given an arbitrary segment ${\mathsf s}=(zq^{2i},\ldots,zq^{2j})$ and $a\in \mathbb C^*$, set
\begin{equation}\label{s-sigma}
{\mathsf s}^{\sigma_a}=(azq^{2i},\ldots,azq^{2j}).
\end{equation}
Similarly we can define ${\bf s}^{\sigma_a}$ and ${\mathfrak s}^{\sigma_a}$
for ${\bf s}\in\mathscr C_r^t$ and ${\mathfrak s}\in\mathscr S_r$.

\begin{lem}\label{lem:Vs-sigma}
Let $a\in\mathbb C^*$ and ${\mathfrak s}\in\mathscr S_r$.
Then
$$
V_{{\mathfrak s}}^{\sigma_a}\cong V_{{\mathfrak s}^{\sigma_a}}.
$$
\end{lem}
\begin{proof}
Take ${\bf s}=({\mathsf s}_1,\ldots,{\mathsf s}_t)\in{\mathfrak s}$.
Note that $\mathbb C_{\widetilde{{\mathsf s}}_k}^{\sigma_a}\cong\mathbb C_{\widetilde{{\mathsf s}_k^{\sigma_a}}}$ for $1\leq k\leq t$ by~\eqref{trivial-sign}
and hence by Lemma~\ref{lem:induced} we obtain
\begin{align*}
I^{\sigma_a}_{{\bf s}}&\cong \big({\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu({\bf s}))}
(\mathbb C_{\widetilde{{\mathsf s}}_1}\otimes\cdots\otimes\mathbb C_{\widetilde{{\mathsf s}}_t})\big)^{\sigma_a}\\
&\cong{\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu({\bf s}))}
(\mathbb C^{\sigma_a}_{\widetilde{{\mathsf s}}_1}\otimes\cdots\otimes\mathbb C^{\sigma_a}_{\widetilde{{\mathsf s}}_t})\; (\text{by Frobenius reciprocity})\\
&\cong {\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu({\bf s}))}
(\mathbb C_{\widetilde{{\mathsf s}^{\sigma_a}_1}}\otimes\cdots\otimes\mathbb C_{\widetilde{{\mathsf s}^{\sigma_a}_t}})\\
&\cong I_{{\bf s}^{\sigma_a}}.
\end{align*}
Therefore, $V^{\sigma_a}_{{\bf s}}$ is a composition factor of $I_{{\bf s}^{\sigma_a}}$.
Meanwhile, since $V^{\sigma_a}_{{\bf s}}\cong V_{{\bf s}}$ as $\mathcal{H}(r)$-modules and hence
$E_{\mu({\bf s}^{\sigma_a})}=E_{\mu({\bf s})}$ appears as a constituent of $V^{\sigma_a}_{{\bf s}}$.
By Proposition~\ref{prop:Ro2}, we get $V^{\sigma_a}_{{\bf s}}\cong V_{{\bf s}^{\sigma_a}}$
and this implies $V^{\sigma_a}_{{\mathfrak s}}\cong V_{{\mathfrak s}^{\sigma_a}}$.
\end{proof}

Now assume that ${\bf s}=({\mathsf s}_1,\ldots,{\mathsf s}_t)$ with ${\mathsf s}_k$ being on the same line $L_a$ for some $a\in\mathbb C^*$.
Then it is easy to check that ${\mathsf s}_k^{\sigma_{a^{-1}}}$ is integral for $1\leq k\leq n$
and moreover by Lemma~\ref{lem:Vs-sigma}
$$
V_{{\bf s}}\cong (V_{{\bf s}^{\sigma_{a^{-1}}}})^{\sigma_a}.
$$
Hence, to understand irreducible modules $V_{{\bf s}}$, it is enough to
consider those ${\bf s}\in\mathscr C_r^t$ with $t\geq1$
such that $\bar{{\bf s}}\in\mathscr S^{\mathbb Z}_r$, where
$$\mathscr S^{\mathbb Z}_{r}=\{\bar{\bf s}\in\mathscr S_r\mid {\bf s} \text{ lies on }L_1\}$$
is the set of all {\it integral multisegments} of total length $r$.

Define $\mathscr C_r^{\mathbb Z}$ similarly to be the set of {\it standard sequences of integral segments} of total length $r$.

For any multisegment $\mathfrak s\in \mathscr S^{\mathbb Z}_r$, there exists a unique standard sequence ${\bf s}$ such that ${\bf s}\in\mathfrak s$. Thus, the map
\begin{equation}
\aligned
 \flat_1:\
 mathscr C_r^{\mathbb Z}&\longrightarrow \mathscr S_r^{\mathbb Z}\\
 {\bf s}&\longmapsto\bar{\bf s}\endaligned
\end{equation}
is a bijection.
  We now use this fact to describe integral multisegments in terms of
 standard words.

Consider the set $\mathcal W$ of nonempty words in the alphabet $\mathbb Z^2=\{{j\choose i}\mid i,j\in\mathbb Z\}$. We always identify such a word ${j_1\choose i_1}\cdots{j_t\choose i_t}$ as ${{j_1\ldots j_t}\choose{i_1\ldots i_t}}$ and define
$\bigl|{{j_1\ldots j_t}\choose{i_1\ldots i_t}}\bigr|:=\sum_{k=1}^t(j_k-i_k+1).$
  Let
 $$\mathcal W(r)=\{w={{j_1\ldots j_t}\choose{i_1\ldots i_t}}\in\mathcal W\mid i_k\leq j_k, 1\leq k\leq t,  |w|=r\}.$$
Clearly, each word in $\mathcal W(r)$ defines a segment sequence on the line $L_1$.  A word
${{j_1\ldots j_t}\choose{i_1\ldots i_t}}$ in $\mathcal W(r)$ is called a {\it standard word} if $j_1\leq j_2\leq\ldots\leq j_{t}$ and $i_k\geq i_{k+1}\text{ whenever }j_k=j_{k+1}$.
 Let $\mathcal W^s(r)$ be the set of standard words in $\mathcal W(r)$.

Clearly, by definition, there is a bijection
\begin{equation}\label{flat2}
\flat_2:\mathcal W^s(r)\longrightarrow\mathscr C_r^{\mathbb Z}
\end{equation}
sending a standard word ${{j_1j_2\ldots j_t}\choose{i_1i_2\ldots i_t}}$ to a standard sequence ${\bf s}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t)$, where,  for $1\leq k\leq t$,
${\mathsf s}_k=(q^{2i_k},q^{2(i_k+1)},\ldots,q^{2j_k})$. This induces a bijection
\begin{equation}\label{flat}
\aligned
\flat=\flat_1\circ\flat_2:\mathcal W^s(r)&\longrightarrow\mathscr S^{\mathbb Z}_r\\
w&\longmapsto \overline{\flat_2(w)}.\endaligned
\end{equation}

We remark that the reduction to the integral case allows us to link irreducible modules $V_{\mathfrak s}$ for $\mathfrak s\in\mathscr S^{\mathbb Z}_r$ with irreducible modules over the Ariki-Koike algebras of integral type considered in \S3.5.

\section{Ariki-Koike algebras and Specht modules}
In this section, we shall recall the construction
of Specht modules for Ariki-Koike algebras (cf. \cite{DJM, DR2}), compute the action of Jucys--Murphy elements on Specht modules, and introduce Kleshchev multipartitions for Ariki--koike algebras of integral type.

\subsection{Basics on multipartitions.}\label{subsec:notation}
Recall that a composition $\mu$ of $r$ is a finite sequence of non-negative integers
$\mu=(\mu_1,\ldots,\mu_\ell)$ such that $|\mu|=\mu_1+\cdots+\mu_\ell=r$.
If, in addition, $\mu_1\geq\mu_2\geq\cdots\geq\mu_\ell$,
then $\mu$ is called a partition of $r$ and write $\mu\vdash r$.
Denote by $\mathcal P(r)$ the set of partitions of $r$.
It is known that a partition $\lambda$ can be identified with its Young diagram
which is formally defined to be set of points $\{(i,j)|1\leq i\leq\ell(\lambda), 1\leq j\leq\lambda_i\}
\subseteq\mathbb N^2$ and the elements $(i,j)$ are called {\em nodes}. Here $\ell(\lambda)$ denotes the number of nonzero parts of $\lambda$.
Given a composition $\mu$, its conjugate $\mu'=(\mu_1',\mu_2',\ldots)$ is the partition
defined via $\mu_i'=\sharp\{k|\mu_k\geq i\}$ for $i\geq 1$.

Suppose $\lambda\in\mathcal P(r)$. A $\lambda$-tableau is a labeling of the nodes in the Young diagram $\lambda$
with integers $1, 2, \ldots, r$.
Let $t^{\lambda}$ (resp. $t_{\lambda}$) be the $\lambda$-tableau in which the numbers
$1,2,\ldots, r$ appear in order along successive rows (resp. column).
For example, if $\lambda=(3,2)$, then
$$
t^{\lambda}=\young(123,45),\quad
t_{\lambda}=\young(135,24).
$$

Observe that the symmetric group $\mathfrak S_r$ naturally acts on $\lambda$-tableaux for $\lambda\in\mathcal P(r)$.
Given $\lambda\in\mathcal P(r)$, let $w_{\lambda}$ be the permutation
satisfying
$$
w_{\lambda}t^{\lambda}=t_{\lambda}.
$$
Observe that
\begin{equation}\label{wlainverse}
w_{\lambda}^{-1}=w_{\lambda'}.
\end{equation}

An $m$-tuple of partitions $\underline{\lambda}=(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(m)})$
with $|\lambda^{(1)}|+\cdots+|\lambda^{(m)}|=r$ is called
a multipartition of $r$. Call $\underline{\lambda}$ a {\it sincere} multipartition if $|\lambda^{(i)}|>0$ for all $i$.
Denote by $\mathcal P_m(r)$ the set of multipartitions
$\underline{\lambda}=(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(m)})$ of $r$.
Given $\underline{\lambda}\in\mathcal P_m(r)$, set
$$
\underline{\lambda}'=(\lambda^{(m)'},\ldots,\lambda^{(2)'},\lambda^{(1)'}).
$$
By concatenating the components of $\underline{\lambda}$, the resulting composition of $r$
will be denoted by
$$
\underline{\lambda}^{\vee}=\lambda^{(1)}\vee\lambda^{(2)}\vee\cdots\vee\lambda^{(m)}.
$$
Define
$$
[\underline{\lambda}]=[a_0,a_1,\ldots,a_m]
$$
such that $a_0=0, a_k=\sum^k_{j=1}|\lambda^{(j)}|$ for $1\leq k\leq m$.
Then all $[\underline{\lambda}]$ form the set
$$
\Lambda[m,r]:=\{{{\bf a}}=[a_0,a_1,\ldots,a_m]~|0=a_0\leq a_1\leq\cdots\leq a_m=r, a_i\in\mathbb Z, 0\leq i\leq m\}.
$$
Note that this set is often associated with the set of compositions of $r$ into $m$ parts:
\begin{equation}\label{Lanr}
\Lambda(m,r):=\{(a_1-a_0,a_2-a_1,\ldots,a_m-a_{m-1})\mid[a_0,a_1,\ldots,a_m]\in\Lambda[m,r]\}.
\end{equation}

For ${{\bf a}}=[a_0,a_2,\ldots,a_m]\in\Lambda[m,r]$, let
$$
{\bf a}'=[r-a_m,r-a_{m-1},\ldots,r-a_1,r-a_0]
$$ and
define $w_{{{\bf a}}}\in\mathfrak S_r$ to be the element satisfying
$$
w_{{\bf a}}(a_{j-1}+k)=r-a_j+k
$$
for $1\leq k\leq a_j-a_{j-1}$ and $1\leq j\leq m$ with $a_{j-1}<a_j$.
For example, if ${\bf a}=[0,4,8,9]$, then $w_{{{\bf a}}}$ has the form
$$
w_{{\bf a}}=\left(
\begin{array}{l}
1~ 2~ 3~ 4\quad 5~ 6~ 7~ 8\quad 9\\
\underline{6~ 7~ 8~ 9}\quad \underline{2~ 3~ 4~ 5}\quad \underline{1}
\end{array}
\right).
$$
In particular, if $\underline{\lambda}=(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(m)})$ is a multipartition
of $r$, then
\begin{equation} \label{lem:wudla}
w_{[\underline{\lambda}]}(|\lambda^{(1)}|+\cdots+|\lambda^{(j-1)}|+b)
=|\lambda^{(j+1)}|+\cdots+|\lambda^{(m)}|+b
\end{equation}
for $1\leq b\leq |\lambda^{(j)}|$ and $1\leq j\leq m$ with $|\lambda^{(j)}|\geq 1$.

A multipartition $\underline{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})$
can also be identified with its Young diagram
which is formally defined to be set
$$\{(i,j,k)|1\leq i\leq\ell(\lambda^{(k)}), 1\leq j\leq\lambda_i^{(k)}, 1\leq k\leq m\}
\subseteq\mathbb N^3.$$ The elements $(i,j,k)$ are called nodes.
For example, the $3$-partition $\underline{\lambda}=((3,1), (2,2), (1))$
is identified with
$$
\underline{\lambda}=\Big(~\young(\,\,\,,\,),\quad \young(\,\,,\,\,),\quad \young(\,)~\Big).
$$
Similarly, a $\underline{\lambda}$-tableau is a labeling of the nodes in the diagrams
with numbers $1,2,\ldots, r$.
Let $t^{\underline{\lambda}}$ be the $\underline{\lambda}$-tableau in
which the numbers $1,2,\ldots, r$ appear
in order along successive rows in the first diagram of $\underline{\lambda}$,
then in the second diagram and so on.
Meanwhile let $t_{\underline{\lambda}}$ be the $\underline{\lambda}$-tableau
in which the numbers $1,2,\ldots, r$ appear
in order along successive columns in the last diagram of $\underline{\lambda}$,
then in the second last diagram and so on.
For example, in the case $\underline{\lambda}=((3,1), (2,2), (1))$,
we have
\begin{equation} \label{tableau}
\aligned
t^{\underline{\lambda}}&=\Big(~\young(123,4),\quad\young(56,78),\quad\young(9)~\Big),\\
t_{\underline{\lambda}}&=\Big(~\young(689,7),\quad\young(24,35),\quad\young(1)~\Big).
\endaligned
\end{equation}
For a multipartition $\underline{\lambda}\in\mathcal P_m(r)$,
let $w_{\underline{\lambda}}\in\mathfrak S_r$ be the element satisfying
$$
w_{\underline{\lambda}} t^{\underline{\lambda}}=t_{\underline{\lambda}}.
$$
Similarly, we have
\begin{equation}\label{wudla-1}
w_{\underline{\lambda}}^{-1}=w_{\underline{\lambda}'}.
\end{equation}

\subsection{\bf The Ariki-Koike algebra $\mc H_{\und{u}}(r)$ and Jucys-Murphy elements.}
Recall $q\in\mathbb C^*$.
Let $\underline{u}=(u_1, u_2,\ldots, u_m)\in\mathbb C^m$ with $m\geq 1$.
The {\em Ariki-Koike} algebra $\mc H_{\und{u}}(r)$ is
the associative algebra over $\mathbb C$ generated by
$T_0, T_1,\ldots, T_{r-1}$ subject to the relations:
\begin{align*}
(T_i-q^2)(T_i+1)&=0,\quad 1\leq i\leq r-1, \\
T_iT_{i+1}T_i&=T_{i+1}T_iT_{i+1},\quad 1\leq i\leq r-2,\\
\quad T_iT_j&=T_jT_i,\quad 1\leq i,j\leq r-1,  |i-j|> 1, \\
(T_0-u_1)(T_0-u_2)&\cdots (T_0-u_m)=0,\\
T_0T_1T_0T_1&=T_1T_0T_1T_0.\\
\end{align*}

Observe that the Hecke algebra $\mathcal{H}(r)$ can be viewed as a special case of Ariki-Koike algebras
obtained by take $m=1$. The {\em Jucys-Murphy} elements $J_k$ $(1\leq k\leq r)$ of $\mathcal{H}(r)$ are defined to be
$J_1=1$ and
\begin{equation}\label{JM1}
J_k=q^{2(1-k)}T_{k-1}\cdots T_1T_1\cdots T_{k-1}
\end{equation}
for $k\geq 2$.
It is known that these elements commute with each other and
they play an important role in the representation theory of $\mathcal{H}(r)$ (cf. \cite{DJ2}).
These elements have been generalized to the Ariki-Koike algebra $\mc H_{\und{u}}(r)$
given by:
\begin{equation}\label{JM2}
L_1=T_0,\quad L_k=q^{2(1-k)}T_{k-1}\cdots T_1T_0T_1\cdots T_{k-1}=q^{-2}T_{k-1}L_{k-1}T_{k-1}
\end{equation}
for $2\leq k\leq r$.
Again, it is easy to check that these elements commute with each other.
By taking advantage of these elements, Ariki-Koike algebras can be regarded as
quotients of affine Hecke algebras as follows, generalising the evaluation map \eqref{ev}.

If $u_1,\ldots,u_m$ are all non-zero,\footnote{This guarantees that all $L_k$ are invertible since all $X_k$ are invertible in ${\mathcal H_\vtg(r)}$.}
there exists a surjective homomorphism
\begin{align}\label{pi}
\varpi_{\underline{u}}: {\mathcal H_\vtg(r)}&\twoheadrightarrow\mc H_{\und{u}}(r)\notag\\
X_k&\mapsto L_k,\quad T_i\mapsto T_i
\end{align}
for $1\leq k\leq r$ and $1\leq i\leq r-1$ and
$\varpi_{\underline{u}}$ induces the algebra isomorphism
$$
\mc H_{\und{u}}(r)\cong{\mathcal H_\vtg(r)}/\langle (X_1-u_1)(X_1-u_2)\cdots(X_1-u_m)\rangle.
$$
This means any $\mc H_{\und{u}}(r)$-module $M$ can be {\it inflated } to an ${\mathcal H_\vtg(r)}$-module in this case. We will use the same letter to denote its inflation.

A direct check of relations shows that there exists an anti-automorphism $\tau$
on $\mc H_{\und{u}}(r)$ defined by
\begin{align}
\tau:\mc H_{\und{u}}(r)&\longrightarrow\mc H_{\und{u}}(r), \label{tau}\\
T_k&\longmapsto T_k\notag
\end{align}
for $0\leq k\leq r-1$.

\subsection{\bf Specht modules for $\mc H_{\und{u}}(r)$.}
We shall mainly follow the constructions of Specht modules
for the Hecke algebra $\mathcal{H}(r)$
and the Ariki-Koike algebras $\mc H_{\und{u}}(r)$ established in \cite{DJ1,DJ2,DR1,DR2},
where right modules are considered.
Observe that by applying the anti-automorphism $\tau$,
the definitions and statements therein
can be reformulated for left modules as follows.

Given a composition $\mu=(\mu_1,\mu_2,\ldots,\mu_{\ell})$ of $r$, set
\begin{align*}
x_{\mu}:=\sum_{\sigma\in \mathfrak S_{\mu}}T_{\sigma},\quad
y_{\mu}:=\sum_{\sigma\in\mathfrak S_\mu}(-q^2)^{l(\sigma)}T_{\sigma}.
\end{align*}
It is easy to check that
\begin{align}\label{xylafin}
T_\sigma x_\mu=x_{\mu}T_\sigma=q^{2l(\sigma)}x_{\mu},\quad
T_\sigma y_\mu=y_{\mu}T_\sigma=(-1)^{l(\sigma)}y_{\mu}
\end{align}
for $\sigma\in\mathfrak S_\mu$.
For $\lambda\in\mathcal P(r)$, define
$$
z_{\lambda}:=y_{\lambda'}T_{w_{\lambda'}}x_{\lambda}.
$$
The {\em Specht} module \cite{DJ1} associated to a partition $\lambda$ of $r$ is the left
ideal $S^{\lambda}$ of $\mathcal{H}(r)$ generated by $z_{\lambda}$, that is,
\begin{equation}\label{Specht1}
S^{\lambda}=\mathcal{H}(r) z_{\lambda}.
\end{equation}
We remark that by \eqref{wlainverse} our $z_\lambda$ is exactly the element obtained by apply
$\tau$ to the elements also denoted by $z_\lambda$ in \cite[P.~34]{DJ1},
where right modules are studied.

Following \cite{DR2}, we shall introduce the notion of Specht modules
for the Ariki-Koike algebras $\mc H_{\und{u}}(r)$ in the rest of this section.
For $u\in\mathbb C$ and a positive integer $k$, let
$$
\pi_0(u)=1,\quad \pi_k(u)=(L_1-u)(L_2-u)\cdots(L_k-u).
$$
Then, for ${\bf a}=[a_0,a_1,\ldots,a_m]\in\Lambda[m,r]$, we define (cf. \cite{DJM, DR1,GL})
\begin{align*}
\pi_{{\bf a}}=&\pi_{a_1}(u_2)\cdots\pi_{a_{m-1}}(u_m),\quad
\widetilde{\pi}_{{\bf a}}=\pi_{a_1}(u_{m-1})\cdots\pi_{a_{m-1}}(u_1),\\
v_{{\bf a}}=&\widetilde{\pi}_{{\bf a}'}T_{w_{{\bf a}'}}\pi_{{\bf a}}.
\end{align*}

For a multipartition $\underline{\lambda}$ of $r$, let
\begin{align*}
x_{\underline{\lambda}}=&\pi_{[\underline{\lambda}]}x_{\underline{\lambda}^\vee}=x_{\underline{\lambda}^\vee}\pi_{[\underline{\lambda}]},\quad
y_{\underline{\lambda}}=\widetilde{\pi}_{[\underline{\lambda}]}y_{\underline{\lambda}^\vee}=y_{\underline{\lambda}^\vee}\widetilde{\pi}_{[\underline{\lambda}]},\\
z_{\underline{\lambda}}=&y_{\underline{\lambda}'}T_{w_{\underline{\lambda}'}}x_{\underline{\lambda}}.
\end{align*}
Again, the element $z_{\underline{\lambda}}$ defined here is the one obtained by applying the anti-automorphism
$\tau$ to the one also denoted by $z_{\underline{\lambda}}$ constructed in \cite[Definition 2.1]{DR2}
since $\tau$ satisfies
$$
\tau(T_{w_{\underline{\lambda}}})=T_{w_{\underline{\lambda}'}}
$$
due to~\eqref{wudla-1}.
By~\eqref{xylafin}, we get
\begin{equation}\label{Tzudla}
T_{\sigma}z_{\underline{\lambda}}=(-1)^{l(\sigma)}z_{\underline{\lambda}}
\end{equation}
for $\sigma\in\mathfrak S_{(\underline{\lambda}')^\vee}$.
\begin{defn}
The Specht module $S^{\underline{\lambda}}$ associated to a multipartition $\underline{\lambda}\in\mathcal P_m(r)$
is defined to be
$$
S^{\underline{\lambda}}=\mc H_{\und{u}}(r) z_{\underline{\lambda}},
$$
that is, the left ideal of $\mc H_{\und{u}}(r)$ generated by $z_{\underline{\lambda}}$.
\end{defn}

\begin{lem}[{\cite[Proposition 3.1]{DR1}}] \label{lem:T&JM}
Suppose ${\bf a}\in\Lambda[m,r]$ and let $\mathfrak S_{{\bf a}}$ be the Young subgroup of $\mathfrak S_r$
associated to the composition $(a_1, a_2-a_1,\ldots, a_m-a_{m-1})$ of $r$. Then
\begin{enumerate}
\item $T_k v_{{\bf a}}=v_{{\bf a}} T_{w_{{\bf a}}^{-1}(k)} $
for $1\leq k\leq r$ such that $s_k\in\mathfrak S_{{\bf a}}$.

\item $L_k v_{{\bf a}}=u_jv_{{\bf a}}$ for $1\leq k\leq r$
satisfying $k=r-a_j+1$ for some $j$ with $a_{j-1}<a_j$.
\end{enumerate}
\end{lem}
Observe that, if ${\bf a}=[\underline{\lambda}]$, then Lemma~\ref{lem:T&JM} implies  (cf. \eqref{lem:wudla})
\begin{align}
L_{r-a_j+1}v_{[\underline{\lambda}]}
=&u_jv_{[\underline{\lambda}]}, \label{Lvla}\\
T_{r-a_j+b}v_{[\underline{\lambda}]}
=&v_{[\underline{\lambda}]}T_{a_{j-1}+b}\label{Tvla}
\end{align}
for $1\leq b\leq |\lambda^{(j)}|$ and $1\leq j\leq m$ with $|\lambda^{(j)}|\geq 1$.

We remark that the Specht module $S^{\underline{\lambda}}$ defined above
is known~\cite[Theorem 2.9]{DR2} to be isomorphic to those cell modules associated with $\underline{\lambda}'$ defined in \cite{DJM} up to a twist by an automorphism
of $\mc H_{\und{u}}(r)$.
The following lemma will be used later on.
\begin{lem}[{\cite[Corollary 2.3]{DR2}}]\label{lem:Specht}
Suppose that $\underline{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})$ is a multipartition of $r$
with  $|\lambda^{(k)}|=r_k$ for $1\leq k\leq m$.
Then
\begin{equation}\label{zv}
z_{\underline{\lambda}}=v_{[\underline{\lambda}]}(z_{\lambda^{(1)}}\otimes\cdots \otimes z_{\lambda^{(m)}})
=(z_{\lambda^{(m)}}\otimes\cdots \otimes z_{\lambda^{(1)}})v_{[\underline{\lambda}]},
\end{equation}
where $z_{\lambda^{(1)}}\otimes\cdots \otimes z_{\lambda^{(m)}}
\in\mathcal H(r_1)\otimes\cdots\otimes\mathcal H(r_m)\subseteq\mathcal{H}(r)$.
Moreover, as $\mathcal{H}(r)$-modules, the following holds
\begin{equation}\label{slaind}
S^{\underline{\lambda}}|_{\mathcal{H}(r)}\cong {\rm ind}^{\mathcal{H}(r)}_{\mathcal H(r_m)\otimes\cdots\otimes\mathcal H(r_1)}
(S^{\lambda^{(m)}}\otimes \cdots\otimes S^{\lambda^{(1)}}).
\end{equation}
\end{lem}

\subsection{\bf The action of Jucys-Murphy elements on $S^{\underline{\lambda}}$.}  For later use, we now study the action of Jucys-Murphy elements on
Specht modules for Ariki-Koike algebras.

In the Hecke algebra $\mathcal{H}(r)$, there is another set of interesting elements
(also called Jucys-Murphy elements in
some contexts (cf. \cite{DJ1}) )
defined by
\begin{align*}
J_1'&=0,\\
J_k'&=q^{-2}T_{(k-1,k)}+q^{-4}T_{(k-2,k)}+\cdots+q^{-2(k-1)}T_{(1,k)}
\end{align*}
for $2\leq k\leq r$, where $(i,j)$ denotes the transposition switching $i$ and $j$
for $1\leq i,j\leq r$.
It is easy to check that $J_k'$ is related to the elements $J_k$ defined in \eqref{JM1} via:
\begin{equation}\label{twoJM}
J_k=(q^2-1)J_k'+1.
\end{equation}
\begin{lem}[{\cite[Theorem 3.14]{DJ2}\label{lem:DJ}}]
Let $\lambda\in\mathcal P(r)$ and $1\leq k\leq r$.
Assume that the node occupied by $k$ in $t_\lambda$ is $(i,j)$
with $1\leq i\leq\ell(\lambda)$ and $1\leq j\leq\lambda_i$.
Then
$$
J_k' z_{\lambda}=(1+q^2+\cdots+q^{2(j-i-1)})z_{\lambda}.
$$
\end{lem}

Given $\lambda\in\mathcal P(r)$ and a node $\rho=(i,j)\in\lambda$,  define the residue of $\rho$ relative to $q$
to be
$$
{\rm res}(\rho)=q^{2(j-i)}.
$$
Accordingly, for a $\lambda$-tableau $t$ and $1\leq k\leq r$,
let ${\rm res}_t(k)$ be the residue of the node occupied by $k$
in $t$. For example, if $t=\young(341,25)$, then ${\rm res}_t(4)=q^2$.
Then by Lemma~\ref{lem:DJ} and~\eqref{twoJM}, we obtain the following.
\begin{cor}\label{cor:finJMact}
Let $\lambda\in\mathcal P(r)$ and $1\leq k\leq r$. Then
\begin{align*}
J_k z_{\lambda}={\rm res}_{t_{\lambda}}(k)z_{\lambda}.
\end{align*}
\end{cor}

We now generalise the formula to Ariki--Koike algebras.

For $\underline{\lambda}=(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(m)})\in\mathcal P_m(r)$
and $\rho=(i,j,k)\in\underline{\lambda}$, define its residue relative to
the given parameters $\{q,u_1,\ldots,u_m\}$
to be
$$
{\rm res}(\rho)=u_kq^{2(j-i)}.
$$
For a $\underline{\lambda}$-tableau $t=(t^{(1)},t^{(2)},\ldots,t^{(m)})$ and $1\leq k\leq r$, let
${\rm res}_t(k)$ be the residue of the node occupied by $k$ in $t$.
For example, for the tableau given in \eqref{tableau}, we have ${\rm res}_t(4)=u_2q^2$.
\begin{prop}\label{prop:JMact}
The following holds for any multipartition
$\underline{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})$ of $r$:
$$
L_kz_{\underline{\lambda}}
={\rm res}_{t_{\underline{\lambda}}}(k)z_{\underline{\lambda}}, \quad 1\leq k\leq r.
$$
\end{prop}
\begin{proof}
Fix $1\leq k\leq r$ and assume $[\underline{\lambda}]={\bf a}=[a_0,a_1,\ldots,a_m]$.
Suppose that $k=r-a_j+b=|\lambda^{(j+1)}|+\cdots+|\lambda^{(m)}|+b$
for $1\leq j\leq m$ with $|\lambda^{(j)}|\geq 1$ and $1\leq b\leq |\lambda^{(j)}|$.
Then note that the node in $t_{\underline{\lambda}}$ occupied by $k$
coincides with the node occupied by $b$ in $t_{\lambda^{(j)}}$ and hence
\begin{equation}\label{tworesidue}
{\rm res}_{t_{\underline{\lambda}}}(k)=u_j{\rm res}_{t_{\lambda^{(j)}}}(b).
\end{equation}
Observe that by~\eqref{JM2} the following holds
$$
L_k=L_{r-a_j+b}=q^{2(1-b)}T_{r-a_j+b-1}\cdots T_{r-a_j+1}L_{r-a_j+1}T_{r-a_j+1}\cdots T_{r-a_j+b-1}.
$$
This together with~\eqref{zv},\eqref{Lvla} and~\eqref{Tvla}
implies that
\begin{align}
L_kz_{\underline{\lambda}}
=&q^{2(1-b)}T_{r-a_j+b-1}\cdots T_{r-a_j+1}L_{r-a_j+1}T_{r-a_j+1}\cdots T_{r-a_j+b-1}z_{\underline{\lambda}}\notag\\
=&q^{2(1-b)}T_{r-a_j+b-1}\cdots T_{r-a_j+1}L_{r-a_j+1}T_{r-a_j+1}\cdots T_{r-a_j+b-1}
v_{[\underline{\lambda}]}(z_{\lambda^{(1)}}\otimes\cdots \otimes z_{\lambda^{(m)}})\notag\\
=&q^{2(1-b)}T_{r-a_j+b-1}\cdots T_{r-a_j+1}L_{r-a_j+1}v_{[\underline{\lambda}]}
T_{a_{j-1}+1}\cdots T_{a_{j-1}+b-1}(z_{\lambda^{(1)}}\otimes\cdots \otimes z_{\lambda^{(m)}})\notag\\
=&u_jq^{2(1-b)}T_{r-a_j+b-1}\cdots T_{r-a_j+1}v_{[\underline{\lambda}]}
T_{a_{j-1}+1}\cdots T_{a_{j-1}+b-1}(z_{\lambda^{(1)}}\otimes\cdots \otimes z_{\lambda^{(m)}})\notag\\
=&u_jq^{2(1-b)}v_{[\underline{\lambda}]}T_{a_{j-1}+b-1}\cdots T_{a_{j-1}+1}
T_{a_{j-1}+1}\cdots T_{a_{j-1}+b-1}(z_{\lambda^{(1)}}\otimes\cdots \otimes z_{\lambda^{(m)}}).
 \label{zL1}
\end{align}
Meanwhile, observe that the element
$q^{1-b}T_{a_{j-1}+b-1}\cdots T_{a_{j-1}+1}T_{a_{j-1}+1}\cdots T_{a_{j-1}+b-1}$ belongs to
the subalgebra $\mathcal H(\mu)$ of the Hecke algebra $\mathcal H(r)$ and moreover it
can be identified with $1^{\otimes j-1}\otimes J_b\otimes1^{\otimes m-j}$ by~\eqref{JM1},
where $\mu=(|\lambda^{(1)}|,\ldots,|\lambda^{m}|)$.
Then by Corollary~\ref{cor:finJMact}, we obtain
\begin{align*}
&q^{2(1-b)}T_{a_{j-1}+b-1}\cdots T_{a_{j-1}+1}T_{a_{j-1}+1}\cdots T_{a_{j-1}+b-1}
(z_{\lambda^{(1)}}\otimes\cdots \otimes z_{\lambda^{(m)}})
\\
&=z_{\lambda^{(1)}}\otimes\cdots \otimes J_b z_{\lambda^{(j)}} \otimes\cdots\otimes z_{\lambda^{(m)}}\\
&={\rm res}_{t_{\lambda^{(j)}}}(b)z_{\lambda^{(1)}}\otimes\cdots \otimes z_{\lambda^{(m)}}.
\end{align*}
Thus by \eqref{zL1} and \eqref{tworesidue} one can deduce that
$$
L_k z_{\underline{\lambda}}=u_j{\rm res}_{t_{\lambda^{(j)}}}(b)
v_{[\underline{\lambda}]}(z_{\lambda^{(1)}}\otimes\cdots \otimes z_{\lambda^{(m)}})
={\rm res}_{t_{\underline{\lambda}}}(k)z_{\underline{\lambda}},
$$
as desired.
\end{proof}

Recall that any $\mc H_{\und{u}}(r)$-module $M$ can be regarded as an ${\mathcal H_\vtg(r)}$-module by inflation
in the case that $u_1,\ldots,u_m$ are all non-zero.
The following result will be used later on.
\begin{prop} \label{prop:induceSpecht}
Suppose that $\underline{u}=(u_1,\ldots,u_m)\in\mathbb (\mathbb C^*)^m$ and
$\underline{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})\in\mathcal P_m(r)$.
Then the inflation of the $\mc H_{\und{u}}(r)$-module $S^{\underline{\lambda}}$
gives an ${\mathcal H_\vtg(r)}$-module isomorphism:
\begin{align*}
S^{\underline{\lambda}}\cong {\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
({\rm ev}^*_{u_m}(S^{\lambda^{(m)}})\otimes\cdots\otimes{\rm ev}^*_{u_1}(S^{\lambda^{(1)}})),
\end{align*}
where $\mu=(|\lambda^{(m)}|,\ldots,|\lambda^{(2)}|,|\lambda^{(1)}|)$.
\end{prop}
\begin{proof}
By~\eqref{pi}, Lemma~\ref{lem:T&JM}(1) and Proposition~\ref{prop:JMact},
it is straightforward to check that the following map
\begin{align*}
\zeta: {\rm ev}^*_{u_m}(S^{\lambda^{(m)}})\otimes\cdots\otimes{\rm ev}^*_{u_1}(S^{\lambda^{(1)}})
&\longrightarrow S^{\underline{\lambda}}\\
z_{\lambda^{(m)}}\otimes\cdots\otimes z_{\lambda^{(1)}}&\mapsto z_{\underline{\lambda}}
\end{align*}
is an $\mathcal H_{{\!\vartriangle\!}}(\mu)$-homomorphism.
Then by Frobenius reciprocity, there exists a nonzero homomorphism
$\bar{\zeta}:{\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
{\rm ev}^*_{u_m}(S^{\lambda^{(m)}})\otimes\cdots\otimes{\rm ev}^*_{u_1}(S^{\lambda^{(1)}})
\rightarrow S^{\underline{\lambda}}$ which is also surjective since $S^{\underline{\lambda}}$ is generated by $z_{\underline{\lambda}}$.
By~\eqref{slaind}, a dimension comparison proves the proposition.
\end{proof}

\begin{rem}
By Proposition~\ref{prop:induceSpecht},
the ${\mathcal H_\vtg(r)}$-module
${\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
({\rm ev}^*_{u_m}(S^{\lambda^{(m)}})\otimes\cdots\otimes{\rm ev}^*_{u_1}(S^{\lambda^{(1)}}))$
affords an $\mc H_{\und{u}}(r)$-module. Actually, it is not hard to check directly that the polynomial
$(X_1-u_1)(X_1-u_2)\cdots(X_1-u_m)$ acts as zero and, hence, the ${\mathcal H_\vtg(r)}$-module structure factors
through the surjective homomorphism $\varpi_{\underline{u}}$ followed by the $\mc H_{\und{u}}(r)$-module structure on  ${\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
({\rm ev}^*_{u_m}(S^{\lambda^{(m)}})\otimes\cdots\otimes{\rm ev}^*_{u_1}(S^{\lambda^{(1)}}))$.
\end{rem}

\subsection{\bf Ariki--Koike algebras of integral type and Kleshchev multipartitions.}
The classification of simple $\mc H_{\und{u}}(r)$-modules has been completed by Ariki in terms of Kleshchev multipartitions;
see \cite[Theorem 4.2]{Ar} and \cite[conjecture 2.12]{AM}.
It is also known from \cite[\S1]{AM} that the classification of irreducible
$\mc H_{\und{u}}(r)$-modules is reduced to the cases where the parameters $u_1,\ldots, u_m$
are either all zero or all powers of $q^2$. For our purpose, we are mainly concerned about the latter one
(see footnote 2).

For $m\geq 1$, let
$$
\mathfrak F_m=\{(f_1,\ldots,f_m)\in\mathbb Z^m\mid f_1\geq\cdots\geq f_m\},
$$
and let
\begin{equation}\label{set F}
\mathfrak F=\bigcup_{m\geq1}\mathfrak F_m.
\end{equation}
For $f=(f_1,\ldots,f_m)\in\mathfrak{F}$, let $f^*=(-f_m,\ldots,-f_1)$.  Clearly, this gives a bijection
 $$(\ )^*:\mathfrak F\longrightarrow\mathfrak F.$$

We continue to assume that $q$ is {\it not a root of unity} and consider the parameter vectors of the form
\begin{equation}\label{parameters for integral AKa}
\underline{u}_f=(q^{2f_1},\ldots,q^{2f_m}) \quad (f\in\mathfrak F_m).
\end{equation}
In other words, $u_1=(q^2)^{f_1},\ldots, u_m=(q^2)^{f_m}$. An Ariki--Koike algebra $\mc H_{\und{u}}(r)$ is of {\it integral type} if $\underline{u}=\underline{u}_f$ for some $f\in\mathfrak F$.
Thus, Kleshchev multipartions for an Ariki--Koike algebra $\mc H_{\und{u}}(r)$ of integral type (and a parameter which is not a root of unity) can be described as follows.
\begin{defn}
 Given a multipartition $\lambda=(\lambda^{(1)},\ldots,\lambda^{(m)})$ of $r$,
we will say that $\underline{\lambda}$ is a
{\em Kleshchev multipartition} with respect to $f\in\mathfrak F_m$ (or  to $\underline{u}_f$) if it satisfies
\begin{equation}\label{Kleshchev}
\lambda^{(k)}_{j+f_k-f_{k+1}}\leq\lambda^{(k+1)}_j
\end{equation}
for $j\geq 1$ and $1\leq k\leq m-1$.
\end{defn}

Note that, when $q$ is a root of unity, the definition of Kleshchev multipartitions is much more complicated.

\begin{rem}\label{reduce to 1} Note that if $\ell(\lambda^{(k)})\leq f_k-f_{k+1}+1$ then \eqref{Kleshchev} is equivalent to $\lambda^{(k)}_{1+f_k-f_{k+1}}\leq\lambda^{(k+1)}_1$.
\end{rem}

Denote by $\mathcal K_{\underline{u}_f}(r)$ or $\mathcal K_f(r)$ the set of Kleshchev multipartitions of $r$
with respect to $\underline{u}_f$.
For $\underline{u}=(u_1,\ldots,u_m)$, set
$$
\underline{u}^{-1}=(u_m^{-1},\ldots,u_1^{-1}).
$$
Then by~\eqref{Kleshchev}, a multipartition $\underline{\gamma}=(\gamma^{(1)},\ldots,\gamma^{(m)})$ belongs to $\mathcal K_{\underline{u}_f^{-1}}(r)=\mathcal K_{f^*}(r)$
if and only if
\begin{equation}\label{Kinv}
\gamma^{(k)}_{j+f_{m-k}-f_{m-k+1}}\leq\gamma^{(k+1)}_j
\end{equation}
for $j\geq 1$ and $1\leq k\leq m-1$.

 It is known \cite[Corollary 2]{BK1} that the Ariki-Koike algebra $\mathcal H_{\underline{u}_f}(r)$
for $f\in\mathfrak F$ and $q$ not root of unity is isomorphic to the corresponding degenerate cyclotomic Hecke algebra
over $\mathbb C$. Then we have the following.
\begin{prop}[{\cite[Theorem 3.18, Lemma 3.13]{BK} (cf. \cite{AM, Ar, Va})}]
\label{prop:BK}
Assume that $f=(f_1,\ldots,f_m)\in\mathfrak F$ with $m\geq 1$.
The following holds for all $r\geq 1$:
\begin{enumerate}
\item For multipartition $\underline{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})\in\mathcal P_m(r)$ such that $\underline{\lambda}'\in\mathcal K_{f^*}(r)$,
the ${\mathcal H_\vtg(r)}$-module
${\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
({\rm ev}^*_{q^{2f_m}}(S^{\lambda^{(m)}})\otimes\cdots\otimes{\rm ev}^*_{q^{2f_1}}(S^{\lambda^{(1)}}))$
has irreducible head denoted by $D^{\underline{\lambda}}$, where $\mu=(|\lambda^{(m)}|,\ldots,|\lambda^{(1)}|)$.
In particular, $D^{\underline{\lambda}}$ affords a simple $\mathcal H_{\underline{u}_f}(r)$-module.

\item The set $\{D^{\underline{\lambda}'}~| \underline{\lambda}\in\mathcal K_{f^*}(r)\}$ is a complete set of non-isomorphic
irreducible $\mathcal H_{\underline{u}_f}(r)$-modules.

\end{enumerate}
\end{prop}

By Proposition~\ref{prop:induceSpecht} and Proposition~\ref{prop:BK}, we have the following.
\begin{cor}\label{cor:hdSpecht}
For $\underline{\lambda}\in\mathcal P_m(r)$ satisfying $\underline{\lambda}'\in\mathcal K_{f^*}(r)$ with $f\in\mathfrak F_m$, the
$\mathcal H_{\underline{u}_f}(r)$-module $S^{\underline{\lambda}}$ has irreducible head isomorphic to $D^{\underline{\lambda}}$.
\end{cor}

We will construct a subset of $\cup_{f\in\mathfrak F}\mathcal K_f(r)$ which labels precisely the irreducible ${\mathcal H_\vtg(r)}$-module
with ``integral weights'', i.e., the irreducible objects in ${\mathcal H_\vtg(r)}$-${\sf mod}^{\mathbb Z}$.

\section{Standard Kleshchev multipartitions and integral multisegments}

At the end of \S 2, we saw that simple objects in the category ${\mathcal H_\vtg(r)}$-${\sf mod}^{\mathbb Z}$ are indexed by the set $\mathscr S_r^\mathbb Z$. The set can be further described in terms of standard sequences of integral segments or standard words. In this section,
we will describe the set in terms of standard Kleshchev multipartitions. In particular, we will characterise the simple objects in the category ${\mathcal H_\vtg(r)}$-${\sf mod}^{\mathbb Z}$ by these standard Kleshchev multipartitions and their associated irreducible modules of Ariki--Koike algebras.

\subsection{Column residual segments of a multipartition.}
We first construct a sequence ${\bf s}$ of segments via the column residual segment of  a multipartition $\underline{\lambda}$ and show that $I_{\bf s}$ maps onto $S^{\underline{\lambda}}$.

For $\underline{\lambda}=(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(m)})\in\mathcal P_m(r)$ and parameter vector $\underline{u}=(u_1,\ldots,u_m)$,
define the $j$th-{\it column residual segment} of $\lambda^{(i)}$
$$
{\mathsf s}^{(i)}_j=(u_iq^{2(j-(\lambda^{(i)'})_j)},\ldots, u_iq^{2(j-2)},u_iq^{2(j-1)})\quad (1\leq i\leq m, 1\leq j\leq\lambda^{(i)}_1)
$$
by reading the residues of the nodes in the $j$-th column of $\lambda^{(i)}$ from bottom to top, and form
the sequence of column residual segments from column 1, then column 2, and so on, starting from the last partition of $\underline{\lambda}$, then the second last partition, and so on,
\begin{equation}\label{scla}
{\bf s}^{{\mathsf c}}_{\underline{\lambda};\underline{u}}=\big({\mathsf s}_1^{(m)},\ldots,{\mathsf s}^{(m)}_{\lambda^{(m)}_1},\ldots,
{\mathsf s}^{(1)}_1,\ldots,{\mathsf s}^{(1)}_{\lambda^{(1)}_1}\big);
\end{equation}
cf. the order of the Young diagram $t_{\underline{\lambda}}$ in \eqref{tableau}.
 We will simply write ${\bf s}^{{\mathsf c}}_{\underline{\lambda};f}$  in the sequel, if $\underline{u}=\underline{u}_f$ for some $f\in \mathfrak F_m$.
\begin{lem}\label{lem:Ibfs}
Assume $\underline{\lambda}=(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(m)})\in\mathcal P_m(r)$ and
$\underline{u}=(u_1\ldots,u_m)\in(\mathbb C^*)^m$.
Write ${\bf s}={\bf s}^{{\mathsf c}}_{\underline{\lambda};\underline{u}}$.
Then
\begin{enumerate}
\item The subspace $\mathbb C z_{\underline{\lambda}}$ of $\mc H_{\und{u}}(r)$
affords an $\mathcal H_{{\!\vartriangle\!}}((\underline{\lambda}')^\vee)$-submodule of the inflation of the $\mc H_{\und{u}}(r)$-module
$S^{\underline{\lambda}}$.

\item There exists a surjective ${\mathcal H_\vtg(r)}$-homomorphism
$\varphi: {\rm Ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}((\underline{\lambda}')^\vee)}\mathbb C z_{\underline{\lambda}}
\twoheadrightarrow S^{\underline{\lambda}}$ such that $\varphi(1\otimes z_{\underline{\lambda}})=z_{\underline{\lambda}}$.

\item There exists a surjective ${\mathcal H_\vtg(r)}$-homomorphism
$\widetilde\varphi:I_{{\bf s}}
\twoheadrightarrow S^{\underline{\lambda}}$ with $\widetilde\varphi\big(C_{w^0_{\mu({\bf s})}}\big)=z_{\underline{\lambda}}$.
\end{enumerate}
\end{lem}
\begin{proof}
By~\eqref{Tzudla}, one can deduce that
$$
T_i z_{\underline{\lambda}}=-z_{\underline{\lambda}}
$$
for any $T_i\in\mathcal H((\underline{\lambda}')^\vee)$.
On the other hand, by Proposition~\ref{prop:JMact} we have
$$
X_k z_{\underline{\lambda}}=L_k z_{\underline{\lambda}}={\rm res}_{t_{\underline{\lambda}}}(k)z_{\underline{\lambda}}
$$
for $1\leq k\leq r$.
Since $\mathcal H_{{\!\vartriangle\!}}((\underline{\lambda}')^\vee)$ is generated by
$\mathcal H((\underline{\lambda}')^\vee)$ and $X^{\pm1}_1,\ldots, X^{\pm1}_r$, Part (1) is verified.

Now, by Frobenius reciprocity, there exists an ${\mathcal H_\vtg(r)}$-homomorphism
$$\varphi: {\rm Ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}((\underline{\lambda}')^\vee)}\mathbb C z_{\underline{\lambda}}
\rightarrow S^{\underline{\lambda}}$$
which sends $h\otimes z_{\underline{\lambda}}$ to $hz_{\underline{\lambda}}$ for all $h\in{\mathcal H_\vtg(r)}$.
Since $S^{\underline{\lambda}}$ is generated by $z_{\underline{\lambda}}$, Part (2) holds.

Observe from~\eqref{scla}, \eqref{tilde s}, and the standard tableau $t_{\underline{\lambda}}$ (see \eqref{tableau}) that the following holds
\begin{align}
\widetilde{{\bf s}}^\vee
&=({\rm res}_{t_{\underline{\lambda}}}(1),{\rm res}_{t_{\underline{\lambda}}}(2),
\ldots,{\rm res}_{t_{\underline{\lambda}}}(r)),\notag\\
\mu({\bf s})
&=(\underline{\lambda}')^\vee.\label{mubfs}
\end{align}
Then by~\eqref{cell} and Proposition~\ref{prop:Ro2}(2) we have
\begin{align*}
T_i C_{w^0_{\mu({\bf s})}}
=&-C_{w^0_{\mu({\bf s})}},\\
X_k C_{w^0_{\mu({\bf s})}}
=&{\rm res}_{t_{\underline{\lambda}}}(k)C_{w^0_{\mu({\bf s})}}
\end{align*}
for $T_i\in\mathcal H((\underline{\lambda}')^\vee)$ and $1\leq k\leq r$.
This means
$$
\mathbb C C_{w^0_{\mu({\bf s})}}\cong\mathbb C z_{\underline{\lambda}}
$$
as $\mathcal H_{{\!\vartriangle\!}}((\underline{\lambda}')^\vee)$-modules by the proof of Part (1).
Hence, by Lemma~\ref{lem:induced}, there exists an isomorphism
$$
\phi: I_{{\bf s}}\overset{\cong}{\longrightarrow}{\rm Ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}((\underline{\lambda}')^\vee)}\mathbb C z_{\underline{\lambda}}.
$$
This together with Part (2) gives Part (3) if we set $\widetilde\varphi=\varphi\circ\phi$.
\end{proof}
\subsection{\bf Kleshchev multipartitions associated with integral multisegments.}
We shall construct a Kleshchev multipartition associated to an integral multisegment as follows.

Given ${{j_1j_2\ldots j_t}\choose{i_1i_2\ldots i_t}}\in\mathcal W^s(r)$ so that
 $\mathfrak s=\flat{{j_1j_2\ldots j_t}\choose{i_1i_2\ldots i_t}}\in\mathscr S^{\mathbb Z}_{r}$ (see \eqref{flat}),
let $1\leq k_1\leq t$ be the right-most column index satisfying $j_a=j_{a-1}+1$, for $2\leq a\leq k_1$, and
$i_1<i_2<\cdots<i_{k_1}$.
Define $\gamma^{(1)}=(\gamma^{(1)}_1,\ldots,\gamma^{(1)}_{k_1})$ by setting
\begin{align*}
\gamma^{(1)}_b=|j_b-i_b+1|
\end{align*}
for $1\leq b\leq k_1$. Note that $k_1=j_{k_1}-j_1+1$ and $\gamma^{(1)}$ is a partition.

Let
$$
\mathfrak s^1=\flat{{j_{k_1+1}j_{k_1+2}\ldots j_t}\choose{i_{k_1+1}i_{k_1+2}\ldots i_t}} \in\mathscr S^{\mathbb Z}_{r-|\gamma^{(1)}|}.
$$
Now applying the same procedure to $\mathfrak s^1$, we obtain
a positive number $1\leq k_2\leq t-k_1$, a partition $\gamma^{(2)}$ and $\mathfrak s^2\in\mathscr S^{\mathbb Z}_{r-|\gamma^{(1)}|-|\gamma^{(2)}|}$.
Continuing in this way, we will end up with $1\leq k_1,\ldots, k_m\leq t$ and a multipartition
\begin{equation}\label{ga_s}
\underline{\gamma}_{\mathfrak s}=(\gamma^{(1)},\gamma^{(2)},\ldots,\gamma^{(m)}),
\end{equation}
where $ \gamma^{(a)}=(\gamma^{(a)}_1,\gamma^{(a)}_2,\ldots,\gamma^{(a)}_{k_a}) $ with
$$
\gamma^{(a)}_b=|j_{k_1+\cdots+k_{a-1}+b}-i_{k_1+\cdots+k_{a-1}+b}+1|
$$
for all $1\leq a\leq m$ and $1\leq b\leq k_a$. Note that
\begin{equation}\label{ka}
k_a=j_{k_1+\cdots+k_a}-j_{k_1+\cdots+k_{a-1}+1}+1.
\end{equation}
Meanwhile, define
\begin{equation}\label{us}
\underline{u}_{\mathfrak s}=(u_1,\ldots,u_m)
\end{equation}
by setting
$$ u_1=q^{2(j_{k_1+\cdots+k_{m-1}+1})},~
u_2=q^{2(j_{k_1+\cdots+k_{m-2}+1})},~
\ldots,
u_m=q^{2j_1}.$$
(So $f_i=j_{k_1+\cdots+k_{m-i}+1}$ in the notation of \eqref{parameters for integral AKa}.)
Observe that
\begin{equation}\label{equiv}
{\mathfrak s}={\mathfrak s}' \text{ if and only if }  \underline{\gamma}_{{\mathfrak s}}=\underline{\gamma}_{{\mathfrak s}'} , \underline{u}_{{\mathfrak s}}=\underline{u}_{{\mathfrak s}'}.
\end{equation}
\begin{Example}
Suppose
$$\mathfrak s=\flat(w),\text{ where }w=\biggl(\begin{matrix}-1&0&1&2&2\\
-4&-5&-2&-1&-2\\\end{matrix}\biggr).$$
Then by the procedure described above, one can deduce that $k_1=1,k_2=3, k_3=1$, and
$\underline\gamma_{\mathfrak s}=(\gamma^{(1)},\gamma^{(2)},\gamma^{(3)})$, where
$$
\gamma^{(1)}=(4),
\gamma^{(2)}=(6,4,4),\text{ and }
\gamma^{(3)}=(5).$$
Moreover, $\underline{u}_{\mathfrak s}=(u_1,u_2,u_3)$ with
$$
u_1=q^4,u_2=q^0,u_3=q^{-2}.
$$
We also observe from this example that $\underline\gamma_{\mathfrak s}\in\mathcal K_{\underline{u}_{\mathfrak s}^{-1}}(r)$ is Kleshchev. Moreover, for multipartition
$$
\underline{\lambda}_{\mathfrak s}=\underline\gamma_{\mathfrak s}'=((1^5),(3^41^2),(1^4))
$$ and the $u_{\mathfrak s}$ above, the sequence
 ${\bf s}^{{\mathsf c}}_{\underline{\lambda}_{\mathfrak s};\underline{u}_{\mathfrak s}}$ constructed in \eqref{scla}
 is the standard element in ${\mathfrak s}$.
\end{Example}
In fact, the last observation holds in general.

\begin{prop}\label{prop:stoK}
For ${\mathfrak s}\in\mathscr S^\mathbb Z_r$, let $\gamma_{\mathfrak s}$ and $u_{\mathfrak s}$ be defined as in \eqref{ga_s} and \eqref{us}.
Then $\underline{\gamma}_{\mathfrak s}$ is a sincere Kleshchev multipartition with respect to
$u_{\mathfrak s}^{-1}$, and moreover ${\bf s}^{{\mathsf c}}_{\underline{\gamma}_{\mathfrak s}';\underline{u}_{\mathfrak s}}$ is the standard element in ${\mathfrak s}$.
\end{prop}
\begin{proof}
Suppose $w={{j_1j_2\ldots j_t}\choose{i_1i_2\ldots i_t}}\in\mathcal W^s(r)$ with $\mathfrak s=\flat(w)$.
Write $\underline{\gamma}_{\mathfrak s}=(\gamma^{(1)},\gamma^{(2)},\ldots,\gamma^{(m)})$
with $\ell(\gamma^{(a)})=k_a=j_{k_1+\cdots+k_a}-j_{k_1+\cdots+k_{a-1}+1}+1$ for $1\leq a\leq m$. So $\underline{\gamma}_{\mathfrak s}$ is sincere.
By~\eqref{Kinv} and~\eqref{us}, it suffices to show that
the following holds
for $1\leq a\leq m-1$ and $b\geq 1$:
\begin{equation}\label{inequality}
\gamma^{(a)}_{b+j_{k_1+\cdots+k_a+1}-j_{k_1+\cdots+k_{a-1}+1}}\leq \gamma^{(a+1)}_b
\end{equation}
Fix $1\leq a\leq m-1$ and $b\geq 1$.
If $\gamma^{(a)}_{b+j_{k_1+\cdots+k_a+1}-j_{k_1+\cdots+k_{a-1}+1}}=0$, then
~\eqref{inequality} automatically holds.

Suppose now $\gamma^{(a)}_{b+j_{k_1+\cdots+k_a+1}-j_{k_1+\cdots+k_{a-1}+1}}>0$.
Since $w$ is standard, the sequence $j_1,\ldots,j_t$ is weakly increasing. By \eqref{ka},
\begin{equation}\label{length ka}
j_{k_1+\cdots+k_a+1}-j_{k_1+\cdots+k_{a-1}+1}\geq
j_{k_1+\cdots+k_a}-j_{k_1+\cdots+k_{a-1}+1}=k_a-1.
\end{equation}
Hence, we obtain from the definition of the partition $\gamma^{(a)}$ that
$$0<\gamma^{(a)}_{b+j_{k_1+\cdots+k_a+1}-j_{k_1+\cdots+k_{a-1}+1}}\leq
\gamma^{(a)}_{b+k_a-1},$$
which forces $b=1$ and $1+j_{k_1+\cdots+k_a+1}-j_{k_1+\cdots+k_{a-1}+1}=k_a$
since $\ell(\gamma^{(a)})=k_a$ (cf. Remark \ref{reduce to 1}).
Thus, \eqref{length ka} must be an equality and consequently,
\begin{equation}\label{eq1}
 j_{k_1+\cdots+k_a+1}=j_{k_1+\cdots+k_a}.
\end{equation}
Hence, the fact that $w$ is standard implies
\begin{equation}\label{eq2}
i_{k_1+\cdots+k_a}\geq i_{k_1+\cdots+k_a+1}.
\end{equation}
Finally, by \eqref{eq1} and \eqref{eq2},
\begin{align*}
\gamma^{(a)}_{b+j_{k_1+\cdots+k_a+1}-j_{k_1+\cdots+k_{a-1}+1}}
&=\gamma^{(a)}_{k_a}\\
&=j_{k_1+\cdots+k_a}-i_{k_1+\cdots+k_a}+1\\
&=j_{k_1+\cdots+k_a+1}-i_{k_1+\cdots+k_a}+1\\
&\leq j_{k_1+\cdots+k_a+1}-i_{k_1+\cdots+k_a+1}+1\\
&=\gamma^{(a+1)}_1=\gamma^{(a+1)}_b
\end{align*}
proving \eqref{inequality}.

The last assertion follows from the definition.
\end{proof}
\begin{thm} \label{thm:VsDla}
For any given multisegment ${\mathfrak s}\in\mathscr S^{\mathbb Z}_{r}$, let $\underline\gamma_{\mathfrak s}$ and $\underline{u}_{\mathfrak s}$ be defined as in \eqref{ga_s} and \eqref{us} and let $\underline{\lambda}_{\mathfrak s}=\underline\gamma_{\mathfrak s}'$. Then the irreducible ${\mathcal H_\vtg(r)}$-module $V_{\mathfrak s}$
is isomorphic to  the inflated ${\mathcal H_\vtg(r)}$-module $D^{\underline{\lambda}_{\mathfrak s}}$. In particular,
 $V_{\mathfrak s}$ affords an $\mathcal H_{\underline{u}_{\mathfrak s}}(r)$-module.\end{thm}
\begin{proof}
Let ${\bf s}\in{\mathfrak s}$ be standard. By Proposition \ref{prop:stoK}, $\underline{\lambda}_{\mathfrak s}'=\underline\gamma_{\mathfrak s}\in\mathcal K_{\underline{u}_{{\mathfrak s}}^{-1}}(r)$ and ${\bf s}={\bf s}^{{\mathsf c}}_{\underline{\lambda}_{\mathfrak s};\underline{u}_{\mathfrak s}}$.
Thus, by Corollary~\ref{cor:hdSpecht}, the Specht module $S^{\underline{\lambda}_{\mathfrak s}}$ has the simple head $D^{\underline{\lambda}_{\mathfrak s}}$. On the other hand, by
Lemma~\ref{lem:Ibfs}, there exists a surjective ${\mathcal H_\vtg(r)}$-homomorphism
$I_{{\bf s}}\twoheadrightarrow S^{\underline{\lambda}_{\mathfrak s}}$, which extends to a surjective
${\mathcal H_\vtg(r)}$-homomorphism $
I_{{\bf s}}\twoheadrightarrow D^{\underline{\lambda}_{\mathfrak s}}$.
This together with Proposition~\ref{prop:Ro3} gives the required isomorphism.

The last assertion is clear since $D^{\underline{\lambda}_{\mathfrak s}}$ is an $\mathcal H_{\underline{u}_{\mathfrak s}}(r)$-module.\end{proof}

In the case $m=1, u_1=1$, Theorem~\ref{thm:VsDla} recovers \cite[Proposition 8.2]{DF}.

\begin{rem}
Fix ${\mathfrak s}\in\mathscr S^\mathbb Z_r$.
It is known \cite{Ar1} that the decomposition numbers for the Ariki-Koike algebras $\mathcal H_{\underline{u}_{{\mathfrak s}}}(r)$
over $\mathbb C$ can be computed in terms of the canonical basis of an associated integral highest weight module of
the Lie algebra $\mathfrak{gl}_{\infty}(\mathbb C)$.
Using Theorem~\ref{thm:VsDla} and Lemma~\ref{lem:Specht}, one can derive a dimension formula for $V_{{\mathfrak s}}$ and the decomposition
of the restriction $V_{{\mathfrak s}}$ to $\mathcal{H}(r)$ in terms of Spech modules $S^\lambda$ for $\lambda\in\mathcal P(r)$.
\end{rem}
\subsection{Standard Kleshchev multipartitions and simple objects in ${\mathcal H_\vtg(r)}$-${\sf mod}^{\mathbb Z}$.}
We now characterise the Kleshchev multipartition $\underline{\gamma}_{\mathfrak s}$ constructed from a standard word $w$ with ${\mathfrak s}=\flat(w)$.

Recall the sets  $\mathfrak F_m,\mathfrak F$ in \eqref{set F} and the elements
$\underline{u}_f=(q^{2f_1},q^{2f_2},\ldots,q^{2f_m})$ for $f=(f_1,\ldots,f_m)\in\mathfrak F_m$
in \eqref{parameters for integral AKa}.
For $f\in\mathfrak F_m$,
let
$$
\|f\|=\text{min}\{f_i-f_{i+1}\mid 1\leq i<m\}.
$$
\begin{defn}\label{SKM}
A multipartitions $\underline{\gamma}=(\gamma^{(1)},\ldots,\gamma^{(m)})$ of $r$
is called a {\it standard Kleshchev multipartition} relative to $f=(f_1,\ldots,f_m)\in\mathfrak F$, if it satisfies
\begin{itemize}
\item[(SK1)] $\underline{\gamma}$ is sincere (or equivalently, $\ell(\gamma^{(a)})\geq 1$ for each $ 1\leq a\leq m$).

\item[(SK2)]$\ell(\gamma^{(a)})\leq f_a-f_{a+1}+1$ and $\gamma^{(a)}_{f_a-f_{a+1}+1}\leq\gamma^{(a+1)}_1$
for each $ 1\leq a\leq m-1$.

\item[(SK3)] If $\ell(\gamma^{(a)})= f_a-f_{a+1}$ for some $1\leq a\leq m-1$,
then $\gamma^{(a)}_{f_a-f_{a+1}}\leq \gamma^{(a+1)}_1-1$.
\end{itemize}
\end{defn}
Observe from the condition (SK2) that a standard Kleshchev multipartition is a Kleshchev multipartition; see Remark \ref{reduce to 1}.

Let $\mathcal K^s_{f}(r)$ be the set of all standard Kleshchev multipartitions of $r$ relative to $f\in\mathfrak F$. Then
$$
\mathcal K^s_{f}(r)\subseteq\mathcal K_{f}(r)=\mathcal K_{\underline{u}_f}(r).
$$

\begin{lem}
Suppose $f=(f_1,\ldots,f_m)\in\mathfrak F$.
Then
\begin{enumerate}
\item For any $c\in\mathbb Z$, we have
$$
\mathcal K^s_{f}(r)=\mathcal K^s_{f+c}(r)
$$
where $f+c=(f_1+c,\ldots,f_m+c)$.

\item If $m=1$, then
$
\mathcal K^s_{f}(r)=\mathcal P(r),
$ the set of partitions of $r$.

\item If $m>1$ and $\|f\|\geq r$,
then $m\leq r$ and
$
\mathcal K^s_{f}(r)$ is the set of all sincere multipartitions in
$\mathcal P_{m}(r)$.

\item If $f_1=f_2=\cdots=f_m$, then $m\leq r$ and
$$\mathcal K^s_{f}(r)\equiv\{(a_1,\ldots,a_m)\in\Lambda(m,r)\mid 1\leq a_1\leq\cdots\leq a_m\}.$$

\end{enumerate}
\end{lem}
\begin{proof}
Observe that the conditions (SK1)-(SK3) depend only on $f_a-f_{a+1}$ for $1\leq a \leq m-1$
and hence Part (1) holds. Clearly Part (2) is true.
Now suppose $f_a-f_{a+1}\geq r$ for all $1\leq a\leq m-1$ and $\underline{\gamma}=(\gamma^{(1)},\ldots,\gamma^{(m)})
\in\mathcal P_m(r)$ is sincere.
Then $|\gamma^{(a)}|\leq r-m+1\leq r-1$ and hence (SK2) and (SK3) automatically hold for $\underline{\gamma}$
and thus $\underline{\gamma}\in\mathcal K^s_{f}(r)$, proving Part (3).
Finally, if $f_1=f_2=\cdots=f_m$, the $\underline{\gamma}=(\gamma^{(1)},\ldots,\gamma^{(m)})
\in\mathcal K^s_{f}(r)$ if and only if $\gamma^{(i)}=(\gamma_1^{(i)})$ for $1\leq i \leq m$ and
$\gamma^{(1)}_1\leq\cdots\leq\gamma^{(m)}_1$ by (SK2). This proves Part (4).\end{proof}

Recall from \S4.2 that associated to each $\mathfrak s\in\mathscr S_r^\mathbb Z$, there exist
$\underline{\gamma}_{\mathfrak s}$ and $\underline{u}_{\mathfrak s}$ such that
$\underline{\gamma}_{\mathfrak s}\in\mathcal K_{\underline{u}_{\mathfrak s}^{-1}}(r)$.
Thus, every $f\in\mathfrak F$ defines a subset
$$\mathscr S_{r,f}^\mathbb Z:=\{\mathfrak s\in\mathscr S_r^\mathbb Z~|~\underline{u}_{\mathfrak s}=\underline{u}_f\},$$
and
a map
\begin{align*}
\eta_{f}:\mathscr S_{r,f}^\mathbb Z
&\longrightarrow \mathcal K_{\underline{u}_f^{-1}}(r)=\mathcal K_{f^*}(r)\\
\mathfrak s&\longmapsto \underline{\gamma}_{\mathfrak s}
\end{align*}
which is injective by~\eqref{equiv}.

\begin{prop}\label{prop:etau} With the notation above we have ${\rm im}(\eta_{f})=\mathcal K^s_{f^*}(r)$
and hence the map
$$
\eta_{f}: \mathscr S_{r,f}^\mathbb Z
\longrightarrow \mathcal K^s_{f^*}(r)
$$
is a bijection.
\end{prop}
\begin{proof}
Suppose $\mathfrak s\in \mathscr S_{r,f}^\mathbb Z$ and
assume that $\mathfrak s =\flat(w)$  with $w={{j_1j_2\ldots j_t}\choose{i_1i_2\ldots i_t}}\in\mathcal W^s(r)$. Then, $\underline{u}_{\mathfrak s}=\underline{u}_f$ and so $f_i=j_{k_1+\cdots+k_{m-i}+1}$
 with $k_a$ as given in \eqref{ka}. In particular,
$$j_{k_1+\cdots+k_{a-1}+1}=f_{m-a+1}, \quad j_{k_1+\cdots+k_a}=f_{m-a+1}+k_a-1$$
for all $1\leq a\leq m$.
We want to show that $\underline{\gamma}_{\mathfrak s}=(\gamma^{(1)},\gamma^{(2)},\ldots,\gamma^{(m)})$  defined in \eqref{ga_s} satisfies the conditions (SK1)--(SK3).

First,
\begin{equation}\label{bijection1}
\ell(\gamma^{(a)})=k_a\geq 1, \text{ for all }1\leq a\leq m,
\end{equation}
 and
 $f^*_a=-f_{m-a+1}=-j_{k_1+\cdots+k_{a-1}+1} $ for $1\leq a\leq m$.
Also,  the hypothesis that the word $w$ is standard implies that
\begin{align*}
k_a-1+f_{m-a+1}=j_{k_1+\cdots+k_a}\leq j_{k_1+\cdots+k_a+1}=f_{m-a}.\quad
\end{align*}
Hence,
\begin{equation}\label{bijection2}
k_a\leq f_{m-a}-f_{m-a+1}+1=f^*_{a}-f^*_{a+1}+1
\end{equation}
for all $1\leq a\leq m-1$.
Thus, $\underline{\gamma}_{\mathfrak s}$ satisfies
(SK1) and (SK2) by \eqref{bijection1} and \eqref{bijection2} and Proposition
\ref{prop:stoK}.
It remains to prove (SK3). If $k_a=f^*_{a}-f^*_{a+1}=f_{m-a}-f_{m-a+1}$ for some $1\leq a\leq m-1$, then
we have
$$
j_{k_1+\cdots+k_a}=f_{m-a+1}+k_a-1=f_{m-a}-1=j_{k_1+\cdots+k_a+1}-1.
$$
This implies, by the choice of $k_a$ in the construction of $\underline{\gamma}_{\mathfrak s}$, that
$$
i_{k_1+\cdots+k_a}\geq i_{k_1+\cdots+k_a+1}.
$$
Hence,
\begin{align*}
\gamma^{(a)}_{k_a}
&=j_{k_1+\cdots+k_a}-i_{k_1+\cdots+k_a}+1\\
&=j_{k_1+\cdots+k_a+1}-i_{k_1+\cdots+k_a}\\
&\leq j_{k_1+\cdots+k_a+1}-i_{k_1+\cdots+k_a+1}=\gamma^{(a+1)}_1-1,
\end{align*}
proving (SK3) and so $\underline{\gamma}_{\mathfrak s}\in\mathcal K^s_{f^*}(r)$. In other words, we have proved ${\rm im}(\eta_{f})\subseteq\mathcal K^s_{f^*}(r)$.

Conversely, let $\underline{\gamma}=(\gamma^{(1)},\gamma^{(2)},\ldots,\gamma^{(m)})\in\mathcal K^s_{f^*}(r)$ be a standard Kleshchev multipartition (or satisfy (SK1)--(SK3)). We now construct an integral multisegment ${\mathfrak s}\in\mathscr S_r^\mathbb Z$ such that $\underline{u}_{\mathfrak s}=\underline{u}_f$ and $\underline{\gamma}_{\mathfrak s}=\underline{\gamma}.$

Set $k_a=\ell(\gamma^{(a)})$ for $1\leq a\leq m$ and let $t=k_1+\cdots+k_m$.
Given $1\leq c\leq t$, assume that
$c=k_1+k_2+\cdots+k_{a-1}+b$ with $1\leq b\leq\ell(\gamma^{(a)})=k_a$ for some $1\leq a\leq m$ and define
\begin{equation}\label{bijectionji}
j_c=f_{m-a+1}+b-1, \quad i_c=f_{m-a+1}-\gamma^{(a)}_b+b.
\end{equation}
We claim that
\begin{equation}\label{standard}
{{j_1j_2\ldots j_t}\choose{i_1i_2\ldots i_t}}\in\mathcal W^s(r).
\end{equation}
Indeed, firstly, since $\underline{\gamma}=(\gamma^{(1)},\gamma^{(2)},\ldots,\gamma^{(m)})\in\mathcal K^s_{f^*}(r)$,
we have $k_a\leq f_{m-a}-f_{m-a+1}+1$ by (SK2). In other words,
$$
f_{m-a+1}+k_a-1\leq f_{m-a},
$$
which together with \eqref{bijectionji} implies that
\begin{equation}\label{bijectionjka}
j_{k_1+\cdots+k_a}\leq j_{k_1+\cdots+k_a+1}
\end{equation}
for $1\leq a\leq m-1$.
If $j_{k_1+\cdots+k_a}= j_{k_1+\cdots+k_a+1}$ for some $1\leq a\leq m-1$,
then $f_{m-a+1}+k_a-1=f_{m-a}$ and so $\ell(\gamma^{(a)})=k_a=f_{m-a}-f_{m-a+1}+1$.
Then, by (SK2),
\begin{align*}
i_{k_1+\cdots+k_a}
=f_{m-a+1}-\gamma^{(a)}_{k_a}+k_a
=f_{m-a}-\gamma^{(a)}_{k_a}+1
\geq f_{m-a}-\gamma^{(a+1)}_1+1=i_{k_1+\cdots+k_a+1}.
\end{align*}
Thus, this together with \eqref{bijectionji} and \eqref{bijectionjka} shows that
$j_1\leq\ldots\leq j_t$ and $i_c\geq i_{c+1}$ whenever $j_c=j_{c+1}$, proving~\eqref{standard}.

Now,
if $j_{k_1+\cdots+k_a}=j_{k_1+\cdots+k_a+1}-1$ for some $1\leq a\leq m-1$, then by~\eqref{bijectionji}
we have $f_{m-a}=f_{m-a+1}+k_a$ and, by (SK3)
\begin{align*}
i_{k_1+\cdots+k_a}
=f_{m-a+1}-\gamma^{(a)}_{k_a}+k_a
=f_{m-a}-\gamma^{(a)}_{k_a}
\geq f_{m-a}-\gamma^{(a+1)}_1+1
=i_{k_1+\cdots+k_a+1}.
\end{align*}
This and \eqref{bijectionji} show that $\underline{u}_{\mathfrak s}=\underline{u}_f$ and $\underline{\gamma}_{\mathfrak s}=\underline{\gamma}$
if we put ${\mathfrak s}=\flat{{j_1j_2\ldots j_n}\choose{i_1i_2\ldots i_n}}$. Hence, ${\rm im}(\eta_{f})\supseteq\mathcal K^s_{f^*}(r)$.
\end{proof}

For $r\geq 1$, set
$$
\mathcal K^s(r)=\dot{\bigcup}_{f\in\mathfrak F}\mathcal K^s_{f}(r)=\dot{\bigcup}_{f\in\mathfrak F}\mathcal K^s_{f^*}(r),
$$
where $\dot{\cup}$ means a disjoint union.
Observe that
$$
\mathscr S^\mathbb Z_r=\dot{\bigcup}_{f\in\mathfrak F}
\mathscr S^\mathbb Z_{r,f}.
$$

For $\underline{\gamma}\in \mathcal K^s_{\underline{u}}(r)$, recall that $D^{\underline{\gamma}'}$ is the irreducible
$\mathcal H_{\underline{u}^{-1}}(r)$-module associated to $\underline{\gamma}'$ defined in Proposition~\ref{prop:BK}.

\begin{thm}\label{thm:standardK}
The bijective maps $\eta_{f}$ ($f\in\mathfrak F$) give rise to a bijection
$$\aligned
\eta:\mathscr S^\mathbb Z_r&\longrightarrow \mathcal K^s(r)\\
\mathfrak s&\longmapsto \underline{\gamma}_{\mathfrak s}.\endaligned$$
Moreover, the set $\{D^{\underline{\gamma}'}~|~\underline{\gamma}\in\mathcal K^s(r)\}$ after inflation forms
 a complete set of non-isomorphic irreducible objects in the category ${\mathcal H_\vtg(r)}$-{\sf mod}$^\mathbb Z$.
\end{thm}
\begin{proof}
By Proposition~\ref{prop:etau}, the first assertion is clear.
For $\underline{\gamma}\in\mathcal K^s_{f}(r)$, suppose $\eta^{-1}(\underline{\gamma})=\mathfrak s$.
Then $\underline{\gamma}=\underline{\gamma}_{{\mathfrak s}},\underline{u}_f= \underline{u}_{\mathfrak s}^{-1}$ and by Theorem~\ref{thm:VsDla} we have
$$
D^{\underline{\gamma}'_{\mathfrak s}}\cong V_{\mathfrak s}.
$$
Hence,
the set of modules $D^{\underline{\gamma}'}$ for $\underline{\gamma}\in\mathcal K^s(r)$
coincides with the set of modules $V_{\mathfrak s}$ with $\mathfrak s\in\mathscr S^\mathbb Z_r$ up to isomorphism.
Then the last assertion follows from Proposition~\ref{prop:Ro2}(5) by restricting to $\mathscr S_r^\mathbb Z$.
\end{proof}

\section{Identification of irreducible representations
of an Ariki-Koike algera of integral type}
In this section, we shall consider the inflation of irreducible representations
of Ariki-Koike algeras of integral type via the canonical surjective homomorphism
$\varpi_{\underline{u}}:{\mathcal H_\vtg(r)}\to\mc H_{\und{u}}(r)$ given in \eqref{pi}
and identify the corresponding integral multisegments. We will extend the isomorphisms
$D^{\underline{\lambda}'}\cong V_{\mathfrak s}$ for $\underline{\lambda}\in\mathcal K_f^s(r)$, where ${\mathfrak s}=\overline{{\bf s}_{\underline{\lambda}';f}^{{\mathsf c}}}$
(see Proposition~\ref{prop:stoK} and Theorem~\ref{thm:VsDla}), to all $\underline{\lambda}\in\mathcal K_f(r)$.

\subsection{Row residual segments of a multipartition.}
Given a sequence of integral segments  ${\bf s}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t)$ ($t\geq 1$)
with ${\mathsf s}_k=(q^{2i_k},\ldots,q^{2j_k})$, by Definition \ref{defn:order}, ${\bf s}$ is anti-standard  if it satisfies
\begin{equation}\label{orderVa}
i_1\geq i_2\geq\cdots\geq i_t,\text{ and } j_k\leq j_{k+1}
\text{ whenever } i_k=i_{k+1}.
\end{equation}
This order is exactly the so-called {\em right order} introduced in \cite[\S 2.3]{Va}.
Observe that any multisegment  ${\mathfrak s}\in\mathscr S^\mathbb Z_r$ contains
a unique anti-standard sequence ${\bf s}=({\mathsf s}_1,\ldots,{\mathsf s}_t)\in{\mathfrak s}$.

For a multisegment ${\mathfrak s}=\bar{{\bf s}}\in\mathscr S_r$, set ${\mathfrak s}^{-1}=\overline{{\bf s}^{-1}}=\overline{({\mathsf s}_1^{-1},\ldots,{\mathsf s}_t^{-1})}$. The function
$$(\;)^{-1}:\mathscr S_r^\mathbb Z\longrightarrow\mathscr S_r^\mathbb Z$$
is a bijection.

For ${\mathfrak s}\in\mathscr S^\mathbb Z_r$, assume that ${\bf s}=({\mathsf s}_1,\ldots,{\mathsf s}_t)\in{\mathfrak s}$ is anti-standard
and define
\begin{equation}\label{Ms}
M_{\mathfrak s}={\rm hd}({\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}\mathbb C_{{\mathsf s}_1}\otimes\cdots\otimes\mathbb C_{{\mathsf s}_t}),
\end{equation}
where $\mu=(|{\mathsf s}_{1}|,\ldots,|{\mathsf s}_{t}|)$. By \cite[Theorem 2.2]{Va}, $M_{\mathfrak s}$ is irreducible. We will reproduce the fact in Lemma \ref{lem:VsMs}.

Given $\underline{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})\in\mathcal P_m(r)$ and $f\in\mathfrak F_m$, define
the {\it i-th row residual segment} of $\lambda^{(k)}$
$$
{\mathsf s}^{(k)}_i=(q^{2(f_k+1-i)},q^{2(f_k+2-i)},\ldots,q^{2(f_k+\lambda^{(k)}_i-i)})\quad(1\leq k\leq m,1\leq i\leq\ell(\lambda^{(k)}))$$
by reading the residues of the nodes in
the $i$th row of $\lambda^{(k)}$ from left to right, and then form the sequence of segments in the order of row 1, row 2, ... of the first partition of $\underline{\lambda}$, of the second partition, and so on:
\begin{equation}\label{srla}
{\bf s}^{{\mathsf r}}_{\underline{\lambda};f}=({\mathsf s}^{(1)}_1,\ldots,{\mathsf s}^{(1)}_{\ell(\lambda^{(1)})},
\ldots,{\mathsf s}^{(m)}_1,\ldots,{\mathsf s}^{(m)}_{\ell(\lambda^{(m)})}).
\end{equation}

By~\eqref{scla} and~\eqref{srla}, a direct calculation shows that the following holds
\begin{equation}\label{c&r}
({\bf s}^{{\mathsf c}}_{\underline{\lambda};f})^{-1}={\bf s}^{{\mathsf r}}_{\underline{\lambda}';f^*}.
\end{equation}
for any $\underline{\lambda}\in\mathcal P_m(r)$ and $f\in\mathfrak F_m$.

\begin{prop}[{\cite[Theorem 3.4]{Va}}]\label{prop:Va}
Let $\underline{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})\in\mathcal K_{\underline{u}}(r)$ with $\underline{u}=\underline{u}_f$ and $f\in\mathfrak F_m$.
Assume that ${\mathfrak s}$ is the multisegment containing ${\bf s}^{{\mathsf r}}_{\underline{\lambda};f}$.
Then the ${\mathcal H_\vtg(r)}$-module ${\rm hd}\,{\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
({\rm ev}^*_{u_1}(S^{\lambda^{(1)}})\otimes\cdots\otimes {\rm ev}^*_{u_m}(S^{\lambda^{(m)}}))$
is irreducible, and moreover
$$
M_{{\mathfrak s}}\cong{\rm hd}\,{\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
({\rm ev}^*_{u_1}(S^{\lambda^{(1)}})\otimes\cdots\otimes {\rm ev}^*_{u_m}(S^{\lambda^{(m)}})),
$$
where $\mu=(|\lambda^{(1)}|,\ldots,|\lambda^{(m)}|)$.
\end{prop}

\begin{rem} (1)
Observe that ${\mathcal H_\vtg(r)}$ possesses an anti-automorphism $\ast$ with $T_i^\ast=T_i$ and $X_k^*=X_k$.
Using this we can introduce a natural duality $\circledast$ on finite dimensional left
${\mathcal H_\vtg(r)}$-modules: $M^{\circledast}$ denotes the dual space $M^*$ with the right action shifted to a left action by the anti-automorphism
$\ast$. It is known \cite[Corollary 3.7.5]{Kl} that
$$
({\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
{\rm ev}^*_{u_1}(S^{\lambda^{(1)}})\otimes\cdots\otimes {\rm ev}^*_{u_m}(S^{\lambda^{(m)}}))^{\circledast}
\cong {\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu^o)}
{\rm ev}^*_{u_m}(S^{\lambda^{(m)}})\otimes\cdots\otimes {\rm ev}^*_{u_1}(S^{\lambda^{(1)}})
$$
for any $\underline{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})\in\mathcal P_m(r)$ and $\underline{u}=(u_1,\ldots,u_m)=\underline{u}_f$
with $f\in\mathfrak F_m$, where $\mu=(|\lambda^{(1)}|,\ldots,|\lambda^{(m)}|)$ and $\mu^o=(|\lambda^{(m)}|,\ldots,|\lambda^{(1)}|)$.

(2)
It is straightforward to check that there exists an automorphism $\sharp$ on ${\mathcal H_\vtg(r)}$ defined by
\begin{align}
\sharp: {\mathcal H_\vtg(r)}&\longrightarrow{\mathcal H_\vtg(r)}\notag\\
T_i&\longmapsto -q^2T_i^{-1},\quad
X_k\longmapsto X_k^{-1}\label{sharp}
\end{align}
for $1\leq i\leq r-1$ and $1\leq k\leq r$.
Given an ${\mathcal H_\vtg(r)}$-module $M$, we denote by $M^{\sharp}$ the ${\mathcal H_\vtg(r)}$-module obtained from $M$ by
twisting the action via the automorphism $\sharp$.
It is easy to deduce that for
an $\mathcal H_{{\!\vartriangle\!}}(a)$-module $M$ and an $\mathcal H_{{\!\vartriangle\!}}(b)$-module $N$ with
$1\leq a,b\leq r$ and $a+b=r$,
we have
\begin{equation}\label{indsharp}
({\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(a)\otimes\mathcal H_{{\!\vartriangle\!}}(b)}M\otimes N)^\sharp
\cong {\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(a)\otimes\mathcal H_{{\!\vartriangle\!}}(b)}M^\sharp\otimes N^\sharp.
\end{equation}
\end{rem}

\subsection{Identification of irreducible representations.} We now prove the main result of this section.
\begin{lem}\label{lem:VsMs}
For any ${\mathfrak s}\in\mathscr S^\mathbb Z_r$,
$$
M^{\sharp}_{{\mathfrak s}^{-1}}\cong V_{\mathfrak s}
$$
and hence $M_{{\mathfrak s}}$ is irreducible.
\end{lem}
\begin{proof} Clearly, the second assertion follows from the first assertion.

Let ${\bf s}=({\mathsf s}_1,\ldots,{\mathsf s}_t)\in{\mathfrak s}$ be standard.
By Definition~\ref{defn:order},
${\bf s}^{-1}=({\mathsf s}_1^{-1},\ldots,{\mathsf s}_t^{-1})$ is the unique anti-standard element in ${\mathfrak s}^{-1}$
and hence by~\eqref{Ms}
$$
M_{{\mathfrak s}^{-1}}\cong
{\rm hd}({\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}\mathbb C_{{\mathsf s}_1^{-1}}\otimes\cdots\otimes\mathbb C_{{\mathsf s}_t^{-1}}).
$$
This together with~\eqref{indsharp} implies
$$
M^\sharp_{{\mathfrak s}^{-1}}\cong
{\rm hd}({\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}\mathbb C^\sharp_{{\mathsf s}_1^{-1}}\otimes\cdots\otimes\mathbb C^\sharp_{{\mathsf s}_t^{-1}}).
$$
Observe that $\mathbb C_{{\mathsf s}^{-1}}^\sharp\cong \mathbb C_{\widetilde{{\mathsf s}}}$ for any segment ${\mathsf s}$
and hence we obtain
\begin{align*}
M^\sharp_{{\mathfrak s}^{-1}}
\cong
{\rm hd}({\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}\mathbb C_{\widetilde{{\mathsf s}}_1}\otimes\cdots\otimes\mathbb C_{\widetilde{{\mathsf s}}_t})
\cong{\rm hd} (I_{{\bf s}})\cong V_{\bf s}\cong V_{{\mathfrak s}},
\end{align*}
where the second isomorphism is due to Lemma~\ref{lem:induced} and the third isomorphism follows from~Proposition \ref{prop:Ro3}
since ${\bf s}$ is standard.
\end{proof}

\begin{rem}
It is easy to check that ${\mathcal H_\vtg(r)}$ affords another automorphism
$\diamond$ which sends $ T_i,X_k$ to $ T_{r-i}, X_{r+1-k}^{-1}$ for $1\leq i\leq r-1$ and $1\leq k\leq r$, respectively, and moreover
the composition $\sharp\circ\diamond$ is exactly the Zelevinsky's involution~\cite{Ze} on ${\mathcal H_\vtg(r)}$.
By~\eqref{Ms}  and \cite[Corollary 6.1]{Va} one can directly compute that $M^{\diamond}_{{\mathfrak s}}\cong M_{{\mathfrak s}^{-1}}$.
Hence, Lemma~\ref{lem:VsMs} implies that $V_{{\mathfrak s}}$ is isomorphic to the twist of $M_{{\mathfrak s}}$
via Zelevinsky's involution.
\end{rem}

Observe also that the automorphism $\sharp$ on ${\mathcal H_\vtg(r)}$ also defines an automorphism
on $\mathcal{H}(r)$ still denoted by $\sharp$ which maps $T_w$ to $(-q^2)^{l(w)}T^{-1}_{w^{-1}}$
for $w\in\mathfrak S_r$. Since $q$ is not a root of unity, each Specht module $S^\lambda$
is self-dual for $\lambda\in\mathcal P(r)$.
Furthermore it is known \cite[Theorem 3.5]{DJ2} that
\begin{equation}\label{Spechtsharp}
(S^{\lambda})^\sharp\cong S^{\lambda'}
\end{equation}
for $\lambda\in\mathcal P(r)$.

Recall that for $\underline{\lambda}\in\mathcal K_{f^*}(r)$ with $f\in\mathfrak F$,
$D^{\underline{\lambda}'}$ is an irreducible $\mathcal H_{\underline{u}_f}(r)$-module, which can be also regarded as an
${\mathcal H_\vtg(r)}$-module by inflation.

We are now ready to generalise Theorem \ref{thm:VsDla} to arbitrary Kleshchev multipartitions.
\begin{thm}\label{thm:identify}
Suppose $\underline{\lambda}\in\mathcal P_m(r)$ such that $\underline{\lambda}'\in\mathcal K_{f^*}(r)$ for some $f\in\mathfrak F_m$.
Then there is an isomorphism of ${\mathcal H_\vtg(r)}$-modules:
$$
D^{\underline{\lambda}}\cong V_{{\bf s}^{{\mathsf c}}_{\underline{\lambda};f}}.
$$

\end{thm}
\begin{proof}
By Lemma~\ref{lem:VsMs} and~\eqref{c&r}, the following holds
\begin{equation}\label{Vscla}
V_{{\bf s}^{\mathsf c}_{\underline{\lambda};f}}
\cong M^\sharp_{{\bf s}^{\mathsf r}_{\underline{\lambda}';f^*}}.
\end{equation}
Since $\underline{\lambda}'\in\mathcal K_{f^*}(r)=\mathcal K_{\underline{u}^{-1}_f}(r)$, by Proposition~\ref{prop:Va} we obtain
$$
M_{{\bf s}^{{\mathsf r}}_{\underline{\lambda}';f^*}}
\cong {\rm hd}\big({\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
{\rm ev}^*_{u_m^{-1}}(S^{\lambda^{(m)'}})\otimes\cdots\otimes
{\rm ev}^*_{u_1^{-1}}(S^{\lambda^{(1)'}})\big),
$$
where $\mu=(|\lambda^{(m)}|,\ldots,|\lambda^{(1)}|)$ and $u_k=q^{2f_k}$ for $1\leq k\leq m$.
This together with~\eqref{Vscla} and~\eqref{indsharp} gives rise to
\begin{align}
V_{{\bf s}^{\mathsf c}_{\underline{\lambda};f}}
&\cong {\rm hd}({\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
{\rm ev}^*_{u_m^{-1}}(S^{\lambda^{(m)'}})\otimes\cdots\otimes
{\rm ev}^*_{u_1^{-1}}(S^{\lambda^{(1)'}}))^\sharp\notag\\
&\cong {\rm hd}~{\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
{\rm ev}^*_{u_m^{-1}}(S^{\lambda^{(m)'}})^\sharp\otimes\cdots\otimes
{\rm ev}^*_{u_1^{-1}}(S^{\lambda^{(1)'}})^\sharp. \label{Vsclaind}
\end{align}
By~\eqref{ev} and~\eqref{sharp}, one can easily check that
$$
{\rm ev}^*_u(M)^\sharp \cong {\rm ev}^*_{u^{-1}}(M^\sharp)
$$
for any $\mathcal{H}(r)$-module $M$ and hence by~\eqref{Spechtsharp}
$$
{\rm ev}^*_u(S^\lambda)^\sharp \cong {\rm ev}^*_{u^{-1}}((S^\lambda)^\sharp)
\cong {\rm ev}^*_{u^{-1}}(S^{\lambda'}).
$$
This means
\begin{align*}
{\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
({\rm ev}^*_{u_m^{-1}}(S^{\lambda^{(m)'}})^\sharp\otimes\cdots\otimes
{\rm ev}^*_{u_1^{-1}}(S^{\lambda^{(1)'}})^\sharp)
\cong {\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
({\rm ev}^*_{u_m}(S^{\lambda^{(m)}})\otimes\cdots\otimes
{\rm ev}^*_{u_1}(S^{\lambda^{(1)}})).
\end{align*}
Then by~\eqref{Vsclaind} we obtain
\begin{align*}
V_{{\bf s}^{{\mathsf c}}_{\underline{\lambda};f}}
\cong {\rm hd}~{\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}
({\rm ev}^*_{u_m}(S^{\lambda^{(m)}})\otimes\cdots\otimes
{\rm ev}^*_{u_1}(S^{\lambda^{(1)}}))
\cong D^{\underline{\lambda}},
\end{align*}
where the second isomorphism is due to Corollary~\ref{cor:hdSpecht}
since $\underline{\lambda}'\in\mathcal K_{\underline{u}_f^{-1}}(r)=\mathcal K_{f^*}(r)$.
This proves the theorem.
\end{proof}
\subsection{A branching property.} When the parameter $q$ is not a root of unity, the Hecke algebra $\mathcal{H}(r)$ is semisimple. Thus, in this case, the restriction to $\mathcal{H}(r)$ of an irreducible ${\mathcal H_\vtg(r)}$-module $V_{\mathfrak s}$ is a direct sum of Specht modules $S^\lambda$, $\lambda\vdash r$. The determination of $S^\lambda$ which appears in $V_{\mathfrak s}$ is called the affine branching rule in \cite[Problem 4.3.6]{DDF}.

We now look at a relatively simple problem. Given $\underline{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})\in\mathcal P_m(r)$,
let $\underline{\lambda}^+=(\underline{\lambda}^+_1,\underline{\lambda}^+_2,\ldots)\in\mathcal P(r)$ with
$$
\underline{\lambda}^+_i=\lambda^{(1)}_i+\lambda^{(2)}_i+\cdots+\lambda^{(m)}_i, \text { for }i\geq 1.
$$
In other words, $\underline{\lambda}^+=\lambda^{(1)}+\cdots+\lambda^{(m)}$.

Generally speaking, by~\eqref{slaind} and the Littlewood-Richardson rule, one can deduce that
the Specht module $S^{\underline{\lambda}^+}$ over $\mathcal{H}(r)$
occurs in the restriction $S^{\underline{\lambda}}|_{\mathcal{H}(r)}$ with multiplicity one. (One can also prove this by an induced cell module argument; see \cite[Theorem 2.8]{Roi}.)

If, in addition, $\underline{\lambda}'$ is a Kleshchev multipartition,
does $S^{\underline{\lambda}^+}$ appear in $D^{\underline{\lambda}}|_{\mathcal{H}(r)}$? This sounds like a simple question but we didn't see a straightforward proof in literature.

\begin{cor}\label{cor:quotient}
Let $\underline{\lambda}\in\mathcal P_m(r)$
such that $\underline{\lambda}'\in\mathcal K_{f^*}(r)$ with $f\in\mathfrak F_m$.
Then
\begin{enumerate}
\item $V_{{\bf s}^{\mathsf c}_{\underline{\lambda};f}}$ is a quotient of $I_{{\bf s}^{\mathsf c}_{\underline{\lambda};f}}$.

\item The Specht $\mathcal{H}(r)$-module $S^{\underline{\lambda}^+}$  occurs
with multiplicity one in the restriction $D^{\underline{\lambda}}|_{\mathcal{H}(r)}$.
\end{enumerate}
\end{cor}
\begin{proof}
Part (1)  follows from Lemma~\ref{lem:Ibfs} and Theorem~\ref{thm:identify}.
By Proposition~\ref{prop:Ro2}(3) and Theorem~\ref{thm:identify},
 the cell module $E_{\mu({\bf s}^{\mathsf c}_{\underline{\lambda};f})}$
occurs with multiplicity one in $D^{\underline{\lambda}}$ as $\mathcal{H}(r)$-modules.
Observe that by~\cite[I, (1.8)]{Mac} and~\eqref{mubfs} we have
$$
\underline{\lambda}^+=((\underline{\lambda}')^\vee)'=\mu({\bf s}^{\mathsf c}_{\underline{\lambda};f})'
$$
and, hence, $S^{\underline{\lambda}^+}\cong E_{\mu({\bf s}^c_{\underline{\lambda};f})}$ occurs with multiplicity one in $D^{\underline{\lambda}}|_{\mathcal{H}(r)}$.
\end{proof}

\begin{rem}\label{rem:Vazirani}
It would be interesting to find a direct approach to verifying that the Specht module
$S^{\underline{\lambda}^+}$ must occur in $D^{\underline{\lambda}}|_{\mathcal{H}(r)}$
whenever $D^{\underline{\lambda}}$ is nonzero.
Then combining this with Lemma~\ref{lem:Ibfs} gives an elementary
way to determine the multisegments corresponding to the irreducible modules associated
to Kleshchev multipartitions without using Proposition~\ref{prop:Va} of  Vazirani.

\end{rem}
\section{Drinfeld polynomials of integral type}
In this section, we shall compute the Drinfeld polynomials
associated with the irreducible ${\rm U}_q(\widehat{\frak{gl}}_n)$-modules which are constructed by the irreducible $\mc H_{\und{u}}(r)$-modules associated with Kleshchev multipartitions.

\subsection{\bf The affine Schur-Weyl duality. }
 The {\it quantum loop algebra} ${\rm U}_q(\widehat{\frak{gl}}_n)$ (or {\it
quantum affine $\mathfrak {gl}_n$}) is the $\mathbb C$-algebra generated by ${\tt x}^\pm_{i,s}$
($1\leq i<n$, $s\in\mathbb Z$), ${\tt k}_i^{\pm1}$ and ${\tt g}_{i,t}$ ($1\leq
i\leq n$, $t\in\mathbb Z\backslash\{0\}$) with the following relations:
\begin{itemize}
 \item[(QLA1)] ${\tt k}_i{\tt k}_i^{-1}=1={\tt k}_i^{-1}{\tt k}_i,\,\;[{\tt k}_i,{\tt k}_j]=0$;
 \item[(QLA2)]
 ${\tt k}_i{\tt x}^\pm_{j,s}=q^{\pm(\delta_{i,j}-\delta_{i,j+1})}{\tt x}^\pm_{j,s}{\tt k}_i,\;
               [{\tt k}_i,{\tt g}_{j,s}]=0$;
 \item[(QLA3)] $[{\tt g}_{i,s},{\tt x}^\pm_{j,t}]
               =\begin{cases}0,\;\;&\text{if $i\not=j,\,j+1$};\\
                  \pm q^{-js}\frac{[s]_q}{s}{\tt x}^\pm_{j,s+t},\;\;\;&\text{if $i=j$};\\
                  \mp q^{-js}\frac{[s]_q}{s}{\tt x}_{j,s+t}^\pm,\;\;\;&\text{if $i=j+1$;}
                \end{cases}$
 \item[(QLA4)] $[{\tt g}_{i,s},{\tt g}_{j,t}]=0$;
 \item[(QLA5)] $[{\tt x}_{i,s}^+,{\tt x}_{j,t}^-]=\delta_{i,j}\frac{\phi^+_{i,s+t}-\phi^-_{i,s+t}}{q-q^{-1}}$;
 \item[(QLA6)] ${\tt x}^\pm_{i,s}{\tt x}^\pm_{j,t}={\tt x}^\pm_{j,t}{\tt x}^\pm_{i,s}$, for $|i-j|>1$, and
 $[{\tt x}_{i,s+1}^\pm,{\tt x}^\pm_{j,t}]_{q^{\pm c_{i,j}}}
               =-[{\tt x}_{j,t+1}^\pm,{\tt x}^\pm_{i,s}]_{q^{\pm c_{i,j}}}$;
 \item[(QLA7)] $[{\tt x}_{i,s}^\pm,[{\tt x}^\pm_{j,t},{\tt x}^\pm_{i,p}]_q]_q
               =-[{\tt x}_{i,p}^\pm,[{\tt x}^\pm_{j,t},{\tt x}^\pm_{i,s}]_q]_q\;$ for
               $|i-j|=1$,
\end{itemize}
 where $[x,y]_a=xy-ayx$, $C=(c_{i,j})_{i,j\in I}$ denotes the generalized Cartan matrix of type $\tilde{A}_{n-1}$,
and $\phi_{i,s}^\pm$ are defined via the generating functions in indeterminate $u$ by
$$\Phi_i^\pm(u):={\tilde{\tt k}}_i^{\pm 1}
\exp\bigl(\pm(q-q^{-1})\sum_{m\geq 1}{\tt h}_{i,\pm m}u^{\pm
m}\bigr)=\sum_{s\geq 0}\phi_{i,\pm s}^\pm u^{\pm s}$$ with
$\tilde{\tt k}_i={\tt k}_i/{\tt k}_{i+1}$ (${\tt k}_{n+1}={\tt k}_1$) and ${\tt h}_{i,\pm
m}=q^{\pm(i-1)m}{\tt g}_{i,\pm m}-q^{\pm(i+1)m}{\tt g}_{i+1,\pm
m}\,(1\leq i<n).$

Consider the $\mathbb C$-space $V_n$
spanned by $\{\omega_1,\ldots,\omega_n\}$ and set
$$\Omega_{(n)}=V_n\otimes_{\mathbb C}\mathbb C[X,X^{-1}]=V_n[X,X^{-1}].$$
Following \cite{VV}, there is a right action of ${\mathcal H_\vtg(r)}_\mathbb C$ on $\Omega_{(n)}^{\otimes r}$.
Then the endomorphism algebra
\begin{align}
{\mathcal S}_{\vtg}(\cycn,r):={\rm End}_{{\mathcal H_\vtg(r)}}(\Omega_{(n)}^{\otimes r})
\end{align}
 is known as the { affine $q$-Schur algebra}.  In particular,
$\Omega_{(n)}^{\otimes r}$ becomes an ${\mathcal S}_{\vtg}(\cycn,r)$-${\mathcal H_\vtg(r)}$-bimodule.
By a double Hall algebra interpretation of ${\rm U}_q(\widehat{\frak{gl}}_n)$
(see \cite[\S 3.5]{DDF}), there is a left action
of ${\rm U}_q(\widehat{\frak{gl}}_n)$-module on $\Omega_{(n)}^{\otimes r}$ which
commutes with the right action of ${\mathcal H_\vtg(r)}$ on $\Omega_{(n)}^{\otimes r}$.
Furthermore, this gives rise to an algebra homomorphism
\begin{align}\label{afglact}
\zeta_r:{\rm U}_q(\widehat{\frak{gl}}_n)\longrightarrow {\rm End}_{{\mathcal H_\vtg(r)}}(\Omega_{(n)}^{\otimes r})={\mathcal S}_{\vtg}(\cycn,r),
\end{align}
which is surjective \cite[Corollary 3.8.4]{DDF}.
We omit the details of the actions of ${\mathcal H_\vtg(r)}$ and ${\rm U}_q(\widehat{\frak{gl}}_n)$
on $\Omega_{(n)}^{\otimes r}$ and the construction for $\zeta_r$ here as it is irrelevant in this paper.

\subsection{\bf Identification of irreducible representations of $\mathcal S_{\!\vartriangle\!}(n,r)$. }
For $1\leq i\leq n$ and $s\in\mathbb Z$, define the elements $\mathscr
Q_{i,s}\in{\rm U}_q(\widehat{\frak{gl}}_n)$ through the generating functions
\begin{equation}\label{exp}
\begin{split}
&\quad\qquad\mathscr Q_i^\pm(x):=\exp\bigg(-\sum_{t\geq
1}\frac{1}{[t]_q}{\tt g}_{i,\pm t} (qx)^{\pm t}\bigg)=\sum_{s\geq
0}\mathscr Q_{i,\pm s} x^{\pm s}\in{\rm U}_q(\widehat{\frak{gl}}_n)[[x,x^{-1}]].
\end{split}
\end{equation}
Let $V$ be a finite dimensional ${\rm U}_q(\widehat{\frak{gl}}_n)$-module and let $\lambda=(\lambda_1,\ldots,\lambda_n)\in\mathbb Z^n$.
A nonzero ($\lambda$-weight) vector $w\in V$ is called a {\it pseudo-highest weight
vector}\index{pseudo-highest weight vector} if there exist
some $Q_{i,s}\in\mathbb C$ such that
\begin{equation}\label{HWvector}
{\tt x}_{j,s}^+w=0,\quad\mathscr Q_{i,s}w=Q_{i,s}w,\quad\text{and}\quad
{\tt k}_iw=q^{\lambda_i}w,
\end{equation}
for all $1\leq i\leq n$, $1\leq j< n$, and $s\in\mathbb Z$. A ${\rm U}_q(\widehat{\frak{gl}}_n)$-module
 $V$ is called a {\it pseudo-highest weight module} if $V={\rm U}_q(\widehat{\frak{gl}}_n) w$
for some pseudo-highest weight vector $w$.

Following \cite{FM}, an $n$-tuple of polynomials
$\mathbf{Q}=(Q_1(x),\ldots,Q_n(x))$ with constant terms $1$ is called {\it
dominant} if, for each $1\leq i\leq n-1$, the ratio
$Q_i(xq^{i-1})/Q_{i+1}(xq^{i+1})$ is a polynomial in $x$. Let
${\mathcal Q}(n)$ be the set of dominant $n$-tuples of polynomials.

For a polynomial $Q(x)=\prod_{1\leq i\leq m}(1-a_ix)\in\mathbb C[x]$
with $a_i\in\mathbb C^*$, put $Q^+(x)=Q(x)$ and define
$$Q^-(x)=\prod_{1\leq i\leq m}(1-a_i^{-1}x^{-1})\in\mathbb C[x^{-1}].$$
 Given a $\mathbf{Q}=(Q_1(x),\ldots,Q_{n}(x))\in{\mathcal Q}(n)$, define
$Q_{i,s}\in\mathbb C$, for $1\leq i\leq n$ and $s\in\mathbb Z$, by the
following formula
\begin{equation}\label{Qis}
Q_i^\pm(x)=\sum_{s\geq 0}Q_{i,\pm s}x^{\pm s}.
\end{equation}
Let $I(\mathbf{Q})$ be the left ideal of ${\rm U}_q(\widehat{\frak{gl}}_n)$ generated by ${\tt x}_{j,s}^+ ,\mathscr
Q_{i,s}-Q_{i,s},$ and ${\tt k}_i-q^{\lambda_i}$, for $1\leq j<n$,
$1\leq i\leq n$, and $s\in\mathbb Z$, where $\lambda_i=\mathrm{deg\,}Q_i(x)$,
and define
\begin{equation}\label{MQ}
M(\mathbf{Q})={\rm U}_q(\widehat{\frak{gl}}_n)/I(\mathbf{Q}).
\end{equation}
Then $M(\mathbf{Q})$ has a unique simple quotient, denoted by $L(\mathbf{Q})$.
The polynomial $Q_i(x)$ are called {\em Drinfeld polynomials} associated to $L(\mathbf{Q})$.
Clearly, $L(\mathbf{Q})$ is a pseudo-highest weight module with pseudo-highest weight
$\lambda=(\lambda_1,\ldots,\lambda_n)\in\Lambda(n,r)$, where $\Lambda(n,r)$ denotes the set
of compositions of $r$ into $n$ parts; see \eqref{Lanr}.

Let
$${\mathcal Q}(n)_r=\big\{\mathbf{Q}=(Q_1(x),\ldots,Q_n(x))\in{\mathcal Q}(n)\mid r=\sum_{1\leq i\leq n}\mathrm{deg}\, Q_i(x)\big\}.$$

Observe that given a left ${\mathcal H_\vtg(r)}$-module $M$, the tensor product
$\Omega_{(n)}^{\otimes r}\otimes_{{\mathcal H_\vtg(r)}} M$ naturally affords a left ${\mathcal S}_{\vtg}(\cycn,r)$-module.
Recall that $\{V_{{\mathfrak s}}|{\mathfrak s}\in\mathscr S_r\}$ is a complete set of simple ${\mathcal H_\vtg(r)}$-modules.
Further, set
$$
\mathscr S_r^{(n)}=\{{\mathfrak s}=
\{{\mathsf s}_1,\ldots,{\mathsf s}_t\}\in\mathscr S_r\mid t\geq 1,\,|{\mathsf s}_i|\leq n,\,\forall i\}.
$$
Clearly, if $n\geq r$, then $\mathscr S_r^{(n)}=\mathscr S_r$.

Given $n,r\geq1$ and $\mathfrak s\in\mathscr S_r^{(n)}$,
take ${\bf s}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t)\in\mathfrak s$
with
$$
{\mathsf s}_k=(z_kq^{2i_k},z_kq^{2(i_k+1)},\ldots,z_kq^{2j_k})\in(\mathbb C^*)^{r_k},
$$
where $r_k=|{\mathsf s}_k|=j_k-i_k+1\in\mathbb Z_+$ for $1\leq k\leq t$.
We set
$$
\partial(\mathfrak s):=\mathbf{Q}_{\mathfrak s}=(Q_1(x),\ldots,Q_n(x))\in {\mathcal Q}(n)_r
$$ by setting recursively
\begin{equation}\label{partialQ}
Q_k(x)
=\prod_{k\leq l\leq n}P_{l}(xq^{l+1-2k})
=P_k(xq^{-k+1})P_{k+1}(xq^{-k+2})\cdots
P_{n-1}(xq^{n-2k})P_n(xq^{n+1-2k}),
\end{equation}
where
\begin{equation}\label{partialP}
P_{k}(x)=\prod_{1\leq b\leq t\atop r_b=k}(1-z_bq^{i_b+j_b}x)
\end{equation}
for $1\leq k\leq n$.
Here we use the convention that $P_i(x)=1$ if there does not exist $1\leq j\leq t$
such that $r_j=i$.
This gives a map
$$\partial: \mathscr S^{(n)}_r\rightarrow {\mathcal Q}(n)_r.$$
Thus, we obtain the associated simple ${\rm U}_q(\widehat{\frak{gl}}_n)$-module $L(\mathbf{Q}_{\mathfrak s})$.
The following result is due to Deng, Du and Fu \cite{DD,DDF,DF}.

\begin{prop}[{\cite[Theorem 6.6]{DD}, \cite[Theorem 4.4]{DF}}] \label{prop:DF&DD}
Assume $n,r\geq 1$. Then
\begin{enumerate}
\item
Both the set $\big\{L(\mathbf{Q})\mid\mathbf{Q}\in{\mathcal Q}(n)_r\big\}$
and the set
$
\{\Omega_{(n)}^{\otimes r}\otimes_{{\mathcal H_\vtg(r)}} V_{\mathfrak s}~|\mathfrak s\in\mathscr S_r^{(n)}\}
$
are complete sets of non-isomorphic irreducible representations of ${\mathcal S}_{\vtg}(\cycn,r)$.
\item
For each $\mathfrak s\in\mathscr S_r^{(n)}$, there is a
${\rm U}_q(\widehat{\frak{gl}}_n)$-module isomorphism (or equivalently, an ${\mathcal S}_{\vtg}(\cycn,r)$-module isomorphism)
$$
L(\mathbf{Q}_{\mathfrak s})\cong \Omega_{(n)}^{\otimes r}\otimes_{{\mathcal H_\vtg(r)}} V_{\mathfrak s}.
$$
Hence, the map $\partial$ is a bijection.
\end{enumerate}
\end{prop}

\subsection{Drinfeld polynomials of integral type.} Recall from \S2.4 the category ${\mathcal H_\vtg(r)}$-${\sf mod}^{\mathbb Z}$
whose simple objects are indexed by the set $\mathscr S_r^{\mathbb Z}$ of integral multisegment and consider
the functor
$${\mathcal F}:{\mathcal H_\vtg(r)}\text{-{\sf mod}}^{\mathbb Z}\longrightarrow{\mathcal S}_{\vtg}(\cycn,r)\text{-{\sf mod}},\quad L\longmapsto\Omega_{(n)}^{\otimes r}\otimes_{{\mathcal H_\vtg(r)}} L.$$
We form full subcategory ${\mathcal S}_{\vtg}(\cycn,r)\text{-{\sf mod}}^{\mathbb Z}$ whose objects are isomorphic to ${\mathcal F}(L)$ for $L\in
\text{Ob}({\mathcal H_\vtg(r)}\text{-{\sf mod}}^{\mathbb Z})$. We now show that simple objects in ${\mathcal S}_{\vtg}(\cycn,r)\text{-{\sf mod}}^{\mathbb Z}$ for all $r\geq0$
determine all irreducible polynomial representations of ${\rm U}_q(\widehat{\frak{gl}}_n)$.

A polynomial $Q(x)\in\mathbb C[x]$ of constant term 1 is said to be of {\em integral type} if its roots are powers of $q^2$.
Let ${\mathcal Q}(n)^{\mathbb Z}$ be the set of dominant $n$-tuples of polynomials of integral type.
Set
\begin{align*}
{\mathcal Q}(n)^{\mathbb Z}_r&={\mathcal Q}(n)^{\mathbb Z}\cap {\mathcal Q}(n)_r\\
\mathscr S_r^{(n),\mathbb Z}&=\mathscr S^\mathbb Z_r\cap\mathscr S^{(n)}_r.
\end{align*}

Given $\mathfrak s\in\mathscr S_r^{(n)}$ with
${\bf s}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t)\in\mathfrak s$
and
$$
{\mathsf s}_k=(z_kq^{2i_k},z_kq^{2(i_k+1)},\ldots,z_kq^{2j_k})
$$
for $1\leq k\leq t$.
Write $\mathbf{Q}_{{\mathfrak s}}=(Q_1(x),\ldots,Q_n(x))$.
Observe that by~\eqref{partialQ} and~\eqref{partialP} we have
\begin{align}
Q_k(x)=\prod_{k\leq l\leq n}P_{l}(xq^{l+1-2k})
=&\prod_{k\leq l\leq n}\prod_{1\leq b\leq t,\atop j_b-i_b+1=l}(1-z_bq^{i_b+j_b+l+1-2k}x)\notag\\
=&\prod_{1\leq b\leq t,\atop j_b-i_b+1\geq k}(1-z_bq^{2(j_b+1-k)}x).
\label{partialQ1}
\end{align}
This implies that $\mathbf{Q}_{\mathfrak s}\in{\mathcal Q}(n)^{\mathbb Z}_r$
if and only if $\mathfrak s\in\mathscr S_r^{(n),\mathbb Z}$.
Hence, the restriction of the map $\partial$ gives a bijection
\begin{equation}\label{partialint}
\partial^\mathbb Z: \mathscr S_r^{(n),\mathbb Z}\longrightarrow{\mathcal Q}(n)^{\mathbb Z}_r.
\end{equation}

For $a\in\mathbb C^*$, analogous to~\eqref{sigma}, there is a Hopf algebra automorphism
$\omega_a:{\rm U}_q(\widehat{\frak{gl}}_n)\longrightarrow {\rm U}_q(\widehat{\frak{gl}}_n)$ defined by
\begin{equation}
{\tt x}^\pm_{i,s}\mapsto a^s{\tt x}^\pm_{i,s},\quad {\tt g}_{j,t}\mapsto a^t{\tt g}_{j,t}, \quad{\tt k}_j^{\pm1}\mapsto {\tt k}_j^{\pm1}\label{omega}.
\end{equation}
Thus, we may twist a ${\rm U}_q(\widehat{\frak{gl}}_n)$-module $M$ by $\omega_a$ to get a
new module denoted by $M^{\omega_a}$.
For $\mathbf{Q}=(Q_1(x),\ldots,Q_n(x))\in{\mathcal Q}(n)_r$, define
$\mathbf{Q}^{\omega_a}=(Q^{\omega_a}_1(x),\ldots,Q^{\omega_a}_n(x))$ by setting
\begin{equation}\label{Qomega}
Q^{\omega_a}_k(x)=Q_k(ax)
\end{equation}
for $1\leq k\leq n$.
Analogous to Lemma~\ref{lem:Vs-sigma}, we have the following.
\begin{lem}\label{LQomega}
Suppose $a\in\mathbb C^*$ and $\mathbf{Q}=(Q_1(x),\ldots,Q_n(x))\in{\mathcal Q}(n)_r$.
Then
$$
L(\mathbf{Q})^{\omega_a}\cong L(\mathbf{Q}^{\omega_a}).
$$
\end{lem}
\begin{proof}
Let $w$ be a pseudo-highest weight vector of $L(\mathbf{Q})$.
By~\eqref{omega}, \eqref{exp} and~\eqref{HWvector},  $\omega_a(\mathscr Q_{i,s})=a^s\mathscr Q_{i,s}$
and hence
$$
\omega_a({\tt x}^+_{j,s})w=0,\quad \omega_a(\mathscr Q_{i,s})w=a^s\mathscr Q_{i,s}w=a^sQ_{i,s}w, \quad \omega_a({\tt k}_i)w=q^{\lambda_i}w,
$$
where $\lambda_i={\rm deg}\,Q_i(x)$ and $Q_{i,s}$ is defined in \eqref{Qis} for
$1\leq i\leq n, 1\leq j<n$ and $s\in\mathbb Z$.
This implies that $w$ is also a pseudo-highest weight vector of $L(\mathbf{Q})^{\omega_a}$ and moreover
$L(\mathbf{Q})^{\omega_a}$ is a quotient of $M(\mathbf{Q}^{\omega_a})$ by~\eqref{MQ} and~\eqref{Qomega}.
Then the lemma is proved since $L(\mathbf{Q})^{\omega_a}$ is irreducible.
\end{proof}

 It is known from \cite[\S4.3]{FM} that $L(\mathbf{Q})$ for $\mathbf{Q}\in{\mathcal Q}(n)$ are all non-isomorphic
finite dimensional irreducible polynomial representations of ${\rm U}_q(\widehat{\frak{gl}}_n)$. (We refer the reader to \cite{DDF,FM}
for the definition of polynomial representations of ${\rm U}_q(\widehat{\frak{gl}}_n)$.)
We now use the discussion in \S2.4 and Proposition 6.2.1 to prove the following.

\begin{theorem} \label{thm:intDrinfeld}
For $\mathbf{Q}\in{\mathcal Q}(n)$, there exist $\mathbf{Q}^{(1)},\ldots,\mathbf{Q}^{(p)}\in{\mathcal Q}(n)^{\mathbb Z}$ and $a_1,\ldots,a_p\in\mathbb C^*$
such that as ${\rm U}_q(\widehat{\frak{gl}}_n)$-modules
$$
L(\mathbf{Q})\cong L(\mathbf{Q}^{(1)})^{\omega_{a_1}}\otimes\cdots\otimes L(\mathbf{Q}^{(p)})^{\omega_{a_p}}.
$$
\end{theorem}
\begin{proof}
Write $\mathbf{Q}=(Q_1(u),\ldots,Q_n(u))$ and let $r=\sum_i{\rm deg}Q_i(u)$.
By Proposition~\ref{prop:DF&DD},
there exists ${\mathfrak s}\in\mathscr S^{(n)}_r$ such that $\mathbf{Q}=\mathbf{Q}_{{\mathfrak s}}$ and
\begin{equation}\label{LQ&Vs}
L(\mathbf{Q})\cong \Omega_{(n)}^{\otimes r}\otimes_{{\mathcal H_\vtg(r)}} V_{{\mathfrak s}}.
\end{equation}
Suppose ${\bf s}\in{\mathfrak s}$ is
standard in the sense of Definition~\ref{defn:order}(2)
with the decomposition
${\bf s}={\bf s}^{(1)}\cup{\bf s}^{(2)}\cup\cdots\cup{\bf s}^{(p)}$ and
${\bf s}^{(k)}$ lying on the line $L_{a_k}$ for $1\leq k\leq p$.
Let ${\mathfrak s}^{(k)}$ be the multisegment containing ${\bf s}^{(k)}$ for $1\leq k\leq p$.
By~\eqref{gen&int}, \eqref{LQ&Vs}, and Proposition~\ref{prop:DF&DD}, we have the following isomorphisms of ${\rm U}_q(\widehat{\frak{gl}}_n)$-modules
\begin{align}
L(\mathbf{Q})&\cong \Omega_{(n)}^{\otimes r}\otimes_{{\mathcal H_\vtg(r)}}
{\rm ind}^{{\mathcal H_\vtg(r)}}_{\mathcal H_{{\!\vartriangle\!}}(\mu)}(V_{{\mathfrak s}^{(1)}}\otimes\cdots\otimes V_{{\mathfrak s}^{(p)}})\notag\\
&\cong\Omega_{(n)}^{\otimes r}\otimes_{\mathcal H_{{\!\vartriangle\!}}(\mu)}\big(V_{{\mathfrak s}^{(1)}}\otimes\cdots\otimes V_{{\mathfrak s}^{(p)}}\big)\notag\\
&\cong \big(\Omega_{(n)}^{\otimes r_1}\otimes_{\mathcal H_{{\!\vartriangle\!}}(r_1)}V_{{\mathfrak s}^{(1)}}\big)
\otimes\cdots\otimes\big(\Omega_{(n)}^{\otimes r_p}\otimes_{\mathcal H_{{\!\vartriangle\!}}(r_p)}V_{{\mathfrak s}^{(p)}}\big)\notag\\
&\cong L(\mathbf{Q}_{{\mathfrak s}^{(1)}})\otimes\cdots\otimes L(\mathbf{Q}_{{\mathfrak s}^{(p)}}), \label{LQtensor}
\end{align}
where $\mu=(r_1,\ldots,r_p)$ and $r_k=\sum_i|{\mathsf s}^{(k)}_i|$ for $1\leq k\leq p$.

Furthermore, set
\begin{equation}\label{intsegment}
{\bf s}^{(k)'}=({\bf s}^{(k)})^{\sigma_{a_k^{-1}}}
\end{equation}
for $1\leq k\leq p$.
Then ${\mathfrak s}^{(k)'}:=\overline{{\bf s}^{(k)'}}\in\mathscr S^\mathbb Z_{r_k}$ and hence
$$
\mathbf{Q}^{(k)}:=\mathbf{Q}_{{\mathfrak s}^{(k)'}}\in{\mathcal Q}(n)_{r_k}^{\mathbb Z}
$$
for $1\leq k\leq p$.
Then by~\eqref{intsegment}, \eqref{partialQ1} and \eqref{s-sigma}, one can easily deduce that
$$
\mathbf{Q}_{{\mathfrak s}^{(k)}}=\mathbf{Q}_{({\mathfrak s}^{(k)'})^{\sigma_{a_k}}}=\mathbf{Q}^{\omega_{a_k}}_{{\mathfrak s}^{(k)'}}=(\mathbf{Q}^{(k)})^{\omega_{a_k}}
$$
for $1\leq k\leq p$.
This together with Lemma~\ref{LQomega} and~\eqref{LQtensor} gives rise to an isomorphism of
${\rm U}_q(\widehat{\frak{gl}}_n)$-modules
$$
L(\mathbf{Q})\cong L(\mathbf{Q}^{(1)})^{\omega_{a_1}}\otimes\cdots\otimes L(\mathbf{Q}^{(p)})^{\omega_{a_p}},
$$
as desired.
\end{proof}
\begin{rem}
Retain the notations in Theorem~\ref{thm:intDrinfeld}.
Observe that ${\mathcal S}_{{\!\vartriangle\!}}({{n}},r)$ can be naturally regarded as a subalgebra of the tensor product
${\mathcal S}_{{\!\vartriangle\!}}({{n}},r_1)\otimes\cdots\otimes{\mathcal S}_{{\!\vartriangle\!}}({{n}},r_p)$.
Then, by~\eqref{LQtensor},
$L(\mathbf{Q})$ is isomorphic to the restriction to ${\mathcal S}_{{\!\vartriangle\!}}({{n}},r)$
of the irreducible
${\mathcal S}_{{\!\vartriangle\!}}({{n}},r_1)\otimes\cdots\otimes{\mathcal S}_{{\!\vartriangle\!}}({{n}},r_p)$-module
$L(\mathbf{Q}_{{\mathfrak s}^{(1)}})\otimes\cdots\otimes L(\mathbf{Q}_{{\mathfrak s}^{(p)}})$.
Hence by Theorem~\ref{thm:intDrinfeld}, to study irreducible ${\mathcal S}_{\vtg}(\cycn,r)$-modules
or finite dimensional irreducible polynomial representation of ${\rm U}_q(\widehat{\frak{gl}}_n)$, it is enough to consider those $L(\mathbf{Q})$
associated with Drinfeld polynomials $\mathbf{Q}$ of integral type.
In other words, every irreducible polynomial representation of ${\rm U}_q(\widehat{\frak{gl}}_n)$  is determined
by the simple objects in the category ${\rm U}_q(\widehat{\frak{gl}}_n)\text{-{\sf mod}}^{\mathbb Z}:=\oplus_{r\geq0}{\mathcal S}_{\vtg}(\cycn,r)\text{-{\sf mod}}^{\mathbb Z}$.
\end{rem}

\subsection{\bf Kleshchev multipartitions and Drinfeld polynomials.}
For $f\in\mathfrak F$ and $r\geq 1$,
let
\begin{align*}
\mathcal K^{s}_{f,(n)}(r)
&=\{\underline{\gamma}\in\mathcal K^s_{f}(r)~|~\gamma^{(k)}_1\leq n, \forall k\geq 1\}\\
\mathcal K^{s}_{(n)}(r)&=\dot{\bigcup}_{f\in\mathfrak F}\mathcal K^{s}_{f,(n)}(r).
\end{align*}
Recall from Theorem \ref{thm:standardK} the bijective map $\theta=\eta^{-1}:\mathcal K^s(r)\to\mathscr S_r^{\mathbb Z}$. Restriction induces a bijection
$$\theta^{\mathbb Z}:\mathcal K^s_{(n)}(r)\to\mathscr S_r^{(n),\mathbb Z}.$$

The two parts of the following result are the counterparts of Theorems \ref{thm:standardK} and
\ref {thm:identify}, respectively.

\begin{thm}\label{thm:K&D}
The following holds for $n,r\geq 1$:
\begin{enumerate}
\item There is a bijection
$$
\partial^{\mathbb Z}\circ\theta^{\mathbb Z}: \mathcal K^{s}_{(n)}(r)\longrightarrow
{\mathcal Q}(n)_r^{\mathbb Z}
$$
such that, for each $\mathbf{Q}\in{\mathcal Q}(n)^{\mathbb Z}_r$, if $\mathbf{Q}=\partial^{\mathbb Z}\circ\theta^{\mathbb Z}(\underline{\gamma})$ for some $\underline{\gamma}\in {\mathcal K}^{s}_{(n)}(r)$, then the following isomorphism of ${\rm U}_q(\widehat{\frak{gl}}_n)$-modules (or ${\mathcal S}_{\vtg}(\cycn,r)$-modules) holds
$$
L(\mathbf{Q})\cong \Omega_{(n)}^{\otimes r}\otimes_{{\mathcal H_\vtg(r)}} D^{\underline{\gamma}'}.
$$

\item Let $\underline{\lambda}\in\mathcal P_m(r)$ and $f\in\mathfrak F$.
Assume that $\underline{\lambda}'\in\mathcal K_{f^*}(r)$ and
$\ell(\lambda^{(k)})\leq n$ for $1\leq k\leq m$
and write ${\bf s}={\bf s}^{{\mathsf c}}_{\underline{\lambda};f}$ and $\mathfrak s=\bar{{\bf s}}$.
Then $\mathbf{Q}_{\mathfrak s}=(Q_1(x),Q_2(x),\ldots,Q_n(x))$ are given by
$$
Q_i(x)=\prod_{1\leq k\leq m, 1\leq j\leq \lambda^{(k)}_i}(1-q^{2(f_k+j-i)}x).
$$
In other words, for every $1\leq i\leq n$,
$Q_i(x)$ is the polynomial whose roots are determined by
the residues (with respect to $\underline{u}_f$)
of the nodes in $i$th row of $\lambda^{(k)}$ for all $1\leq k\leq m$.
Moreover, the following isomorphism of ${\rm U}_q(\widehat{\frak{gl}}_n)$-modules holds:
$$
\Omega_{(n)}^{\otimes r}\otimes_{{\mathcal H_\vtg(r)}} D^{\underline{\lambda}}\cong L(\mathbf{Q}_{\mathfrak s}).
$$
\end{enumerate}
\end{thm}
\begin{proof}
Suppose $\mathbf{Q}\in{\mathcal Q}(n)^{\mathbb Z}_r$.
By Proposition~\ref{prop:DF&DD} and~\eqref{partialint}, there exists a unique
$\mathfrak s\in\mathscr S^{(n),\mathbb Z}_r$ such that $\mathbf{Q}=\partial^{\mathbb Z}(\mathfrak s)$ and
$L(\mathbf{Q})\cong \Omega_{(n)}^{\otimes r}\otimes_{{\mathcal H_\vtg(r)}} V_{\mathfrak s}$.
Then, by Theorem~\ref{thm:standardK}, we have $\underline{\gamma}=\eta(\mathfrak s)\in\mathcal K^{s}_{(n)}(r)$ and,
by Theorem~\ref{thm:VsDla}, $V_{\mathfrak s}\cong D^{\underline{\gamma}'}$ and so
$$
L(\mathbf{Q})\cong \Omega_{(n)}^{\otimes r}\otimes_{{\mathcal H_\vtg(r)}}D^{\underline{\gamma}'},
$$
proving Part (1).

Fix $1\leq i\leq n$.
Recall from~\eqref{scla} that we have
$$
{\bf s}^{{\mathsf c}}_{\underline{\lambda};f}=({\mathsf s}_1^{(m)},\ldots,{\mathsf s}^{(m)}_{\lambda^{(m)}_1},\ldots,
{\mathsf s}^{(1)}_1,\ldots,{\mathsf s}^{(1)}_{\lambda^{(1)}_1}),
$$
where
$$
{\mathsf s}^{(k)}_j=(q^{2(f_k+j-\lambda^{(k)'}_j)},\ldots, q^{2(f_k+j-2)},q^{2(f_k+j-1)})
$$
for $1\leq k\leq m$.
Then by~\eqref{partialQ1}, we obtain
\begin{align*}
Q_i(x)
&=\prod_{1\leq k\leq m}\prod_{1\leq j\leq\lambda^{(k)}_1, \lambda^{(k)'}_j\geq i}(1-q^{2(f_k+j-i)}x)\\
&=\prod_{1\leq k\leq m}\prod_{1\leq j\leq\lambda^{(k)}_i}(1-q^{2(f_k+j-i)}x).
\end{align*}
Now, Part (2) follows from Theorem~\ref{thm:VsDla}
and Proposition~\ref{prop:DF&DD}.
\end{proof}

Note that,
in the case $m=1$ and $u_1=1$, Theorem~\ref{thm:K&D} recovers \cite[Theorem 7.2]{DF}.

\section{Skew shape representations of ${\mathcal H_\vtg(r)}$ and ${\rm U}_q(\widehat{\frak{gl}}_n)$}
We now give an application of our theory.
We will identify Ram's skew shape representations of the affine Hecke algebra in terms of (column residual) multisegments  and compute the Drinfeld polynomials
of their induced skew shape representations of ${\rm U}_q(\widehat{\frak{gl}}_n)$.

Throughout the section, we repeatedly use the notation
$\lambda'=(\lambda_1',\lambda_2',\ldots)$ for the conjugate (or dual) partition of $\lambda=(\lambda_1,\lambda_2,\ldots)\vdash r$.

\subsection{\bf Skew shape representations of ${\mathcal H_\vtg(r)}$ and multisegments.}
If $\lambda$ and $\nu$ are
partitions such that $\nu_k\leq \lambda_k$ for all $k$, we write
$\nu\subseteq\lambda$. A \textit{skew shape}
$\lambda/\nu$ is defined to be the diagram obtained by removing
the Young diagram $\nu$ from $\lambda$ for some pair of partitions $\nu\subseteq\lambda$.
Recall that the residue of a node $\rho=(i,j)$ in $\lambda$ is defined by
${\rm res}(\rho)=q^{2(j-i)}$. Accordingly the residue of nodes in a skew shape
is defined.  A standard $\lambda/\nu$-tableau is a labeling of the
skew Young diagram $\lambda/\nu$ with the numbers $1,
2,\ldots, |\lambda|-|\nu|$ such that the numbers strictly increase
from left to right along each row and down each column.
If $T$ is a
$\lambda/\nu$-tableau, denote by ${\rm res}_T(k)$ the residue of the node of
$\lambda/\nu$ labeled by $k$ in $T$ for  $1\leq k\leq
|\lambda|-|\nu|$.
Let $\mathcal F(\lambda/\nu)$ be the set of standard tableau of shape $\lambda/\nu$.

Let $\lambda/\nu$ be a skew shape such that $|\lambda|-|\nu|=r$ and set
$$
D^{\lambda/\nu}=\mathbb C{-\rm span}\{v_T~|~T\in\mathcal F(\lambda/\nu)\}.
$$
It is known \cite[Theorem 4.1]{Ram} the $D^{\lambda/\nu}$ affords an irreducible
${\mathcal H_\vtg(r)}$-module via
\begin{align}
\displaystyle X_kv_T&={\rm res}_T(k)v_T,\label{skewX}\\
\displaystyle T_iv_T&=\frac{q^2-1}{1-\frac{{\rm res}_T(i)}{{\rm res}_T(i+1)}}v_T
+\Big(1+\frac{q^2-1}{1-\frac{{\rm res}_T(i)}{{\rm res}_T(i+1)}}\Big)v_{s_iT}, \label{skewT}
\end{align}
where $1\leq k\leq r$, $s_iT$ denotes the tableau obtained by switching $i,i+1$ in
$T$ and $v_{s_iT}=0$ if $s_iT$ is not standard for $1\leq i\leq r-1$.

Given a skew shape $\lambda/\nu$, there exist $1\leq j_1<j_2<\cdots<j_t\leq\lambda_1$ with $t\geq 1$
such that $\lambda'_{j_k}>\nu'_{j_k}$
for $1\leq k\leq t$, and $\lambda'_{j}=\nu_j'$ for $1\leq j\leq\lambda_1$ with
$j\neq j_1,j_2,\ldots,j_t$.
Set
\begin{align}\label{segmentskew}
{\mathsf s}_k=(q^{2(j_k-\lambda_{j_k}')},q^{2(j_k-\lambda_{j_k}'+1)},\ldots,q^{2(j_k-\nu_{j_k}'-1)})
\end{align}
for $1\leq k\leq t$, that is, ${\mathsf s}_k$ is the segment obtained by reading the residues of the nodes
in the $k$th nonempty column of the skew shape $\lambda/\nu$ from bottom to top.
Let
\begin{align*}
{\bf s}_{\lambda/\nu}=({\mathsf s}_1,{\mathsf s}_2,\ldots,{\mathsf s}_t).
\end{align*}

\begin{thm}\label{thm:skew}
Suppose that $\lambda/\nu$ is a skew shape with $|\lambda|-|\nu|=r$.
Then
$$
D^{\lambda/\nu}\cong V_{{\bf s}_{\lambda/\nu}}
$$
as ${\mathcal H_\vtg(r)}$-modules.
\end{thm}
\begin{proof}
Let $t_{\lambda/\nu}$ be the standard $\lambda/\nu$-tableau in which
the numbers $1,2,\ldots, r$ appear in order along successive columns from top to bottom.
By~\eqref{skewT}, one can deduce that
\begin{equation}\label{skewsign}
T_iv_{t_{\lambda/\nu}}=-v_{t_{\lambda/\nu}}.
\end{equation}
for $1\leq i\leq r-1$ such that $T_i\in\mathcal H_{{\!\vartriangle\!}}(\mu({\bf s}_{\lambda/\nu}))$.
Meanwhile it is easy to see that
$$
\widetilde{{\bf s}}_{\lambda/\nu}^\vee
=({\rm res}_{t_{\lambda/\nu}}(1),{\rm res}_{t_{\lambda/\nu}}(2),\ldots,{\rm res}_{t_{\lambda/\nu}}(r))
$$
Then by~\eqref{skewX} and~\eqref{skewsign}, we obtain that
$\mathbb C v_{t_{\lambda/\nu}}$ affords an $\mathcal H_{{\!\vartriangle\!}}(\mu({\bf s}_{\lambda/\nu}))$-submodule of $D^{\lambda/\nu}$ and
moreover by~\eqref{cell} and Proposition~\ref{prop:Ro2}(2) we have
\begin{align*}
\mathbb C C_{w^0_{\mu({\bf s}_{\lambda/\nu})}}\cong \mathbb C v_{t_{\lambda/\nu}}
\end{align*}
as $\mathcal H_{{\!\vartriangle\!}}(\mu({\bf s}_{\lambda/\nu}))$-modules.
Then by Frobenius reciprocity and Lemma~\ref{lem:induced}, there exists an ${\mathcal H_\vtg(r)}$-homomorphism
\begin{equation}\label{mapxi}
\xi: I_{{\bf s}_{\lambda/\nu}}
\longrightarrow D^{\lambda/\nu}, \quad C_{w^0_{\mu({\bf s}_{\lambda/\nu})}}\mapsto v_{t_{\lambda/\nu}}.
\end{equation}
By~\eqref{segmentskew} and Definition~\ref{defn:order}(2),
we observe that ${\bf s}_{\lambda/\nu}$ is standard.
Then by Proposition~\ref{prop:Ro3},
the head of $I_{{\bf s}_{\lambda/\nu}}$ is isomorphic to $V_{{\bf s}_{\lambda/\nu}}$
and therefore, \eqref{mapxi} implies the isomorphism
$
D^{\lambda/\nu}\cong V_{{\bf s}_{\lambda/\nu}}.
$
\end{proof}

\begin{cor}
Suppose that $\lambda/\nu$ is a skew shape.
Let $\rho=(\rho_1,\rho_2,\ldots)$ be the partition given by
$\rho_i=\#\{j~|~\lambda'_j-\nu'_j\geq i\}$ for $i\geq 1$.
Then the Littlewood-Richardson coefficients $c^\lambda_{\nu\gamma}$
satisfy
$$
c^\lambda_{\nu\rho}=1.
$$
\end{cor}
\begin{proof}
Assume that $|\lambda|-|\nu|=r$.
It is known \cite[Theorem 6.1]{Ram} that as an $\mathcal{H}(r)$-module, $D^{\lambda/\nu}$ can be decomposed as
\begin{equation}\label{skewdecomp}
D^{\lambda/\nu}\cong\oplus_{\gamma\in\mathcal P(r)}c^{\lambda}_{\nu\gamma}S^{\gamma},
\end{equation}
where $c^{\lambda}_{\nu\gamma}$ are Littlewood-Richardson coefficients.
Then by Theorem~\ref{thm:skew} and Proposition~\ref{prop:Ro2}(3), we obtain that
the left cell module $E_{\mu(s_{\lambda/\nu})}$ occurs with multiplicity one
in $D^{\lambda/\nu}$.
Observe that $\mu(s_{\lambda/\nu})'=\rho$ and hence $E_{\mu(s_{\lambda/\nu})}\cong S^\rho$.
This together with~\eqref{skewdecomp} proves the corollary.
\end{proof}
\subsection{\bf Skew shape representations of ${\rm U}_q(\widehat{\frak{gl}}_n)$.}
Recall that given a left ${\mathcal H_\vtg(r)}$-module $M$, the tensor product
$\Omega^{\otimes r}_{(n)}\otimes_{{\mathcal H_\vtg(r)}}M$ naturally affords a left
${\rm U}_q(\widehat{\frak{gl}}_n)$-module.
Suppose $\lambda/\nu$ is a skew shape such that $|\lambda|-|\nu|=r$ and
$\lambda_j'-\nu_j'\leq n$ for $1\leq j\leq \lambda_1$.
The tensor product $\Omega^{\otimes r}_{(n)}\otimes_{{\mathcal H_\vtg(r)}}D^{\lambda/\nu}$
will be called a skew shape representation of ${\rm U}_q(\widehat{\frak{gl}}_n)$.

\begin{thm}
Suppose that $\lambda/\nu$ is a skew shape such that $|\lambda|-|\nu|=r$ and
$\lambda_j'-\nu_j'\leq n$ for $1\leq j\leq \lambda_1$.
Write ${\mathfrak s}=\bar{{\bf s}}_{\lambda/\nu}$ and $\mathbf{Q}_{{\mathfrak s}}=(Q_1(x),Q_2(x),\ldots,Q_n(x))$.
Then
\begin{enumerate}
\item For $1\leq i\leq n$, we have
$$
Q_i(x)=\prod_{1\leq j\leq\lambda_1, \lambda'_j-\nu'_j\geq i}(1-q^{2(j-(i+\nu'_j))}x),
$$
that is, $Q_i(x)$ is the polynomial whose roots are determined by
 the residues of the $i$th nodes in each column of the skew shape $\lambda/\nu$.

\item The following isomorphism of ${\rm U}_q(\widehat{\frak{gl}}_n)$-modules holds:
$$
\Omega^{\otimes r}_{(n)}\otimes_{{\mathcal H_\vtg(r)}}D^{\lambda/\nu}
\cong L(\mathbf{Q}_{{\mathfrak s}}),
$$
\end{enumerate}
\end{thm}
\begin{proof}

By~\eqref{segmentskew} and~\eqref{partialQ1} we obtain
\begin{align*}
Q_i(x)=\prod_{1\leq j\leq\lambda_1, \lambda'_j-\nu'_j\geq i}
(1-q^{2(j-(i+\nu'_j))}x).
\end{align*}
This proves part (1).

By Theorem~\ref{thm:skew} and Proposition~\ref{prop:DF&DD},
we have
$$
\Omega^{\otimes r}_{(n)}\otimes_{{\mathcal H_\vtg(r)}}D^{\lambda/\nu}
\cong \Omega^{\otimes r}_{(n)}\otimes_{{\mathcal H_\vtg(r)}} V_{{\mathfrak s}}
\cong L(\mathbf{Q}_{{\mathfrak s}}).
$$
Hence part (2) of the theorem is proved.
\end{proof}

\begin{rem} Skew shape (or skew) representations of ${\rm U}_q(\widehat{\frak{gl}}_n)$ have been considered
in~\cite{HM} (see also \cite{Ch}) and Drinfeld polynomials $P_i(x)=Q_i(xq^{i-1})/Q_{i+1}(xq^{i+1})$ ($1\leq i\leq n-1$) are computed. A comparison shows that a shift is required between the two sets of the $P_i(x)$'s.
More precisely, if $g$ denotes  the bijection $\mathscr S_r\to\mathscr S_r$ sending ${\mathfrak s}=\{{\mathsf s}_1,\ldots,{\mathsf s}_t\}$ to  $g({\mathfrak s})=\{{\mathsf s}_1q^{-|{\mathsf s}_1|},\ldots,{\mathsf s}_tq^{-|{\mathsf s}_t|}\}$, then our $P_i(x)$ are defined relative to ${\mathfrak s}$ and their $P_i(x)$ are defined relative to $g({\mathfrak s})$.

\end{rem}
\noindent
{\bf Acknowledgement.} The second author would like to thank Weiqiang Wang, Alexander Kleshchev, and Jun Hu for some helpful discussions.

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