text:
\section{Introduction}

In the papers \cite{VV2} and \cite{STU} it was shown that the $q$-deformed Fock space module of the quantum affine algebra $U_q(\widehat{ \frak{s}\frak{l}}_n )$ admits an action of a new remarkable object -- the so-called quantum toroidal algebra introduced in \cite{GKV} and \cite{VV} as a $q$-deformation of the universal central extension of the $ {{\frak s} {\frak l}}_n$-valued double-loop Lie algebra. The action of the quantum toroidal algebra on the $q$-Fock space depends on two parameters: the deformation parameter $q$ and an extra parameter $p,$ when  values of these parameters are taken to be generic complex numbers, the $q$-Fock space is known to be irreducible with respect to this action.

The quantum toroidal algebra has two subalgebras, $U'_q (\widehat{ \frak{s}\frak{l}}_n )^{(1)}$ and $U'_q (\widehat{ \frak{s}\frak{l}}_n )^{(2)},$ both isomorphic to $U'_q (\widehat{ \frak{s}\frak{l}}_n ).$ Accordingly, the $q$-Fock space admits two $U'_q (\widehat{ \frak{s}\frak{l}}_n )$-actions.

The first of these actions has level 1, and coincides with the action originally introduced  by Hayashi in \cite{H}. The irreducible decomposition of the   $q$-Fock space with respect to this action was  given in \cite{KMS} by using the semi-infinite $q$-wedge construction due to \cite{St}.

The second of the $U'_q (\widehat{ \frak{s}\frak{l}}_n )$-actions has level 0, the irreducible decomposition of the $q$-Fock space with respect to this action was constructed in \cite{TU} at generic values of the parameters $q$ and $p.$

Kashiwara, Miwa and Stern \cite{KMS} have shown, that the level-1 action of $U'_q (\widehat{ \frak{s}\frak{l}}_n )$ on the $q$-Fock space is centralized by the action of the Heisenberg algebra. In the paper \cite{STU} it was proven that the proper ideal of the $q$-Fock space  generated by the negative-frequency part of the Heisenberg algebra is invariant under the action of the  quantum toroidal algebra provided the value of the parameter $p$ in the latter is set to be equal to 1. The quotient of the $q$-Fock space  by this ideal is isomorphic to one of the irreducible level-1 highest weight modules of  $U_q (\widehat{ \frak{s}\frak{l}}_n ).$ As a consequence, each of these modules admits an action of the quantum toroidal algebra.

The corresponding action of the subalgebra $U'_q (\widehat{ \frak{s}\frak{l}}_n )^{(1)}$ is irreducible, it is just the standard level-1 action on the highest weight irreducible module of $U_q (\widehat{ \frak{s}\frak{l}}_n ).$ On the other hand, the action of the subalgebra $U'_q (\widehat{ \frak{s}\frak{l}}_n )^{(2)}$  has level 0 and is completely reducible. The construction of the irreducible  decomposition of the level-1 $U_q (\widehat{ \frak{s}\frak{l}}_n )$-modules  relative to the level-0 action is the problem which we address in the present paper. To solve this problem we utilize, as our main tools, the semi-infinite wedges of \cite{KMS} and the Non-symmetric Macdonald polynomials  of \cite{Cheann}.

As a result we obtain a parameterization of the irreducible components of the level-0 action by the skew Young diagrams of the border strip type, proving thereby a $q$-analogue of the conjecture made in \cite{KKN} in the classical setting.

\section{The actions of the quantum affine algebra $U'_q(\widehat{ \frak{s}\frak{l}}_n )$}

\subsection{Definition of the quantum affine algebra $U'_q(\widehat{ \frak{s}\frak{l}}_n )$}

\begin{df}
The quantum affine algebra $U'_q(\widehat{ \frak{s}\frak{l}}_n )$
is the unital
associative algebra over ${\Bbb C} $ with generators $E_i$, $F_i$, $K_i^{{}\pm
1}$\/ ($i\in I:=\{0,1, \dots ,n-1\}$) and the following defining relations:
\begin{eqnarray}K_iK_i^{- 1} =&1& =  K_i^{ -1}K_i,\\
K_iK_j&=&K_jK_i\;,\\
K_iE_jK_i^{-1}&=& q^{ a_{ij}}E_j \; ,\\
K_iF_jK_i^{-1}&=& q^{ -a_{ij}}F_j \; ,\\
{}[E_i , F_j]&=&\delta_{ij}\frac{K_i -K_i^{-1}}{q-q^{-1}}\;,
\end{eqnarray}
\vspace{-.15in}
\begin{equation}
\sum_{r=0}^{1-a_{ij}}
 (-1)^r\left[{1-a_{ij}\atop r}\right]_{q}\;
(E_i)^rE_j (E_i)^{1-a_{ij}-r}= 0\;,\;i\ne j .
\end{equation}
\vspace{-.2in}
\begin{equation}
\sum_{r=0}^{1-a_{ij}}
 (-1)^r\left[{1-a_{ij}\atop r}\right]_{q}\;
(F_i)^rF_j (F_i)^{1-a_{ij}-r}= 0\;,\;i\ne j .
\end{equation}
\vspace{-.15in}
\begin{equation}
 \mbox{where }
[n]_{q} :=\frac{q^n -q^{-n}}{q -q^{-1}},\; \;
{}\left[{n\atop r}\right]_{q}:=\frac{[n]_{q}[n-1]_{q}\ldots {}[n-r+1]_{q}}
{[r]_{q}[r-1]_{q}\ldots [1]_{q}},
\end{equation}
\vspace{-.2in}
\begin{equation}
 a_{ij}= \left\{
\begin{array}{cc}
2 & (i=j) \\
-1 &(|i-j|=1 ,(i,j)=(1,n) ,(n,1)) \\
0 & (\mbox{otherwise})
\end{array} \right.
 \; \; n \geq 3 ,
\end{equation}
\vspace{-.2in}
\begin{equation}
 a_{ij}= \left\{
\begin{array}{cc}
2 & (i=j) \\
-2 & (i\neq j)
\end{array} \right.
 \; \; n =2 .
\end{equation}
\end{df}
The coproduct $\Delta $ is given by
\begin{eqnarray}
\Delta(E_{i})& =&  E_{i} \otimes K_{i} + 1 \otimes E_{i}, \label{cp1} \\
\Delta(F_{i})& =&  F_{i} \otimes 1 + K_{i}^{-1} \otimes F_{i}, \label{cp2} \\
\Delta(K_{i})& =&  K_{i} \otimes K_{i}.\label{cp3}
\end{eqnarray}

We put $c':=K_{0}K_{1} \dots K_{n-1}$ in $U'_{q}(\widehat{ \frak{s}\frak{l}}_n)$,
 then $c'$ is the central in $U'_{q}(\widehat{ \frak{s}\frak{l}}_n)$.

\subsection{q--wedge product and semi--infinite q--wedge product}

The affine Hecke algebra of type $ {{\frak g} {\frak l}}_N$, $\widehat{H}_N(q) $ is a unital associative algebra over ${\Bbb C}[q^{\pm 1}]$ with generators $T_i^{\pm 1},$ $Y_j^{\pm 1},$ $i=1,2,\dots,N-1,$ $j=1,2,\dots
,N$ and relations
\begin{gather*}
T_i T_i^{-1} =  T_i^{-1} T_i = 1, \quad (T_i + 1) (T_i - q^2) =0, \\
T_i T_{i+1} T_i =  T_{i+1} T_i T_{i+1},\\
T_i T_j = T_j T_i \quad \text{ if $ |j-i| > 1,$} \\
Y_i Y_j = Y_j Y_i, \quad T_i^{-1} Y_i T_i^{-1} = q^{-2} Y_{i+1}\\
Y_j T_i = T_i Y_j \quad \text{ if $ j \neq i,i+1,$}
\end{gather*}
The subalgebra $H_N(q)$ generated by $ T_i^{\pm 1}$ is isomorphic to the Hecke algebra of type $ {{\frak g} {\frak l}}_N.$

Let $p \in {\Bbb C}^{\times}$ and consider the following operators in ${\mathrm E}{\mathrm n}{\mathrm d}(\cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}])$
\begin{eqnarray}
& g_{i,j} = \frac{q^{-1}z_i - q z_j}{z_i - z_j}(K_{i,j} - 1) + q, \;\; & 1\leq i \neq j \leq N,\nonumber \\
& Y_i^{(N)} = g_{i,i+1}^{-1}K_{i,i+1} \cdots g_{i,N}^{-1} K_{i,N} p^{D_i} K_{1,i} g_{1,i} \cdots K_{i-1,i}g_{i-1,i}, \;\; & i=1,2,\dots,N, \nonumber
\end{eqnarray}
where $K_{i,j}$ acts on $\cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}]$ by permuting variables $z_i,z_j$ and $p^{D_i}$ is the difference operator
\begin{equation}
p^{D_i}f(z_1,\dots,z_i,\dots,z_N)  =  f(z_1,\dots,pz_i,\dots,z_N) , \quad f \in {\Bbb C}[z_1^{\pm 1},\dots, z_N^{\pm 1}]. \nonumber
\end{equation}
Then the assignment
\begin{equation}
T_i \mapsto \overset{c}{T_i} = -q g_{i,i+1}^{-1}, \quad Y_i \mapsto q^{1-N}Y_i^{(N)}
\end{equation}
defines a right action of $\widehat{H}_N(q) $  on $\cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}]$.

The commuting difference operators $Y_1^{(N)}, \dots, Y_N^{(N)}$ are called Cherednik's operators.

Moreover, the assignment
\begin{equation}
T_i \mapsto \overset{c}{T_i} = -q g_{i,i+1}^{-1}, \quad Y_i \mapsto z_i^{-1} \; \mbox{(multiplication)}
\end{equation}
defines another right action of $\widehat{H}_N(q) $  on $\cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}]$. \\
\vspace{.1in}
\begin{rmk}
The actions of $-q g_{i,i+1}^{-1}, \; q^{1-N}Y_i^{(N)}, \; z_i^{-1}$ are related to the toroidal Hecke algebra \cite{VV} or the double affine Hecke algebra \cite{Cheann}.
\end{rmk}
\vspace{.1in}

Let $V = \cplx^{n}$, with basis $\{ v_1,\dots,v_n \}.$ Then $\otimes^N V$ admits a left $H_N(q)$-action given by
\begin{gather}
T_i \mapsto \overset{s}{T_i} = 1^{\otimes^{i-1}}\otimes \overset{s}{T} \otimes 1^{\otimes^{N-i-1}}, \quad \text{ where } \quad  \overset{s}{T}  \in {\mathrm E}{\mathrm n}{\mathrm d}(\otimes^2 V) \\
\text{ and } \quad   \overset{s}{T}(v_{\epsilon_1}\otimes v_{\epsilon_2}) = \begin{cases} q^2 v_{\epsilon_1}\otimes v_{\epsilon_2} & \text{ if $ \epsilon_1 = \epsilon_2,$} \\
 q v_{\epsilon_2}\otimes v_{\epsilon_1} & \text{ if $ \epsilon_1 < \epsilon_2,$ }\\
q v_{\epsilon_2}\otimes v_{\epsilon_1} + (q^2  - 1) v_{\epsilon_1}\otimes v_{\epsilon_2}& \text{ if $ \epsilon_1 > \epsilon_2.$ }  \end{cases}
\end{gather}

Let $V(z)= {\Bbb C} [z^{\pm 1}]\otimes V,$ with  basis $ \{ z^m \otimes v_{\epsilon} \}$, $ m\in {\Bbb Z} $ , $ \epsilon \in \{1,2,\dots,n\} $. Often it will be convenient to set $ k = \epsilon - nm $ and $ u_k = z^m \otimes v_{\epsilon} $.
Then $\{ u_k \}, $  $k \in {\Bbb Z} $ is a basis of $V(z)$. In what follows we will write $ z^m v_{\epsilon} $ as a short-hand for $ z^m \otimes v_{\epsilon} ,$ and use both  notations: $ u_k $ and $ z^m  v_{\epsilon} $ switching between them according to convenience.
The two actions of the Hecke algebra are naturally extended on the tensor product $\cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}]\otimes(\otimes^N V)$ so that $\overset{c}T_i$ acts trivially on $\otimes^N V$ and  $\overset{s}T_i$ acts trivially on $\cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}].$ The vector space $\otimes^N V(z)$ is identified with  $\cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}]\otimes(\otimes^N V)$ and the $q$-wedge product \cite{KMS} is defined as the following quotient space:
\begin{equation}
\wedge^N V(z) = \otimes^N V(z)/ \sum_{i=1}^{N-1}{\mathrm K}{\mathrm e}{\mathrm r}\left( \overset{c}T_i + q^2 (\overset{s}T_i)^{-1} \right). \label{e: q-wedgeprod}
\end{equation}

Let $\Lambda: \otimes^N V(z) \rightarrow \wedge^N V(z) $ be the quotient map specified by (\ref{e: q-wedgeprod}). The image of a pure tensor $ u_{k_1}\otimes u_{k_2}\otimes \dots \otimes u_{k_N} $ under this map is called a wedge and is denoted by
\begin{equation}
u_{k_1}\wedge u_{k_2}\wedge \dots \wedge u_{k_N} :=\Lambda ( u_{k_1}\otimes u_{k_2}\otimes \dots \otimes u_{k_N}) . \label{e: wedge}
\end{equation}
A wedge is normally ordered if $k_1 > k_2 > \cdots > k_N.$ In \cite{KMS} it is proven that normally ordered wedges form a basis in $\wedge^N V(z)$.
\vspace{.2in}

Let us now define the semi-infinite q--wedge product $\wedge ^{\frac{\infty}{2}} V(z)$ and for any integer $M$ its subspace $F_M$, following \cite{KMS}.

Let $\otimes ^{\frac{\infty}{2}} V(z)$ be the space spanned by the vectors
$u_{k_{1} } \otimes u_{k_{2}} \otimes \dots , \; ( k_{i+1} = k_{i} -1 , \; i >>1 )$. We define the space $\wedge ^{\frac{\infty}{2}} V(z)$ as the quotient of $\otimes ^{\frac{\infty}{2}} V(z)$:
\begin{equation}
\wedge ^{\frac{\infty}{2}} V(z) :=  \otimes ^{\frac{\infty}{2}} V(z) / \sum_{i=1}^{\infty }{\mathrm K}{\mathrm e}{\mathrm r}\left( \overset{c}T_i + q^2 (\overset{s}T_i)^{-1}\right). \label{e: inf-wedgeprod}
\end{equation}
Let $\Lambda: \otimes ^{\frac{\infty}{2}} V(z) \rightarrow \wedge ^{\frac{\infty}{2}} V(z)$ be the quotient map specified by (\ref{e: inf-wedgeprod}). The image of a pure tensor $ u_{k_1}\otimes u_{k_2}\otimes \dots  $ under this map is called a semi-infinite wedge and is denoted by
\begin{equation}
u_{k_1}\wedge u_{k_2}\wedge \dots  :=\Lambda ( u_{k_1}\otimes u_{k_2}\otimes \dots ) . \label{e: infwedge}
\end{equation}
A semi-infinite wedge is normally ordered if $k_1 > k_2 > \cdots $ and $k_{i+1}= k_i -1 \; (i >> 1)$. In \cite{KMS} it is proven that normally ordered semi-infinite wedges form a basis in $\wedge ^{\frac{\infty}{2}} V(z)$.
\vspace{.1in}

Let $U_{M}$ be the subspace of $\otimes ^{\frac{\infty}{2}} V(z)$ spanned by the vectors $u_{k_{1} } \otimes u_{k_{2}} \otimes \dots , \; ( k_{i} = M-i+1 , \; i >>1 )$.
Let $F_{M}$ be the quotient space of $U_{M}$ defined by the map (\ref{e: infwedge}).
 Then $F_{M}$ is a subspace of $\wedge ^{\frac{\infty}{2}} V(z)$, and the vectors $u_{k_1}\wedge u_{k_2}\wedge \dots $, ($k_1 > k_2 > \cdots $,  $k_{i} = M-i+1, \; i >>1 )$ form a basis of $F_{M}$. We will  call the space $F_{M}$ the q--deformed Fock space.

\subsection{Actions of the quantum affine algebra on the q--wedge product}
\label{fwrep}

We will define two actions of $U'_q (\widehat{ \frak{s}\frak{l}}_n )$ on the space $\wedge ^N V(z)$.

The first one is defined as follows.
\begin{align}
& E_i ( m \otimes v)  = \sum_{j=1}^N m \otimes E_j^{i,i+1} K_{j+1}^{i}\dots K_N^{i} v,  \label{e: Efin}\\
& F_i ( m \otimes v)  = \sum_{j=1}^N m \otimes (K_{1}^{i})^{-1}\dots (K_{j-1}^{i})^{-1}E_j^{i+1,i} v,  \label{e: Ffin}\\
& K_i( m \otimes v)   = m \otimes   K^{i}_1K^{i}_2\dots K^{i}_N v ,\quad    (i=1,2,\dots,n-1)  \label{e: Kfin} \\
& E_0( m \otimes v) =  \sum_{j=1}^N m Y_j^{-1}\otimes E_j^{n,1} K_{j+1}^{0}\dots K_N^{0} v,  \label{e: E0} \\
& F_0 ( m \otimes v)    =   \sum_{j=1}^N m Y_j \otimes (K_{1}^{0})^{-1}\dots (K_{j-1}^{0})^{-1}E_j^{1,n} v,  \label{e: F0} \\
&  K_0  = (K_1 K_2 \cdots K_{n-1})^{-1}. \label{e: K0}
\end{align}
Here $ E_j^{i,k} = 1^{\otimes^{j-1}}\otimes E^{i,k} \otimes 1^{\otimes^{N-j}},$ where $E^{i,k} \in {\mathrm E}{\mathrm n}{\mathrm d}(V)$ is the matrix unit in the basis $ v_1,\dots,v_n,$ and $ K_j^i = q^{E_j^{i,i} - E_j^{i+1,i+1}},$ $ K_j^0 = (K_j^1 K_j^2 \cdots K_j^{n-1})^{-1}.$

We will denote this action by $U_0^{(N)}$. Note that  it is well defined on the quotient space $\wedge ^N V(z)$ in view of the relations of the affine Hecke algebra.

The second one is defined as follows.
\begin{align}
 E_0( m \otimes v) =  \sum_{j=1}^N m z_j \otimes E_j^{n,1} K_{j+1}^{0}\dots K_N^{0} v,   \\
 F_0 ( m \otimes v)    =   \sum_{j=1}^N m z_j^{-1} \otimes (K_{1}^{0})^{-1}\dots (K_{j-1}^{0})^{-1}E_j^{1,n} v.
\end{align}
The actions of other generators are the same as in (\ref{e: Efin}--\ref{e: Kfin}, \ref{e: K0}).

We will denote this action by $U_1^{(N)}$. Again, this action is well defined on the quotient space $\wedge ^N V(z)$ in view of the relations of the affine Hecke algebra.

\subsection{Level--0 action of the quantum affine algebra on the q--deformed Fock space} \label{sec:L-0}

We will define a level-0 action of $U_q'(\widehat{ \frak{s}\frak{l}}_n)$ on $F_M$ $(M\in {\Bbb Z})$ following the paper \cite{TU,STU}.

Let $ \end{enumerate} := (\epsilon_1,\epsilon_2,\dots,\epsilon_N) $ where $ \epsilon_i \in \{1,2,\dots,n\} $. For a sequence $\end{enumerate} $ we set
\begin{equation}
{\bold v}_{\end{enumerate}}:= v_{\epsilon_1}\otimes v_{\epsilon_2}\otimes \dots \otimes v_{\epsilon_N}\quad  ( \in \otimes^N \cplx^{n} ).
\end{equation}
A sequence ${\bold m} := (m_1,m_2,\dots,m_N)$ from ${\Bbb Z}^N$ is called $n$-strict if it contains no more than $n$ equal elements of any given value.
Let us define the sets ${\cal M}_N^n$ and ${\cal E}({\bold m})$ by
\begin{align}
 & {\cal M}_N^n := \{{\bold m} = (m_1,m_2,\dots,m_N) \in {\Bbb Z}^N \; | \; m_1\leq m_2 \leq \dots \leq m_N,\; \text{${\bold m} $  is $n$-strict } \}, \label{e: Mm}\\
 \intertext{ and for $ {\bold m} \in {\cal M}_N^n $ }
& {\cal E}({\bold m}) := \{ \end{enumerate} = (\epsilon_1,\epsilon_2,\dots,\epsilon_N) \in \{1,2,\dots,n\}^N \; | \; \epsilon_i > \epsilon_{i+1} \; \text{for all $i$ s.t. $m_i =m_{i+1}$ } \}.   \label{e: Em}
\end{align}
In these notations the set
\begin{equation}
\{ w({\bold m},\end{enumerate}):= \Lambda( {\bold z}^{{\bold m}} \otimes {\bold v}_{\end{enumerate}}) = z^{m_1}v_{\epsilon_1}\wedge z^{m_2}v_{\epsilon_2}\wedge\dots\wedge z^{m_N}v_{\epsilon_N} \quad | \quad {\bold m} \in {\cal M}_N^n , \end{enumerate} \in {\cal E}({\bold m}) \}.
\end{equation}
is nothing but the base of the normally ordered wedges in $\wedge^N V(z)$. We  will use the notation $w({\bold m},\end{enumerate})$ {\em exclusively} for normally ordered wedges.

Similarly for a semi-infinite wedge $w = u_{k_1}\wedge u_{k_2} \wedge \dots \; $ $=$ $z^{m_1}v_{\epsilon_1}\wedge z^{m_2}v_{\epsilon_2} \wedge \dots \;,$ such that $w \in F_M,$ the semi-infinite sequences ${\bold m} = (m_1,m_2,\dots \;\;)$ and $\end{enumerate} = (\epsilon_1,\epsilon_2,\dots \;\;)$ are defined by $ k_i = \epsilon_i - nm_i $, $\epsilon_i \in \{1,2,\dots,n\},$ $m_i \in {\Bbb Z}.$ In particular the ${\bold m}$- and $\end{enumerate}$- sequences of the vacuum vector in $F_M$ will be denoted by ${\bold m}^0$ and $\end{enumerate}^0$:
\begin{equation}
 |M\rangle = u_M\wedge u_{M-1} \wedge u_{M-2}\wedge  \dots \; = z^{m_1^0}v_{\epsilon_1^0}\wedge z^{m_2^0}v_{\epsilon_2^0} \wedge z^{m_3^0}v_{\epsilon_3^0} \wedge \cdots \;. \end{equation}

The Fock space $F_M$ is ${\Bbb Z}_{\geq 0}$-graded. For any semi-infinite wedge $w$ $=$ $u_{k_1}\wedge u_{k_2} \wedge \dots \;$ $=$  $z^{m_1}v_{\epsilon_1}\wedge z^{m_2}v_{\epsilon_2} \wedge \dots \;$ $\in F_M$ the degree $| w |$ is defined by
\begin{equation}
| w | = \sum_{i \geq 1} m_i^0 - m_i . \label{wdeg}
\end{equation}
Let us denote by $F_M^k$ $\subset$ $F_M$ the homogeneous component of degree $k$.

We will define a level--0 action of $U_q'(\widehat{ \frak{s}\frak{l}}_n)$ on the Fock space $F_M$ in such a way that each homogeneous component $F_M^k$ will be invariant with respect to this action. Throughout this section we fix an integer $M$ and  $s \in \{0,1,2,\dots,n-1 \} $ such that  $M = s\bmod n.$

Let $l$ be a non-negative integer and define $ V_M^{s+nl} \subset \wedge^{s+nl}V(z) $ as follows:
\begin{equation}
V_M^{s+nl}  = \bigoplus\begin{Sb} {\bold m} \in {\cal M}_{s+nl}^n, \end{enumerate} \in {\cal E}({\bold m}) \\ m_{s+nl} \leq m_{s+nl}^0 \end{Sb} {\Bbb C} w({\bold m},\end{enumerate}) . \label{e: VM}
\end{equation}

The vector space $V_M^{s+nl}$ has a grading similar to the grading of the Fock space $F_M$. In this case the degree $|w |$ of a wedge  $w$ $=$ $u_{k_1}\wedge u_{k_2} \wedge \dots \wedge u_{k_{s+nl}}$ $=$  $z^{m_1}v_{\epsilon_1}\wedge z^{m_2}v_{\epsilon_2} \wedge \dots \wedge z^{m_{s+nl}}v_{\epsilon_{s+nl}} $ $\in$  $V_M^{s+nl}$ is defined by
\begin{equation}
| w | = \sum_{i = 1}^{s+nl} m_i^0 - m_i .
\label{degifw}
\end{equation}
The degree is a non-negative integer, and for $k\in {\Bbb Z}_{\geq 0}$ we denote by $ V_M^{s+nl,k} $ the homogeneous component of degree $k.$

The following result is contained in the paper \cite{TU}:
\begin{prop}
For each $k\in {\Bbb Z}_{\geq 0}$ the homogeneous component $ V_M^{s+nl,k} $ $\subset$ $\wedge^{s+nl}V(z)$ is invariant under the $U_q'(\widehat{ \frak{s}\frak{l}}_n) $-action  $U_0^{(s+nl)}$ defined in section $\ref{fwrep}$.
\end{prop}
We have $|\overline{\rho}^{M,k}_{l}(w)|$ $=$ $|w|$ and hence $\overline{\rho}^{M,k}_{l}$ $:$ $V_M^{s+nl,k} $ $\rightarrow$ $F_M^k$ for all $k \in {\Bbb Z}_{\geq 0}$. In the paper \cite{STU} the following propositions are shown.
\begin{prop}
When $l \geq k$ the map $\overline{\rho}^{M,k}_{l}$ is an isomorphism of vector spaces. \label{p: isom}
\end{prop}
\begin{prop} \label{u1isom}
For each triple of non-negative integers $k,l,m$ such that $k\leq l < m$ the map $\overline{\rho}^{M,k}_{l,m}$ $:$ $ V_M^{s+nl,k}$ $\rightarrow$ $ V_M^{s+nm,k},$ defined for any $w$ $\in$  $ V_M^{s+nl,k}$ by
\begin{equation}
\overline{\rho}^{M,k}_{l,m}(w) = w\wedge u_{M-s-nl}\wedge u_{M-s-nl-1}\wedge \dots \wedge u_{M-s-nm+1},
\end{equation}
is an isomorphism of the $U_q'(\widehat{ \frak{s}\frak{l}}_n)$-modules.
\end{prop}
We define on the vector space $F_M^k$ a level--0 action of $U_q'(\widehat{ \frak{s}\frak{l}}_n)$ by using Propositions \ref{p: isom} and \ref{u1isom}:
\begin{df}
The vector space $F_M^k$ is a level-$0$ module of $U_q'(\widehat{ \frak{s}\frak{l}}_n)$ with the action $U_0$ defined by
\begin{equation}
 U_0 = \overline{\rho}^{M,k}_{l} U_0^{(s+nl)} {\overline{\rho}^{M,k}_{l}}^{-1} \quad \text{ where $ l \geq k .$ }  \label{eq:U0def}
\end{equation}
\end{df}
This definition does not depend on the choice of $l$ as long as $l$ is greater or equal to $k.$ Since we have
\begin{equation}
F_M = \bigoplus_{k \geq 0} F_{M}^k
\end{equation}
the level-0 action $U_0$ extends to the entire Fock space $F_M.$

\subsection{Level--1 action of the quantum affine algebra on the q--deformed Fock space}

In this section we review the level--1 action of $U_q'(\widehat{ \frak{s}\frak{l}}_n)$ on the Fock space $F_M$ \cite{KMS}.

First we define the action of $U'_q(\widehat{ \frak{s}\frak{l}}_n)$ (generated by $E _{i}, \; F _i , \; K_i, \; $ $i=0, \dots ,n-1$) on the vector $|M' \rangle $ as follows.
\begin{equation}
E_i| M' \rangle =0,
\label{eq79}
\end{equation}
\vspace{-.18in}
\begin{equation}
F_i| M' \rangle = \left\{
\begin{array}{ll}
u_{M' +1}\wedge u_{M'-1}\wedge u_{M'-2}\wedge \cdots &  \mbox { if } i\equiv M' \mbox{ mod }n; \\
0 & \mbox{ otherwise},
\end{array}
\right.
\end{equation}
\vspace{-.18in}
\begin{equation}
K_i | M' \rangle = \left\{
\begin{array}{ll}
q| M' \rangle & \mbox { if } i\equiv M' \mbox{ mod }n; \\
| M' \rangle & \mbox{ otherwise},
\end{array}
\right.
\end{equation}
For every element $v \in F_{M}$, there exists $N$ such that
\begin{equation}
v = v^{(N)} \wedge | M -N \rangle , \; \; \; \; \;
v^{(N)} \in \wedge ^N V(z).
\end{equation}
We define the actions of $E _{i}, \; F _i , \; K_i, \; $ $i=0, \dots ,n-1$ on the vector $v$ as follows.
\begin{eqnarray}
& E _i v := E _i v^{(N)} \wedge K_i | M -N \rangle +  v^{(N)} \wedge E_i | M -N
\rangle ,& \\
& F _i v := F _i v^{(N)} \wedge | M -N \rangle +  K_i^{-1} v^{(N)} \wedge F_i
M -N \rangle ,& \\
& K _i v := K _i  v^{(N)} \wedge K _i | M -N \rangle. &
\end{eqnarray}
The actions of $E _{i}, \; F _i , \; K_i, \; $ $i=0, \dots ,n-1$ on $v^{(N)}$ are determined in Section \ref{fwrep}. The definition of the actions on
$v$ does not depend on $N$ and is well--defined, and we can easily check that the $U'_q (\widehat{ \frak{s}\frak{l}}_n)$-module defined in this section is level-1. We will use the notation $U_1$ for this  $U'_q (\widehat{ \frak{s}\frak{l}}_n)$-action on the Fock space. \\
\begin{rmk}
The two actions $U_0$ and $U_1$ appear as the representations of the subalgebras of the quantum toroidal algebra. For details, see \cite{STU}.
\end{rmk}

\subsection{The $p=1$ case}

In the paper \cite{KMS} it was demonstrated that the Fock space $F_M$ admits an action of the Heisenberg algebra $H$ which commutes with the level-1 action  $U_1$ of the algebra $U_q'(\widehat{ \frak{s}\frak{l}}_n).$ The Heisenberg algebra is a unital ${\Bbb C}$-algebra generated by elements $1, B_a$ with $a \in {\Bbb Z}_{\neq 0}$ which are subject to  relations
\begin{equation}
[ B_a , B_b ] = \delta_{a+b,0}a\frac{1 - q^{2n a}}{1 - q^{2 a}} . \label{e: Heisenbergrel} \end{equation}
The Fock space $F_M$ is an $H$-module with the action of the generators given by \cite{KMS}
\begin{equation}
B_a = \sum_{i=1}^{\infty} z_i^{a}. \label{e: Bact}
\end{equation}

Let ${\Bbb C}[H_{-}]$ be the Fock space of $H$, i.e.,  ${\Bbb C}[H_{-}]= {\Bbb C}[B_{-1},B_{-2},\dots,\:].$ The element $B_{-a}$ $ ( a = 1,2,\dots\:)$ acts on  ${\Bbb C}[H_{-}]$ by multiplication. The action of  $B_{a}$ $ ( a = 1,2,\dots\:)$ is given by (\ref{e: Heisenbergrel}) together with the relation
\begin{equation}
 B_a \cdot 1 = 0 \qquad \text{ for $ a \geq 1. $}
\end{equation}

Let $\Lambda_i$ $(i\in \{ 0,1,\dots,n-1\}) $ be the fundamental weights of $\widehat{ \frak{s}\frak{l}}_n'.$ And let $V(\Lambda_i)$ be the irreducible (level-1) highest weight module of $U_q'(\widehat{ \frak{s}\frak{l}}_n)$ with highest weight vector $V_{\Lambda_i}$ and highest weight $\Lambda_i.$

The following results are proven in \cite{KMS}:
\begin{itemize}
\item The action of the Heisenberg algebra on $F_M$ and the action $U_1$ of $U'_q (\widehat{ \frak{s}\frak{l}}_n )$ commute.
\item There is an isomorphism
\begin{equation}
 \iota_M : F_M \cong V(\Lambda_i)\otimes {\Bbb C}[H_{-}] \qquad ( M= i\bmod n )
\label{e: iota} \end{equation}
of $U_q'(\widehat{ \frak{s}\frak{l}}_n)\otimes H$-modules normalized so that $\iota_M( |M\rangle )$ $ =$ $V(\Lambda_i)\otimes 1.$
\end{itemize}
In general the level-0 $U_q'(\widehat{ \frak{s}\frak{l}}_n)$-action $U_0$   does not commute with the Heisenberg algebra. However if we choose the parameter $p$ in $U_0$ in a special way, then $U_0$ commute with the negative frequency part of $H$. Precisely, we have the following proposition, proved in \cite{STU}:
\begin{prop} \label{p: U0comH-}
At $p=1$ we have
\begin{equation}
[ U_0 , H_{-}] = 0.
\end{equation}
\end{prop}

Let $H_{-}'$ be the non-unital subalgebra in $H$ generated by  $ B_{-1},B_{-2},\dots\;.$ Proposition \ref{p: U0comH-} allows us to define a level-0 $U_q'(\widehat{ \frak{s}\frak{l}}_n)$-module structure on the irreducible level-1 module $V(\Lambda_i)$ $(i \in \{0,1,\dots,n-1\}).$ Indeed from this proposition it follows that the subspace
\begin{equation}
  H_{-}'F_M \subset F_M
\end{equation}
is invariant with respect to the action $U_0$ at $p=1$ and therefore a level-0 action of $U_q'(\widehat{ \frak{s}\frak{l}}_n)$ is defined on the quotient space
\begin{equation}
F_M/ (H_{-}'F_M)  \label{eq: jjj}
\end{equation}
which in view of (\ref{e: iota}) is isomorphic to  $V(\Lambda_i)$ with $ i \equiv M$  mod $n$.

\section{Skew Young diagrams and the level--$0$ representations of $U'_q(\widehat{ \frak{s}\frak{l}}_n )$}

\subsection{Skew Young diagrams}
Let us  recall, following the book \cite{Macbook}, the definitions of the skew (Young) diagrams, their semi-standard   tableaux and the associated skew Schur functions.

Let $\lambda ,\mu $ be partitions i.e. sequences of non--negative integers.
We assume $\l _{i} \geq \mu _i $ for all possible $i$, and if $\mu _j < i \leq \l_j $ then we draw a square whose edges are $(i-1,j-1), \; (i-1, j), \; (i,j)$ and $(i,j-1)$. (For example, see Figure 1.) This diagram is called a skew (Young) diagram and is denoted as  $\l \setminus \mu $.
We define the degree of the skew Young diagram $\l \setminus \mu $ as $| \l \setminus \mu |$ = $\sum _{i} (\l _{i} - \mu _i)$.

A skew diagram is called a border strip if it is connected and contains no $2 \times 2$ blocks of boxes. Let $\langle m_1 , \dots , m_r \rangle $ denote the border strip of $r$ columns such that the length of $i$--th column (from the right) is $m_i$. (Figure 1)

\begin{figure}
\begin{center}
\begin{picture}(180,140)(0,-130)
\put(60,0){\line(1,0){20}}
\put(20,-20){\line(1,0){60}}
\put(20,-40){\line(1,0){60}}
\put(0,-60){\line(1,0){40}}
\put(0,-80){\line(1,0){40}}
\put(0,-100){\line(1,0){20}}
\put(0,-60){\line(0,-1){40}}
\put(20,-20){\line(0,-1){80}}
\put(40,-20){\line(0,-1){60}}
\put(60,0){\line(0,-1){40}}
\put(80,0){\line(0,-1){40}}
\multiput(0,0)(10,0){6}{\line(1,0){3}}
\multiput(0,0)(0,-10){6}{\line(0,-1){3}}
\multiput(80,0)(10,0){10}{\line(1,0){3}}
\multiput(80,-20)(10,0){10}{\line(1,0){3}}
\multiput(80,-40)(10,0){10}{\line(1,0){3}}
\multiput(40,-60)(10,0){14}{\line(1,0){3}}
\multiput(40,-80)(10,0){14}{\line(1,0){3}}
\multiput(20,-100)(10,0){16}{\line(1,0){3}}
\put(90,-20){\vector(0,1){20}}
\put(90,-20){\vector(0,-1){20}}
\put(92,-20){$m_1$}
\put(115,-30){\vector(0,1){10}}
\put(115,-30){\vector(0,-1){10}}
\put(117,-30){$m_2$}
\put(140,-50){\vector(0,1){30}}
\put(140,-50){\vector(0,-1){30}}
\put(142,-50){$m_3$}
\put(165,-80){\vector(0,1){20}}
\put(165,-80){\vector(0,-1){20}}
\put(167,-80){$m_4$}
\put(-15,-10){\vector(0,-1){20}}
\put(-28,-20){$y$}
\put(10,10){\vector(1,0){20}}
\put(15,15){$x$}
\end{picture}
\end{center}
\caption{$\lambda = (4,4,2,2,1), \; \mu= (3,1,1) $.}
\end{figure}

A semi--standard tableau (s.s.t.) of the skew diagram $\l \setminus \mu $ is obtained by inscribing  integers $1,2 ,\dots ,n$ in  each square of the skew diagram. The rule of the semi--standard tableau is as follows. The numbers are strictly increasing along the column and weakly increasing along the row.
For each semi--standard tableau $T$, let  $n_i(T)$ be the multiplicity  of $i$ in $T.$
\begin{df}
For each skew diagram $ \l \setminus \mu $, the skew Schur function $s_{\l \setminus \mu}$ is defined as follows.
\begin{equation}
s_{\l \setminus \mu} (z)= \sum_{T} z_1^{n_{1}(T)} z_2^{n_{2}(T)} \dots z_N^{n_{N}(T)} .
\end{equation}
Here the summmation  is over the set of semi--standard tableaux of the skew diagram $\l \setminus \mu $.
\end{df}

\subsection{The level--$0$ representations of $U'_q(\widehat{ \frak{s}\frak{l}}_n )$ associated with the skew diagrams}

Fix a skew diagram $\l \setminus \mu $ of the border strip type and degree $N$. We put a number $(1, \dots ,N)$ on each box such that if $l>k$ then $x_l >x_k$ or ($x_l = x_k$ and $y_l > y_k$), where $(x,y)$ is a box contained in the skew diagram, and set $a_l = -2x_l + 2y_l+a$ ($a$ is fixed). (Figure 2)

\begin{figure}
\begin{center}
\begin{picture}(100,140)(0,-130)
\put(60,0){\line(1,0){30}}
\put(0,-30){\line(1,0){90}}
\put(0,-60){\line(1,0){90}}
\put(0,-90){\line(1,0){30}}
\put(0,-120){\line(1,0){30}}
\put(0,-30){\line(0,-1){90}}
\put(30,-30){\line(0,-1){90}}
\put(60,0){\line(0,-1){60}}
\put(90,0){\line(0,-1){60}}
\multiput(0,0)(10,0){6}{\line(1,0){3}}
\multiput(0,0)(0,-10){3}{\line(0,-1){3}}
\put(1,-53){$a+2$}
\put(1,-83){$a+4$}
\put(1,-113){$a+6$}
\put(45,-53){$a$}
\put(61,-23){$a-4$}
\put(61,-53){$a-2$}
\put(0,-30){\oval(25,25)[br]}
\put(0,-60){\oval(25,25)[br]}
\put(0,-90){\oval(25,25)[br]}
\put(30,-30){\oval(25,25)[br]}
\put(60,0){\oval(25,25)[br]}
\put(60,-30){\oval(25,25)[br]}
\put(2,-38){$1$}
\put(2,-68){$2$}
\put(2,-98){$3$}
\put(32,-38){$4$}
\put(62,-8){$5$}
\put(62,-38){$6$}
\put(-15,-10){\vector(0,-1){20}}
\put(-28,-20){$y$}
\put(10,10){\vector(1,0){20}}
\put(15,15){$x$}

\end{picture}
\end{center}
\caption{The case $\langle 2,1,3 \rangle $.}
\end{figure}

On the space $\otimes ^{N} V$, we define the  evaluation action $ \pi_{a_1,\dots,a_N}^{(N)} $ of $U_q'(\widehat{ \frak{s}\frak{l}}_n)$
\begin{align}
& \pi_{a_1,\dots,a_N}^{(N)}( E_i ) = \sum_{j=1}^N E_j^{i,i+1} K_{j+1}^{i}\dots K_N^{i} ,\\
& \pi_{a_1,\dots,a_N}^{(N)}( F_i ) = \sum_{j=1}^N (K_{1}^{i})^{-1}\dots (K_{j-1}^{i})^{-1}E_j^{i+1,i} ,\\
& \pi_{a_1,\dots,a_N}^{(N)}( K_i ) =  K^{i}_1K^{i}_2\dots K^{i}_N , \quad (i=1,2,\dots,n-1)  \\
& \pi_{a_1,\dots,a_N}^{(N)}( E_0 ) = \sum_{j=1}^N q^{a_j} E_j^{n,1} K_{j+1}^{0}\dots K_N^{0} ,\\
& \pi_{a_1,\dots,a_N}^{(N)}( F_0 ) = \sum_{j=1}^N q^{-a_j} (K_{1}^{0})^{-1}\dots (K_{j-1}^{0})^{-1}E_j^{1,n} , \\
& \pi_{a_1,\dots,a_N}^{(N)}( K_0 ) = (\pi_{a_1,\dots,a_N}^{(N)}(K_1 K_2 \cdots K_{n-1}))^{-1},
\end{align}
and on the same space, we consider the following operators:
\begin{equation}
R_{i,j} (x) = \frac{x S_{i,j}^{-1}- S_{i,j}}{x-1} P_{i,j}, \; \;
\check{R}_{i,j} (x) = \frac{x S_{i,j}^{-1}- S_{i,j}}{x-1},
\label{Rcheck}
\end{equation}
where $P_{i,j}(\dots \otimes \stackrel{i}{u} \otimes \dots \otimes \stackrel{j}{v} \otimes \dots ) = \dots \otimes \stackrel{i}{v} \otimes \dots \otimes \stackrel{j}{u} \otimes \dots$.
We define
\begin{align}
& R_{\l \setminus \mu } = \prod _{1\leq i< j \leq N} R_{i,j} (q^{a_i-a_j}), \\
& \bar{R}_{\l \setminus \mu } = \prod _{1\leq i< j \leq N} R_{j,i} (q^{a_i-a_j}), \\
& \check{R}_{\l \setminus \mu } = \prod _{1\leq i< j \leq N} \check{R}_{N+i-j,N+i-j+1} (q^{a_i-a_j}),
\end{align}
where $(i,j) $ is on the right to $(i',j')$ in the product if $i<i'$ or ($j<j'$ and $i=i'$). As a special case of \cite{CheDuke} Proposition 1.5., we have
\begin{prop}[\cite{CheDuke}] \label{skewrep}
The subspace $\mbox{Im} R_{\l \setminus \mu }(\otimes ^N V) (= \mbox{Im} \check{R}_{\l \setminus \mu }(\otimes ^N V)) $ with the action $\pi_{a_1,\dots,a_N}^{(N)} $ is an irreducible $U_q '(\widehat{ \frak{s}\frak{l}}_n )$--module, and the map $\check{R}_{\l \setminus \mu }$ : $(\pi_{a_N,\dots,a_1}^{(N)} , \; \otimes ^N V / $ Ker $\bar{R}_{\l \setminus \mu })=  (\pi_{a_N,\dots,a_1}^{(N)} , \;\otimes ^N V / $ Ker $\check{R}_{\l \setminus \mu }))$ $\rightarrow $ $(\pi_{a_1,\dots,a_N}^{(N)} , \; $ Im $\check{R}_{\l \setminus \mu })$ is an isomorphism of the $U_q '(\widehat{ \frak{s}\frak{l}}_n )$--modules.
\end{prop}
\begin{rmk}
In \cite{CheDuke}, this proposition is proved in the $U_q '(\widehat{ {\frak{g}} {\frak{l}}}_n)$--module case and the normalizations of $q$ and $x$ are different. The irreducibility as the $U_q '(\widehat{ \frak{s}\frak{l}}_n )$--module follows from the result of \cite{AK}.
\end{rmk}

\subsection{Character formulas}
Let $\overline{\Lambda }_i $ $(i=1, \dots ,n-1)$ be the fundamental weights of $ {{\frak s} {\frak l}}_n $ and let $\epsilon _i = \overline{\Lambda }_i - \overline{\Lambda }_{i-1} $ $(i=1, \dots ,n)$ with $\overline{\Lambda }_0 = \overline{\Lambda }_n :=0$.

The subalgebra  of $U'_q (\widehat{ \frak{s}\frak{l}}_n )$ generated by $E_i , \; F_i ,\; K_i^{\pm }$ $(i=1,\dots n-1)$ is isomorphic to the algebra $U_q ( {{\frak s} {\frak l}}_n)$.  In the paper \cite{KKN} the $ {{\frak s} {\frak l}}_n$--character of the irreducible $Y( {{\frak s} {\frak l}}_n )$--representation associated with a skew diagram was shown to be given by the corresponding skew Schur function. This result  is immediately generalized to the $q$-deformed situation. Precisely we have

\begin{prop}[\cite{KKN}] The skew Schur function $s_{\l \setminus \mu }(z)$ where $z_i = e ^{\epsilon _i}$ is equal to the
 $U_q ( {{\frak s} {\frak l}}_n)$-character of the irreducible $U_q '(\widehat{ \frak{s}\frak{l}}_n )$--module described by Proposition \ref{skewrep}.
\end{prop}
As a corollary we get
\begin{cor} \label{dimbs}
The dimension of the space $\mbox{Im} R_{\l \setminus \mu } \subset (\otimes ^N V)$ is equal to the total number of the semi--standard tableaux of the skew diagram $\l \setminus \mu $.
\end{cor}

Let $V(\Lambda _k )$ be the level--$1$ irreducible module of $U_q (\widehat{ \frak{s}\frak{l}}_n)$ whose highest weight is the $k$--th fundamental weight $\Lambda _k$ of $\widehat{ \frak{s}\frak{l}}_n '$. We set $ch (V(\Lambda _k )) = \sum _{i,\l } ($dim$V_{\l , i}) e ^{\l} q^{i}$, where $V_{\l ,i}$ is the weight subspace with  $U_q ( {{\frak s} {\frak l}}_n)$-weight $\l$ and homogeneous degree $i.$
The following proposition is proved in \cite{KKN}.
\begin{prop} [\cite{KKN}] \label{chara}
Setting  $z_i = e ^{\epsilon _i}$ we have
\begin{equation}
ch (V(\Lambda _k )) =   q^{\frac{1-n}{24}- \frac{k(n-k)}{2n}} \sum_{\theta \in BS \atop {| \theta | \equiv k \; mod \; n}} q ^{\frac{1}{2n} | \theta | (n- | \theta |)+t(\theta )} s_\theta (z).
\end{equation}
where BS is the set of all the border strips $\theta = \langle m_1 , \dots , m_r \rangle $ and $t(\theta ) = \sum_{i=1}^{r-1} (r-i)m_i$ with $m_r <n$.
\end{prop}
Note that if $m_i >n$ for some $i$, then the skew Schur function $s_{\theta }$ is equal to $0$, moreover, for  the border strip of the form $\theta _l = \langle m_1, \dots , m_r , \overbrace{n, \dots n}^{l} \rangle $ the number $\frac{1}{2n} | \theta _l | (n- | \theta _l |)+t(\theta_l  )$ does not depend on $l$.

\section{Non--symmetric Macdonald polynomials and the decomposition}

\subsection{Non--symmetric Macdonald polynomials}
\label{nsMac}

We will define the non--symmetric Macdonald polynomials as the joint eigenfunctions of the Cherednik's operators $Y_i^{(N)}$ $(i=1 , \dots , N)$ \cite{U1,TU}. It will be convenient for our  purposes to label these polynomials by the set of pairs $(\l , \sigma)$ which we now describe.

Let $\tilde {\cal M}_{N} $ be the a set of all non--decreasing sequences of integers
$ \l =  (\lambda _{1} \: , \: \lambda _{2} \: , \dots , \:  \lambda _{N}), $
and  let  $\tilde {\cal M}_{N} ^n $ be the subset of $\tilde {\cal M}_{N} $ which consists of all  $n$--strict non--decreasing sequences (cf. Section  \ref{sec:L-0}).
For each $\lambda \in \tilde {\cal M}_{N} $ we set $|\lambda |$ :=$ \sum_{i=1}^{N} \lambda _{i}. $
 For $\lambda , \mu \in \tilde {\cal M}_{N} $ such that $|\lambda|=|\mu|$  we define the dominance (partial) ordering:
\begin{equation}
 \lambda \succeq \mu \hspace{.4in} \Leftrightarrow
\hspace{.4in} \sum_{j=1}^{i} \lambda_{j} \geq \sum_{j=1}^{i} \mu_{j} \; \;
 (i\in \{1,2,\dots,N \}).
\end{equation}
 Let $S^{\lambda } \subset {{\frak{S}}_{N}} $ be the set of
 elements ${\sigma }$ such that
 if $\lambda_{\sigma (i)} = \lambda_{\sigma (j)}$ and $\sigma (i) < \sigma (j) $ then $i<j$.
 We define the total ordering on  $S^{\lambda }$:
\begin{equation}
 \sigma \succ \sigma ' \; \; \Leftrightarrow \; \;
\mbox{the last nonzero element of }
 (\lambda _{\sigma (i)} -\lambda _{\sigma' (i) } )_{i=1}^N   \mbox{ is } < 0.
\label{Sord}
\end{equation}
Then the following properties are satisfied. (In what follows the  $\sigma (i,i+1) $ denotes  the composition of $\sigma $ and a transposition $(i,i+1)$)\\
a) $S^{\lambda }$ has the unique minimal element with respect to the ordering (\ref{Sord}). We denote this element  by $min.$ Note that we one has $\l _{min(i)} \leq \l _{min(i+1)} $ $(i=1, \dots ,N-1)$. \\
b) $S^{\lambda }$ is connected, i.e. for any $\sigma \in S^{\l }$, there exist $i_1 , \dots ,i_r$ such that if we put $\sigma _{l} = \sigma (i_1, i_1+1) \dots (i_l, i_l +1) $ then $\sigma _{r}= min $,  $\sigma _{l} \in  S^{\lambda }$, $\sigma _{l} \succ \sigma _{l+1} \; (l=1, \dots ,r)$ . \\
c) Suppose $\sigma \in S^{\lambda }$, then $\sigma (i,i+1) \in S^{\lambda } \Leftrightarrow \l _{\sigma (i)} \neq  \l _{\sigma (i+1)}$. \\
d) If $\l _{\sigma (i)} > \l _{\sigma (i+1)}$ and $\sigma \in S^{\lambda }$ then $\sigma \succ \sigma (i,i+1)$.

We define the partial ordering of the set $\{ (\lambda , \sigma )\: | \: \l \in \tilde {\cal M}_{N}, \sigma \in S^{\lambda }\}$:
\begin{equation}
  (\lambda , \sigma ) \succ ( \tilde{\lambda } , \tilde{\sigma })
  \; \; \; \Leftrightarrow \; \; \;
|\lambda |= |\tilde{\lambda } | \mbox{ and }
\left\{
\begin{array}{l}
\lambda \succ \tilde{\lambda }   \\
\lambda = \tilde{\lambda } , \; \; \sigma \succ \tilde{\sigma} .
\end{array} \right.
\end{equation}

Then $Y_{i}^{(N)}$ act triangularly on $\cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}] $
 with respect to this ordering \cite{U1}:
\begin{equation}
 Y_{i}^{(N)} z^{\lambda ^{\sigma }} = \xi _{i}^{\lambda}(\sigma)
z^{\lambda ^{\sigma }} + \mbox{``lower terms''} ,
\label{Ytri}
\end{equation}
\vspace{-.2in}
\begin{equation}
\xi _{i}^{\lambda}(\sigma) =
p^{\lambda _{\sigma (i)}} q^{2\sigma (i) -N-1}  \; \; \; \;
(\sigma \in S^{\lambda } ).
\end{equation}
In the above notation, we identify the ordering of  monomials
$z^{\lambda ^{\sigma }}:= z_{1}^{\l _{\sigma_{(1)}}} z_{2}^{\l _{\sigma_{(2)}}}
\dots z_{N}^{\l _{\sigma_{(N)}}}$ with the ordering on the set of pairs $(\l, \sigma)$.

For generic $q$ and  $p$  the pair $(\lambda , \sigma )$ is uniquely determined from the ordered set
 $ (\xi _{1}^{\lambda}(\sigma) , \xi _{2}^{\lambda}(\sigma) , \dots
\xi _{N}^{\lambda}(\sigma) )$:
 \begin{equation}
 (\lambda , \sigma ) \neq  (\tilde{\lambda } , \tilde{\sigma})
 \; \; \; \Leftrightarrow \; \; \;
( \xi _{1}^{\lambda}(\sigma) , \xi _{2}^{\lambda}(\sigma) , \dots
\xi _{N}^{\lambda}(\sigma) ) \neq
( \xi _{1}^{\tilde{\lambda}}(\tilde{\sigma }) ,
\xi _{2}^{\tilde{\lambda }}(\tilde{\sigma }) , \dots
\xi _{N}^{\tilde{\lambda }}(\tilde{\sigma }) ).
\end{equation}
 Therefore one can simultaneously diagonalize
 the operators $Y_{i}^{(N)} \; \; (1 \leq i \leq N)$.
\begin{equation}
 Y_{i}^{(N)} \Phi_{\sigma}^{\lambda } (z)=
\xi _{i}^{\lambda}(\sigma) \Phi_{\sigma}^{\lambda } (z),\; \; \; \;
\Phi_{\sigma}^{\lambda } (z)=z^{\lambda ^{\sigma }}
+ \mbox{``lower terms''} .
\label{Ydia}
\end{equation}
 The Laurent polynomial $\Phi_{\sigma}^{\l} (z)  $ is known as  the non--symmetric Macdonald polynomial.

 The action of $g_{i,i+1}$ on the non--symmetric Macdonald polynomial is
as follows \cite{U1}.
\begin{equation}
 g_{i,i+1} \Phi _{\sigma }^{\lambda } (z)=
A_{i}(\sigma ) \Phi _{\sigma }^{\lambda } (z)+
B_{i}(\sigma ) \Phi _{\sigma (i,i+1)}^{\lambda }(z),
\end{equation}
\vspace{-.3in}
\begin{equation}
 A_{i}(\sigma ):=\frac{(q-q^{-1})x}{x-1} , \; \; \; \;
B_{i}(\sigma ):=
\left\{
\begin{array}{cc}
 q^{-1} \{ x \} \; \; & (\lambda _{\sigma (i)} > \lambda _{\sigma (i+1)}); \\
 0 & (\lambda _{\sigma (i)} = \lambda _{\sigma (i+1)}); \\
 q^{-1} & (\lambda _{\sigma (i)} < \lambda _{\sigma (i+1)}),
\end{array} \right.
\end{equation}
\vspace{-.2in}
\begin{equation}
\{ x \} :=\frac{(x-q^{2})(q^{2}x-1)}{(x-1)^{2}} ,\; \; \; \;
x:=\frac{\xi _{i+1}^{\lambda}(\sigma)}{\xi _{i}^{\lambda}(\sigma)}.
\label{Ydiae}
\end{equation}

The case $p=1$ is not generic. However, from  the  results of  \cite{Knop} it follows that the coefficients of $\Phi_{\sigma}^{\l} (z) $ have no poles at $p=1.$ Therefore  the non--symmetric Macdonald polynomials $\Phi_{\sigma}^{\l} (z)  $ are still well--defined at $p=1$ and the formulas (\ref{Ydia}--\ref{Ydiae}) are still satisfied.

In what follows we will let  $\tilde{\Phi }_{\sigma}^{\l} (z)  $ denote the non--symmetric Macdonald polynomial at $p=1$.
In virtue of the triangularity (\ref{Ytri}) the non--symmetric Macdonald polynomials $\tilde{\Phi }_{\sigma}^{\l} (z)  \; ( \lambda \in \tilde {\cal M}_{N} $, $ \sigma \in S^{\lambda })$ form a base of $\cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}] $. We put
\begin{equation}
E^{\lambda }= \bigoplus _{\sigma \in S^{\lambda }} {\Bbb C} \tilde{\Phi }_{\sigma}^{\l} (z)  .
\end{equation}
Then  $\cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}] = \oplus _{\lambda } E^{\lambda }$.
In Section \ref{dc} we will use the following lemma.
\begin{lemma} \label{p1nsMac}
Let  $e_{-k}= \sum _{1 \leq n_1 < \dots < n_k \leq N} z_{n_1}^{-1} \dots z_{n_k}^{-1}.$ Suppose that  $\l \in \tilde {\cal M}_{N} $ satisfies  $\l _i - \l _{i+1} = 0$ or $1.$ Then we have
\begin{equation}
e_{-k} \tilde{\Phi }_{\varsigma } ^{\l } (z) = \tilde{\Phi }_{\varsigma } ^{\tilde{\l }} (z).
\end{equation}
Here $\tilde{\l } = (\l _{1} ,\dots ,\l _{N-k} ,\dots ,\l_{N-k+1}-1 ,\dots ,\l _{N}-1)$ and $\varsigma (\in S^{\l }, S^{\tilde{\l }})$ is the minimal element of $S^{\l }$.
\end{lemma}
\begin{pf}
By the triangularity of the non--symmetric Macdonald polynomial (\ref{Ytri}), we have
\begin{eqnarray}
& & e_{-k} \tilde{\Phi }_{\varsigma } ^{\l } (z)  = e_{-k} ( z^{\l ^{\varsigma }} + \sum_{\mu \prec \l , \; \sigma \in S^{\mu }} c _{\mu , \sigma } z^{\mu ^{\sigma }})
\label{eP} \\
& & = z^{\tilde{\l } ^{\varsigma }} + \sum_{(\mu , \sigma ) \prec (\tilde{\l }, \varsigma )} c' _{\mu , \sigma } z^{\mu ^{\sigma }}
= \tilde{\Phi }^{\tilde{\l }} _{\varsigma } (z) + \sum_{(\mu , \sigma ) \prec (\tilde{\l }, \varsigma )} c'' _{\mu , \sigma } \tilde{\Phi } ^{\mu } _{\sigma } (z).
\nonumber
\end{eqnarray}
At $p=1$, the operators $Y_{i}^{(N)}$ commute with symmetric Laurent polynomials considered as multiplication operators on $\cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}]. $ Hence  we have
\begin{equation}
Y_{i}^{(N)} e_{-k} \tilde{\Phi }_{\varsigma } ^{\l } (z) = e_{-k}Y_{i}^{(N)} \tilde{\Phi }_{\varsigma } ^{\l } (z) = q^{2 \varsigma (i)-N-1}e_{-k} \tilde{\Phi }_{\varsigma } ^{\l } (z) .
\label{jev}
\end{equation}
The ordered set of eigenvalues $\{q^{2 \sigma (i)-N-1} \}_{i=1}^N$ determines the element $\sigma \in S^{\tilde{\l}}$ uniquely. Hence (\ref{eP}) and (\ref{jev}) lead to
\begin{equation}
e_{-k} \tilde{\Phi }_{\varsigma } ^{\l } (z) =\tilde{\Phi }^{\tilde{\l }} _{\varsigma } (z) + \sum_{\mu \prec \tilde{\l } } c'' _{\mu  } \tilde{\Phi } ^{\mu } _{\varsigma } (z).
\label{eP2}
\end{equation}
Now let us  consider any  $\mu $ which appears in the sum (\ref{eP2}).
If there exists $i (< N-k )$ such that $\mu _i < \l _i$ then for $j >i$ we necessarily  have $\mu _j < \l _j $ because of the assumption $\l _i - \l _{i+1}=0 $ or $1$ and the fact that $\l _i < \l _j $ implies  $\mu _i < \mu _j $, which follows since  $\varsigma \in S^{\mu }$.
But  $\mu _j < \l _j \; (j >i )$ leads to $|\tilde{\l }| > |\mu |$ which is in contradiction with  $ \mu \prec \tilde{\l }.$
Thus we have $\mu _i \geq \l _i \; (i <N-k).$ If $\mu _i > \l _i $ for some $i$, then necessarily $ \mu \not \preceq \tilde{\l }$ which is again a contradiction. Hence  we get $\mu _i = \l _i \; (i <N-k)$.
By the condition $ \mu \prec \tilde{\l }$, we have $\mu _{N-k} = \tilde{\l } _{N-k}$. Because of the assumption on $\l $ and the fact that $\l _i < \l _j $ implies  $\mu _i < \mu _j $, we get $\mu _j \leq \tilde{\l } _j \; (j >N-k)$.
Combining this  with $|\tilde{\l }| = |\mu |$ we conclude that  $\mu _j = \tilde{\l } _j$ $\forall j$. That is  the second term in  the r.h.s. of (\ref{eP2}) is zero.
\end{pf}

\subsection{The decomposition} \label{dc}
Let us consider the quotient space  $F_{M} / H_{-}' F_{M}$ and for each $k \geq 0$ its subspace $F^k_{M} / (H_{-}' F_{M} \cap F^k_{M}).$

It is straightforward to establish the necessary and sufficient condition for the  vector $\omega = \sum _{\lambda } \sum _{\sigma \in S^{\lambda }} \tilde{\Phi }_{\sigma}^{\l} (z)  \otimes \psi ^{\lambda } _{\sigma }$ $ ( \in \cplx[z_1^{\pm 1},\dots,z_N^{\pm 1}] \otimes (\otimes ^{N} V))$ to be equivalent to $0$ in the quotient space $\wedge ^N V(z)$. The result is
\begin{equation}
\forall \lambda
\left\{
\begin{array}{ll}
\psi ^{\lambda } _{\sigma (i,i+1)} = - \check{R}_{i,i+1} (q^{2(\sigma (i)-\sigma (i+1))})\psi ^{\lambda } _{\sigma }, &
\forall \sigma \mbox{ s.t. } \lambda _{\sigma (i)} > \lambda _{\sigma (i+1)} ; \\
(q^{-2}S_{i,i+1}^{-1} - S_{i,i+1} ) \psi ^{\lambda } _{\sigma }=0  , &
\forall \sigma \mbox{ s.t. } \lambda _{\sigma (i)} = \lambda _{\sigma (i+1)} ,
\end{array}
\right.
\label{phirel1}
\end{equation}
where $\check{R}_{i,i+1} (x)$ is defined in (\ref{Rcheck}). In view of  the properties of the set $S^{\lambda }$, in the space $\wedge ^{N} V(z)$ we have
\begin{equation}
\tilde{\Phi }_{\sigma}^{\l} (z)  \otimes \psi  _{\sigma }
\label{Pp1Pmin}
\end{equation}
\vspace{-.2in}
\begin{equation}
\sim  \tilde{\Phi }_{min}^{\l}(z)   \otimes \check{R}_{i_r,i_r +1} (q^{2(\sigma _{r}(i_r +1) - \sigma _{r}(i_r))}) \dots \check{R}_{i_1,i_1 +1} (q^{2(\sigma _{1}(i_1 +1) - \sigma _{1}(i_1))}) \psi _{\sigma }.
\nonumber
\end{equation}
Here we used the notations of Section \ref{nsMac}.
\vspace{.2in}

By the triangularity of the non--symmetric Macdonald polynomial (\ref{Ytri}) and the relation (\ref{Pp1Pmin}), we get
\begin{equation}
V_{M}^{s+nk,k} = \bigoplus _{\l } (E^{\l } \otimes (\otimes ^{N} V)) / \Omega \cap (E^{\l } \otimes (\otimes ^{N} V))
\label{red0}
\end{equation}
\vspace{-.2in}
\begin{equation}
= \bigoplus _{\l }(\tilde{\Phi }_{min}^{\l}(z)   \otimes (\otimes ^{N} V)) / \Omega \cap (\tilde{\Phi }_{min}^{\l}(z)  \otimes (\otimes ^{N} V)),
\nonumber
\end{equation}
where the summation  is over $\lambda \in \tilde {\cal M}_{N} ^n$, such that  $\l_{1} \leq m_{s+nk}^0$, $| {\bold m} ^0 - \l ^{min} | =k$.

\begin{prop} \label{red1}
Define the set $\tilde{\cal M}_{s+nk}^{n,k}$ as
\begin{equation}
\tilde{\cal M}_{s+nk}^{n,k} = \{ \l \in \tilde{\cal M}_{s+nk}^n | \:  \l_{1} \leq m_{s+nk}^0, \; | {\bold m} ^0 - \l ^{min} | =k \mbox{ and }\l _i -\l _{i+1} =0 \mbox{ or } 1 \} .
\end{equation}
Every vector from the linear space  $F^k_{M} / (H_{-}' F_{M} \cap F^k_{M}) $ can be expressed as a linear combination of vectors of the form $\wedge (\tilde{\Phi }_{min}^{\l}(z)   \otimes \psi ^{\l }) |M-s-nk \rangle $, where  $\lambda \in \tilde{\cal M}_{s+nk}^{n,k}$ and $\psi ^{\l } \in  \otimes ^{N} V.$
\end{prop}
\begin{pf}
By the equation (\ref{red0}), it is sufficient to show that $\wedge( \tilde{\Phi }_{min}^{\l}(z)  \otimes \psi ^{\l}) |M-s-nk \rangle $ ($\l \in \tilde{\cal M}_{s+nk}$, $\l_{1} \leq m_{s+nk}^0, \;  | {\bold m} ^0 - \l ^{min} | =k $, $\psi ^{\l } \in  \otimes ^{N} V$ ) is equivalent to 0 in the space $F^k_{M} / (H_{-}' F_{M} \cap F^k_{M})$ unless $\l _i -\l _{i+1} =0 $ or $1$ for all $ i=1,\dots,N-1$  . We will prove this by  induction with respect to the ordering of the set $\tilde{\cal M}_{s+nk}$. (Note  that if $\l $ is not $n$--strict then $\wedge (\tilde{\Phi }_{min}^{\l}(z)  \otimes \psi ^{\l} )= 0 $.)

Since $\l _i - \l _{i+1} \neq 0,1 $ implies that  $(\l_1, \dots ,\l_i -1, \l_{i+1}+1, \dots )$ is lower with respect to the ordering of $\tilde{\cal M}_{s+nk}$,
the minimal element satisfies the condition of  Proposition \ref{red1}.

Fix $\l $ and assume that the proposition is  proved for all $\mu $ such that  $\mu \prec \l $. Define $\tilde {\l } \in \tilde{\cal M}_{s+nk}$ as follows:
\begin{eqnarray}
& \l _i = \l _{i+1} \; \Leftrightarrow \; \tilde {\l }_i  = \tilde {\l }_{i+1}, & \\
& \l _1 = \tilde {\l }_1 ,& \\
& \tilde {\l }_i  \neq \tilde {\l }_{i+1} \; \Rightarrow \; \tilde {\l }_i  = \tilde {\l }_{i+1} +1 .&
\end{eqnarray}
For each positive integer $l$, define $n_l = \# \{ i \: | \:  \tilde {\l }_i - \l _i =l \}$. If $n_l =0$ for all $l( \geq 1)$ then either $\wedge(\tilde{\Phi }_{min}^{\l}(z)  \otimes \psi ^{\l} )| M-s-nk \rangle$ itself satisfies the condition of the Proposition \ref{red1}, or  else the element $\prod _{l \geq 1} B_{-l}^{n_l} \cdot \wedge(\tilde{\Phi }^{\tilde{\l}}_{min} (z)\otimes \psi _{\l }) | M-s-nk \rangle $ is in $H_{-}' F_M \cap F_M^k$.
Expanding the last element, in the space $F^k_{M} / (H_{-}' F_{M} \cap F^k_{M}) $ we get
\begin{equation}
\wedge (\tilde{\Phi }_{min}^{\l}(z)  \otimes \psi ^{\l } ) | M-s-nk \rangle+ \sum_{\mu \prec \l , \; \sigma \in S^{\mu }} \wedge (\tilde{\Phi }^{\mu }_{\sigma } (z)\otimes \psi ^{\mu }_{\sigma })| M-s-nk \rangle \sim 0,
\end{equation}
for some $\psi ^{\mu }_{\sigma }$. By (\ref{Pp1Pmin}) and the induction assumption the proposition is proven.
\end{pf}

\begin{prop} \label{MNnk}
For each $\lambda \in \tilde{\cal M}_{s+nk}^{n,k}$, define $J$ and $r_j$ such that
$\l _1 = \dots = \l _{r_J} >  \l _{r_J +1} = \dots =\l _{r_J +r_{J-1}} > \dots \geq \l _{r_1 + \dots +r_{J}(=N)}$, then in the space $F^k_{M} / (H_{-}' F_{M} \cap F^k_{M}) $, for each $\psi \in \otimes ^N V$ we have
\begin{equation}
\wedge (\tilde{\Phi }_{min}^{\l}(z)   \otimes R_{i,i+1}(q^{2} )\psi ) |M-s-nk \rangle
\sim 0 , \; \; \; \; ( \l _{min (i)} = \l _{min (i+1)})
\end{equation}
\vspace{-.2in}
\begin{equation}
\wedge (\tilde{\Phi }_{min}^{\l}(z)   \otimes \prod_{1 \leq a \leq r_j \atop{ 0 \leq b \leq r_{j+1}-1}} R_{l_j +a, l_j+r_j+r_{j+1}-b}(q^{-2(a+b)})  \psi ) |M-s-nk \rangle \sim 0 ,
\label{jrel}
\end{equation}
where $(a,b)$ is on the right to $(a',b')$ in the product if $a<a'$ or ($a=a'$ and $b<b'$), $l_j = \sum_{i=1}^{j-1} r_j$ and $l_0 =0$.
\end{prop}
\begin{pf}
The first relation follows from  (\ref{phirel1}) and the identity
\begin{equation}
\mbox{Im} (q^{2} S_{i,i+1}^{-1} - S_{i,i+1} ) = \mbox{Ker} (q^{-2} S_{i,i+1}^{-1} - S_{i,i+1} ) .
\end{equation}
Consider the second relation. We define
\begin{equation}
\bar{\l } = ( \l_1 , \dots , \l _{r_{j+1}+ \dots +r_{J}}, \l _{1+r_{j+1}+ \dots +r_{J}}+1, \dots ,\l_{N}+1).
\end{equation}
By Lemma \ref{p1nsMac}, the definition of the space $F^k_{M} / (H_{-}' F_{M} \cap F^k_{M}) $ and the relation (6.51) in \cite{STU}, we get
\begin{equation}
f( B_{-1}, \dots , B_{-(r_1 + \dots +r_j)} )\cdot \wedge (\tilde{\Phi }^{\bar{\l }}_{min} (z)\otimes \psi ) |M-s-nk \rangle
\end{equation}
\vspace{-.2in}
\begin{equation}
= ( f( B_{-1}, \dots , B_{-(r_1 + \dots +r_j)} ) \cdot \wedge (\tilde{\Phi }^{\bar{\l }}_{min} (z) \otimes \psi ))|M-s-nk \rangle
\nonumber
\end{equation}
\vspace{-.2in}
\begin{equation}
=\wedge (\tilde {\Phi }^{\l }_{\varsigma } (z) \otimes \psi )| M-s-nk \rangle \sim 0.
\nonumber
\end{equation}
Here $f( x_1 , \dots , x_l)$ is a polynomial such that
\begin{equation}
f (\sum _{i=1}^{N} z_i , \sum _{i=1}^{N} z_i^2 ,\dots ,\sum _{i=1}^{N} z_i^{l} ) = \sum _{i_1 < \dots < i_l } z_{i_1} \dots z_{i_l},
\end{equation}
 and $\varsigma \in S^{\l} $ is the minimal element of $S^{\bar{\l}}$.

If we apply the formula (\ref{Pp1Pmin}), we get
\begin{equation}
\wedge (\tilde {\Phi }^{\l }_{min} (z) \otimes  \prod_{0 \leq a \leq r_j-1 \atop{ 0 \leq b \leq r_{j+1}-1}} \check{R}_{l_j +r_{j+1}-b+a, l_j +r_{j+1}-b+a+1}(q^{-2(a+b+1)}) \psi )| M-s-nk \rangle \sim 0,
\end{equation}
where $(a,b)$ on the right to $(a',b')$ in the product if $a<a'$ or ($a=a'$ and $b<b'$).
Finally, taking into account the relation
\begin{equation}
 \prod_{0 \leq a \leq r_j-1 \atop{ 0 \leq b \leq r_{j+1}-1}} \check{R}_{l_j +r_{j+1}-b+a, l_j +r_{j+1}-b+a+1}(q^{-2(a+b+1)}) \cdot \prod_{1 \leq a \leq r_j \atop{ 0 \leq b \leq r_{j+1}-1}} P_{l_j +a, l_j+r_j+r_{j+1}-b}
\end{equation}
\vspace{-.2in}
\begin{equation}
 = \prod_{1 \leq a \leq r_j \atop{ 0 \leq b \leq r_{j+1}-1}} R_{l_j +a, l_j+r_j+r_{j+1}-b}(q^{-2(a+b)}),
\nonumber
\end{equation}
we obtain  (\ref{jrel}).
\end{pf}

With notations of  Proposition \ref{MNnk}, for each  $\l \in \tilde{\cal M}_{s+nk}^{n,k}$ define the linear subspace of $\otimes ^N V$
\begin{equation}
V^{\l } = \sum _{\l _{min(i)} = \l _{min(i+1)}} \mbox{Im} R_{i,i+1}(q^2) \; + \sum_{j=1}^{J-1} \mbox{Im} (\prod_{1 \leq a \leq r_j \atop{ 1 \leq b \leq r_{j+1}-1}} R_{l_j +a, l_j+r_j+r_{j+1}-b}(q^{-2(a+b)})).
\end{equation}

By Proposition \ref{MNnk}, we get
\begin{prop} \label{Psurj}
Consider the map
\begin{eqnarray}
\psi_k : \bigoplus _{\l \in  \tilde{\cal M}_{s+nk}^{n,k}} \tilde{\Phi }_{min}^{\l}(z)  \otimes (\otimes ^{s+nk} V  / V^{\l }) & \rightarrow & F_M^k / ( H_{-}'F_M \cap F_M^k ) \\
\label{psik}
v & \mapsto & v \wedge |M-s-nk \rangle . \nonumber
\end{eqnarray}
Define the action of $U'_q (\widehat{ \frak{s}\frak{l}}_n)$ on the l.h.s. of {\em (\ref{psik})} by {\em (\ref{e: Efin}-\ref{e: K0})}.

Then  the map $\psi_k $ is well-defined, surjective and is a  $U'_q (\widehat{ \frak{s}\frak{l}}_n)$--intertwiner.
\end{prop}

\begin{prop} \label{lteq}
Let $\l \in \tilde{\cal M}_{s+nk(=N)}^{n,k}.$ For any such  $\l$ we define  $J$, $r_j$ and $l_j$ in the same way  as in Proposition \ref{MNnk}.
 Let $\theta $ be the border strip characterized by $\langle r_J, r_{J-1} , \dots , r_1 \rangle $. We have
\begin{equation}
\otimes ^N V / V^{\l } = \otimes ^N V / Ker \bar{R}_{\theta }.
\end{equation}
\end{prop}
\begin{pf}
First we will show that $\otimes ^N V / V^{\l } \supset \otimes ^N V / Ker \bar{R}_{\theta }$.
Applying repeatedly the Yang--Baxter equation $R_{a,b} (x) R_{a,c}(xy) R_{b,c}(y)= R_{b,c}(y)R_{a,c}(xy)R_{a,b} (x)$, we can move some special elements to the right in the product
\begin{equation}
\bar{R}_{\theta } = \cdots R_{i+1,i}(q^{-2})
 = \cdots \prod_{ 1 \leq a \leq r_{j+1} \atop{ 0 \leq b \leq r_j -1}} R_{ l_j+r_j+a, l_j +r_j-b }(q^{2(a+b)}),
\end{equation}
where $(a,b)$ on the right to $(a',b')$ in the product if $a<a'$ or ($a=a'$ and $b<b'$), and $\l _{min(i)} = \l _{min(i+1)}$.
By the  formula
\begin{equation}
R_{b,a} (x) R_{a,b} (x^{-1}) = \frac{(x-q^2)(x^{-1}q^{-2}-1)}{(x-1)(x^{-1}-1)} id,
\end{equation}
we get $\otimes ^N V / V^{\l } \supset \otimes ^N V / Ker \bar{R}_{\theta }$.

Next we will show that $\otimes ^N V / V^{\l } \subset \otimes ^N V / Ker \bar{R}_{\theta }$.
We show that the vectors $\otimes  _{i=1}^N v_{e_i}$ such that $e_i <e_{i+1} $ if $\l _{min{(i)
}} = \l _{min{(i+1)}}$, $e_i \geq e_{i+1} $ if $\l _{min{(i)}} \neq \l _{min{(i+1)}}$  span the space $\otimes ^N V / V^{\l }$.

Using the relations $R_{i',i'+1}(q^{2})( \otimes ^N v_{e_i} ) \sim 0$ $(\l _{min(i')}= \l _{min(i'+1)})$ we find that the set of vectors $\{ \otimes _{i=1} ^N v_{e_i}\: |\: \text{if $\l _{min{(i)}} = \l _{min{(i+1)}}$ then $e_i <e_{i+1}$ }\} $ spans the space  $\otimes ^N V / V^{\l }$. For each  vector of the form $ \otimes  _{i=1}^N v_{e_i}$ we define $\tilde{N} (\otimes  _{i=1}^N v_{e_i} ) = \# \{ (i,j)| \; i<j,\; e_i \geq e_j $ and $\l _{min{(i)}} \neq \l _{min{(j)}} \} $.
 Consider the vector $\otimes  _{i=1}^N v_{e_i} $ such that if $\l _{min{(i)}} = \l _{min{(i+1)}}$ then $e_i <e_{i+1} $.
Assume that there is an $i$ such that $e_i < e_{i+1} $ (if $\l _{min{(i)}} \neq \l _{min{(i+1)}}$), then we get the relation
\begin{equation}
\prod_{1 \leq a \leq r_j \atop{ 1 \leq b \leq r_{j+1}-1}} R_{l_j +a, l_j+r_j+r_{j+1}-b} ( q^{-2(a+b)} )   (  \otimes ^N v_{e_i} )  \sim 0,
\label{re0}
\end{equation}
for all possible $j$. By (\ref{re0}), the vector $\otimes ^N v_{e_i} $ is equivalent to a linear combination  of the vectors $\otimes ^N v_{e_{i'}} $ such that $\tilde{N} (\otimes ^N v_{e_{i'}} ) < \tilde{N} (\otimes ^N v_{e_i} )$.
Because $\otimes ^N v_{e_{i'}} $ is invariant by the relations $R_{i',i'+1}(q^{2})( \otimes ^N v_{e_i} )$ ($ l_j +1 \leq i' \leq l_{j+1}-1, \;  l_{j+1} +1 \leq i' \leq l_{j+2}-1$), if we use these relations we get that the vector  $\otimes ^N v_{e_i} $ ($e_i <e_{i+1} $ if $\l _{min{(i)}} = \l _{min{(i+1)}}$) is expressed by the sum of $\otimes ^N v_{e_{i'}} $ such that $\tilde{N} (\otimes ^N v_{e_{i'}} ) < \tilde{N} (\otimes ^N v_{e_i} )$ and $e_{i'} < e_{i'+1} $ (if $\l _{min{(i')}} \neq \l _{min{(i'+1)}}$).

By the induction on $\tilde{N}$ we find  that the vectors $\otimes ^N v_{e_i}$ ($e_i <e_{i+1} $ if $\l _{min{(i)}} = \l _{min{(i+1)}}$, $e_i \geq e_{i+1} $ if $\l _{min{(i)}} \neq \l _{min{(i+1)}}$) span the space $\otimes ^N V / V^{\l }$.

The number of these vectors is equal to the number of semi--standard tableaux  of $\theta $. By Corollary \ref{dimbs}, we get $\otimes ^N V / V^{\l } \subset \otimes ^N V / Ker \bar{R}_{\theta }$.
\end{pf}
\vspace{.2in}

In what follows we  identify the  border strips $\langle m_1, \dots , m_r \rangle $ and $\langle m_1, \dots , m_r ,n, \dots ,n \rangle $, and identify $(\l_1 , \dots ,\l _{s+nk} )$ and $(\overbrace{\l_1 +1, \dots ,\l_1 +1}^{n} ,\l_1 , \dots ,\l _{s+nk})$ which are elements of $\coprod _k \tilde{{\cal M} }^{n,k}_{s+nk}$. Proposition \ref{lteq} gives a one to one correspondence of  $\coprod _k \tilde{{\cal M} }^{n,k}_{s+nk}$ and the set of all skew Young diagrams of the border strip type  $\langle m_1, \dots , m_r \rangle $  which satisfy $m_i \leq n $ for all $i$ and $\sum_{i=1}^{r} m_i$ $\equiv$ $s$ mod $n$.
On this correspondence the degree of the semi--infinite wedge $ \wedge( \tilde{\Phi }_{min}^{\l}(z)  \otimes \psi ) | M-s-nk \rangle$ is equal to $\frac{1-n}{24} - \frac{k(n-k)}{2n} + \frac{1}{2n} | \theta |(n- | \theta |)+t(\theta ) $, where $\theta $ is the  border strip which corresponds to $\l $ and $t(\theta ) = \sum_{i=1}^{r-1} (r-i) m_i$.

We define $ch (F_M / H_- ' F_M)= \sum _{\mu ,i } dim(V_{\mu ,i}) e^{\mu } q^i$, where $V_{\mu ,i}$ is the subspace of $F_M / H_- 'F_M$ of the degree (\ref{degifw}) $i$ and of the  $U_q( {{\frak s} {\frak l}}_n )$--weight $\mu $.
We put $ \sum _{\mu ,i } a_{\mu ,i} e^{\mu } q^i \leq \sum _{\mu ,i } b_{\mu ,i} e^{\mu } q^i$ iff $a_{\mu ,i} \leq b_{\mu ,i}$ for all $\mu $ and $i$.
By Proposition \ref{Psurj} we have
\begin{equation}
ch (F_M / H_- ' F_M) \leq q^{\frac{1-n}{24}- \frac{k(n-k)}{2n}} \sum_{\theta \in BS \atop {| \theta | \equiv k \; mod \; n}} q ^{\frac{1}{2n} | \theta | (n- | \theta |)+t(\theta )} s_\theta (z).
\label{chFM}
\end{equation}
$F_M / H_- ' F_M$ with $U_1$--action is isomorphic to $V(\Lambda _k) $, this isomorphism is degree preserving with respect to the degree (\ref{degifw}) on $F_M / H_- ' F_M$ and the homogeneous degree on $V(\Lambda _k) $, and the character formula of $V(\Lambda _k)$ is given in Proposition \ref{chara}. Hence the inequality of (\ref{chFM}) is, in fact, an  equality and, therefore,  the map (\ref{psik}) must is  bijective. Thus have the following theorem
\begin{thm}
We have the isomorphism of  $U'_q (\widehat{ \frak{s}\frak{l}}_n )$--modules:
\begin{equation}
F_M / H_- ' F_M \simeq \bigoplus _{\theta } V_{\theta }, \label{deco}
\end{equation}
where the sum is over all border strips $\langle m_1, \dots m_r \rangle $, $(m_i \leq n $, $m_r < n$ and $N\equiv M $ mod $n)$, the space $V_{\theta }$ and the level--$0$ $U'_q (\widehat{ \frak{s}\frak{l}}_n )$--action is defined by $( \pi ^{(N)} _{a_1, \dots , a_{N}} ,\; R_{\theta } \cdot \otimes ^{N} V )$ where $N= \sum_{i=1}^{r} m_i$ and $a_{l+\sum_{i=j}^{r} m_i} = 2(l-1+\sum_{i=1}^{j-1} m_i)$.

\end{thm}

\subsection{$ \frak{s}\frak{l}_2 $ case}

In this section we will discuss the $ \frak{s}\frak{l}_2 $ case in a somewhat more detail.

Let $W_n$ be the $(n+1)$--dimensional irreducible module of $U_q( \frak{s}\frak{l}_2 )$, and $W_n(b)$ be the evaluation module with the parameter $b$ whose $U_q'(\widehat{ \frak{s}\frak{l}}_{2}) $--module structure is given by
\begin{equation}
E_0 = q^b F_1, \; \; \; F_0 = q^{-b}E_1, \; \; \; K_{0}= K_{1}^{-1}.
\end{equation}

It is known that every finite dimensional irreducible $U_q'(\widehat{ \frak{s}\frak{l}}_{2}) $--module is isomorphic to $\otimes _{\mu } W_{n_{\mu }} (b_{\mu})$ for some $n_{\mu }$ and $ b_{\mu}$. We will represent the $U_q'(\widehat{ \frak{s}\frak{l}}_{2}) $--module described by a skew Young diagram as the tensor product of the form  $\otimes _{\mu } W_{n_{\mu }} (b_{\mu})$.

\begin{prop} \label{sltP}
Let $\theta $ be the skew Young diagram of border strip $\langle m_1, \dots, m_r \rangle $ such that $m_i = 1$ or $2$, and $N= \sum_{i=1}^{r} m_i$, $a_{l+\sum_{i=j}^{r} m_i} = 2(l-1+\sum_{i=1}^{j-1} m_i)$.
We put $I= \{ i \: | \: m_i =1 $ and $ m_{i-1} =2\} = \{ l_1 , \; l_2 , \dots , l_{r'} \}$ $(m_0=2, l_i< l_{i+1})$ and let $n_i$ be the integer such that $m_{l_i} = \dots = m_{l_i +n_i -1} =1, \; m_{l_i +n_i} =2$ and $b_i= 2l_i+n_i -3$.

The $U_q'(\widehat{ \frak{s}\frak{l}}_{2}) $--module $(  \pi ^{(N)} _{a_1, \dots , a_{N}} , \: R_\theta \cdot \otimes^N {\Bbb C} ^2  )$ is isomorphic to $W_{n_1}(b_1) \otimes W_{n_2}(b_2) \otimes \dots \otimes W_{n_{r'}}(b_{r'})$.
\end{prop}

\begin{pf}
By Proposition \ref{skewrep}, we find that  $( \pi ^{(N)} _{a_1, \dots , a_{N}},\: R_\theta  \cdot \otimes^N {\Bbb C} ^2 )$ is isomorphic to $( \pi ^{(N)} _{a_N, \dots , a_1} , \: \otimes^N {\Bbb C} ^2 / $Ker$\bar{R}_{\theta })$.

As in  the proof of Proposition \ref{lteq}, we get Im$\check{R}_{i,i+1} (q^{2}) \subset $Ker$\bar{R}_{\theta }$ if $a_{i+1} = a_i -2$ and Im$ \check{R}_{i,i+1} (q^{-2}) \subset $Ker$\bar{R}_{\theta }$ if $a_{i+1} = a_i +2$.

We can directly confirm that the $U_q'(\widehat{ \frak{s}\frak{l}}_{2}) $--module $( \pi _{a,a-2}, \: {\Bbb C} ^2 \otimes {\Bbb C} ^2 / $Im$ \check{R}_{1,2} (q^{2}) )$ is $1$--dimensional and the module $( \pi _{a,a+2,\dots ,a+2(l-1)}, \: \otimes ^l {\Bbb C} ^2 / \sum_{i=1}^{l-1} $Im$ \check{R}_{i,i+1} (q^{-2}))$ is isomorphic to $W_{l}(a+l-1)$, where $(l)$ is the Young diagram of degree $l$ which has only one row.

If we put
\begin{equation}
\tilde{V} = \sum _{ i | a_{i+1} = a_i +2 } Im \check{R}_{i,i+1} (q^{2})
+ \sum _{ i | a_{i+1} = a_i -2  } Im \check{R}_{i,i+1} (q^{-2}),
\end{equation}
then  $(\pi ^{(N)} _{a_N, \dots , a_1}, \: \otimes^N {\Bbb C} ^2 / \tilde{V}) \simeq W_{n_1}(b_1) \otimes W_{n_2}(b_2) \otimes \dots \otimes W_{n_{r'}}(b_{r'})$.

Since the dimension of $W_{n_1}(b_1) \otimes W_{n_2}(b_2) \otimes \dots \otimes W_{n_{r'}}(b_{r'})$ is equal to the dimension $\otimes^N {\Bbb C} ^2 / $Ker$\bar{R}_{\theta }$ the proof is finished.
\end{pf}
\vspace{.2in}

By Proposition \ref{sltP}, we can rewrite the decomposition (\ref{deco}) for $ \frak{s}\frak{l}_2 $ case. In fact we get the same decomposition as \cite{J}. More precisely we get
\begin{prop}
If we change the coproduct of the level--0 $U_q'(\widehat{ \frak{s}\frak{l}}_{2}) $--module defined in \cite{J} to fit our coproduct (\ref{cp1}--\ref{cp3}), The level--0 $U_q'(\widehat{ \frak{s}\frak{l}}_{2}) $--module of $V(\Lambda _s)$ $(s \equiv M $ mod $2, \; s=0$ or $1)$ defined in \cite{J} is isomorphic to the level--0 $U_q'(\widehat{ \frak{s}\frak{l}}_{2}) $--module of $V(\Lambda _s)$ defined in this paper, and the degree is preserved under this isomorphism.
\end{prop}

{\bf Acknowledgment}
I am grateful to D. Uglov with whom we collaborated on the papers \cite{TU,STU}. I also thank M. Jimbo, M. Kashiwara, T. Miwa, M. Nazarov, Y. Saito, V. Tarasov and colleagues at  R.I.M.S. for discussions and support.

redpajama_set_name: RedPajamaArXiv


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Cayman Islands Economy Stats
With no direct taxation, the islands are a thriving offshore financial center. More than 93,000 companies were registered in the Cayman Islands as of 2008, including almost 300 banks, 800 insurers, and 10,000 mutual funds. A stock exchange was opened in 1997. Tourism is also a mainstay, accounting for about 70% of GDP and 75% of foreign currency earnings. The tourist industry is aimed at the luxury market and caters mainly to visitors from North America. Total tourist arrivals exceeded 1.9 million in 2008, with about half from the US. About 90% of the islands' food and consumer goods must be imported. The Caymanians enjoy one of the highest outputs per capita and one of the highest standards of living in the world.
Exports per capita: This entry provides the total US dollar amount of merchandise exports on an f.o.b. (free on board) basis. These figures are calculated on an exchange rate basis, i.e., not in purchasing power parity (PPP) terms. Figures expressed per capita for the same year.
Fiscal year: The beginning and ending months for a country's accounting period of 12 months, which often is the calendar year but which may begin in any month. All yearly references are for the calendar year (CY) unless indicated as a noncalendar fiscal year (FY).
GDP > Purchasing power parity: This entry gives the gross domestic product (GDP) or value of all final goods and services produced within a nation in a given year. A nation's GDP at purchasing power parity (PPP) exchange rates is the sum value of all goods and services produced in the country valued at prices prevailing in the United States. This is the measure most economists prefer when looking at per-capita welfare and when comparing living conditions or use of resources across countries. The measure is difficult to compute, as a US dollar value has to be assigned to all goods and services in the country regardless of whether these goods and services have a direct equivalent in the United States (for example, the value of an ox-cart or non-US military equipment); as a result, PPP estimates for some countries are based on a small and sometimes different set of goods and services. In addition, many countries do not formally participate in the World Bank's PPP project that calculates these measures, so the resulting GDP estimates for these countries may lack precision. For many developing countries, PPP-based GDP measures are multiples of the official exchange rate (OER) measure. The difference between the OER- and PPP-denominated GDP values for most of the weathly industrialized countries are generally much smaller.
Tourist arrivals > Per capita: International inbound tourists (overnight visitors) are the number of tourists who travel to a country other than that in which they have their usual residence, but outside their usual environment, for a period not exceeding 12 months and whose main purpose in visiting is other than an activity remunerated from within the country visited. When data on number of tourists are not available, the number of visitors, which includes tourists, same-day visitors, cruise passengers, and crew members, is shown instead. Sources and collection methods for arrivals differ across countries. In some cases data are from border statistics (police, immigration, and the like) and supplemented by border surveys. In other cases data are from tourism accommodation establishments. For some countries number of arrivals is limited to arrivals by air and for others to arrivals staying in hotels. Some countries include arrivals of nationals residing abroad while others do not. Caution should thus be used in comparing arrivals across countries. The data on inbound tourists refer to the number of arrivals, not to the number of people traveling. Thus a person who makes several trips to a country during a given period is counted each time as a new arrival." Per capita figures expressed per 1,000 population.
Select year 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013
Budget > Revenues $680.10 million 2013 173th out of 223
Budget surplus > + or deficit > - -2.7% of GDP 2012 87th out of 182
Exports $15.00 million 2012 184th out of 189
Exports per capita $260.55 2012 147th out of 189
Fiscal year 1 2013
GDP > Composition, by sector of origin > Services 72.4% 2012 38th out of 189
GDP > Per capita $47,010.15 per capita 2008 3rd out of 9
GDP > Per capita > PPP $43,800.00 2004 2nd out of 7
GDP > Purchasing power parity $2.25 billion 2008 177th out of 203
GDP > Purchasing power parity per capita $42,523.44 2008 3rd out of 7
Inflation rate > Consumer prices 1.2% 2012 182nd out of 199
Tourist arrivals > Per capita 6,330.7 per 1,000 people 2008 4th out of 144
Unemployment rate 4% 2008 18th out of 26
SOURCES: CIA World Factbooks 18 December 2003 to 28 March 2011; CIA World Factbooks 2010, 2011, 2012, 2013; CIA World Factbooks 2010, 2011, 2012, 2013. Population figures from World Bank: (1) United Nations Population Division. World Population Prospects, (2) United Nations Statistical Division. Population and Vital Statistics Report (various years), (3) Census reports and other statistical publications from national statistical offices, (4) Eurostat: Demographic Statistics, (5) Secretariat of the Pacific Community: Statistics and Demography Programme, and (6) U.S. Census Bureau: International Database.; All CIA World Factbooks 18 December 2003 to 18 December 2008; World Bank national accounts data, and OECD National Accounts data files.; CIA World Factbook 2010, 2011, 2012, 2013; CIA World Factbooks 18 December 2003 to 28 March 2011. Population figures from World Bank: (1) United Nations Population Division. World Population Prospects, (2) United Nations Statistical Division. Population and Vital Statistics Report (various years), (3) Census reports and other statistical publications from national statistical offices, (4) Eurostat: Demographic Statistics, (5) Secretariat of the Pacific Community: Statistics and Demography Programme, and (6) U.S. Census Bureau: International Database.; World Bank national accounts data, and OECD National Accounts data files. Population figures from World Bank: (1) United Nations Population Division. World Population Prospects, (2) United Nations Statistical Division. Population and Vital Statistics Report (various years), (3) Census reports and other statistical publications from national statistical offices, (4) Eurostat: Demographic Statistics, (5) Secretariat of the Pacific Community: Statistics and Demography Programme, and (6) U.S. Census Bureau: International Database.; World Tourism Organisation, Yearbook of Tourism Statistics, Compendium of Tourism Statistics and data files.
Three of the top ten countries for GDP per capita are island nations: Bermuda, Cayman Islands, and Iceland.
Cayman Islands Economy Profiles (Subcategories)
Budget 6 Market capitalization of listed companies 4
Companies 8 National accounts 4
Currency 6 Natural gas 8
Debt 4 Official development assistance and official aid 4
Electricity 8 Oil 10
Financial sector 7 Savings 9
Foreign aid 39 Tax 12
GDP 22 Tourism 7
Government 6 Tourism receipts 4
International tourism 5 Tourist arrivals by region of origin 8
Labor force 3 Trade 350
Cayman Islands categories
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Crime Government People
Culture Health Religion
Disasters Industry Sports
Education Language
France is the top destination in the world for tourists, accounting for 11 percent of all tourist arrivals worldwide.

redpajama_set_name: RedPajamaCommonCrawl


text:
Lord of the Dance | Waterford News and Star
Waterford News
Well! >
Ronan Morrissey of the Waterford News & Star with 98-year-old Jim McManus, who spoke about why he loves dancing and his time serving in World War Two.
TONIGHT, World War Two veteran Jim McManus will don on his dancing shoes and travel to Treacy's Hotel to tango the night away. Tomorrow he will spin to the Brandon House for another date with a dance hall while he's booked in to strut his stuff in Wellington Bridge and the Rhu Glen on Friday and Saturday night respectively. 99 next month, he's showing no signs of slowing down.
Jim's gentle Glasgow accent fails to betray his upbringing in the town of Paisley but he grew up conscious of his Irishness. He attended Catholic school where most if not all his classmates were of similar heritage and the McManus clan would visit the local Hibernian Society where they could engage in several facets of Irish culture. With a Fermanaghman as his father and a mother from Sligo, neither a Tango stronghold, he would leave Scotland in the summer to visit family and be roped into helping on his uncle's crop farm.
Any geographical distance a Waterford native would have with him is rendered obsolete when he describes his own childhood – he would walk out towards the countryside like kids would venture out to Skibbereen to pick blackberries and if his Mam was too busy doing the washing he would have to take his younger sisters for a stroll in their buggy despite only being seven or eight.
In his teenage years, Jim joined the Territorial Army, or the reserves, although the primary reason for this was to enjoy a two week camp they went on each summer which he saw as a free holiday. Armed with youthful naivety, he was unperturbed at being mobilised two months before World War Two began and can recall tensions with the Germans heightening gradually from there.
"On September 1, 1939, I was on guard duty on the banks of the Clyde at a fuel dump which was a prime target for somebody to hit, when a mate of mine came up and said 'I've just heard on the wireless, we're at war with Germany'. I said 'oh, fantastic' and threw my rifle up in the air!"
"We got whisked right down to the south of England to a village on the seafront just west of Brighton. All the locals had been moved away and the Government had taken over their row of bungalows. We were barracked in there and had to do guard duties along the coast because an invasion was expected from Mr. Hitler. If they had we'd have had it. All we had was rifles. We wouldn't have had a chance had Hitler invaded straight away but he made a big blunder."
The German planes would fly so low over the beach on reconnaissance missions that their facial features could be seen by the soldiers and he was later sent to France as part of the Northumberland Fusiliers after the Argyle and Sutherland Highlanders were split up.
It was after this move that he was sometimes exposed to the grim realities of war, something he still laments today.
Jim, who today lives in Slieverue, states: "What a bloody waste. All of those men, I had some good friends killed. I seen a bloke killed, he had his head almost blown off, he was sliced. It was a German mortar – they had about 8 barrels coming over us at once and if they hit television wires, which they did in this case, then it exploded immediately and shrapnel would go everywhere."
"Things happen so quickly that you get into this routine. Firstly you just jump to the ground and search for a bit of cover somewhere. Let's be honest, you get scared then. There's something wrong with you if you don't."
Following the war, his training as a wireless operator came in handy and he had his fees paid for a radio officers course which ended up taking him all over the world, exposing him to how other people in the world lived.
"First of all I did a stint for a few months on the weather ships. Two of them used to leave the port of Greenock and they would spend two weeks at sea before coming back to port. They had weather observers on board, radar which was new at the time. Balloons would be sent up that would observe the wind speed – there had these little transmitters which sent back data to the shop. We'd send them back to London and it would be used for the weather forecast."
Following this, he worked on board a passenger ship to East Africa, Mombasa and a Royal Navy crude oil tanker which sailed to Jan Mayen Island in the Arctic where he was met by piercingly cold 100kmph gusts on deck.
Later in life, Jim would holiday along the south and west coasts of Ireland and following retirement, he fancied a move back to the land of his roots. Waterford, which he had passed through many times, was the preference.
On January 10, 1994, he left Plymouth with his partner Patricia Lusby and having landed in Rosslare, the couple remained in Wexford for a year – although it didn't offer enough to two nimble movers with no intention of slowing down.
His interest in all styles of jiving was borne from visits to Burton's Tailors in Paisley as a 16-year-old where youngsters would try wear in their newly purchased stylish suits on the dancefloors which rested over the store.
"There's more dancing in Waterford," Jim remembers saying, "why don't we move down there?"
In 1995 they moved to Ferrybank which made it easier to make up for the seven years of dancing he lost to the War. Tragically, Patricia died of cancer in 1998.
"That was a big blow. If she was alive we'd still be into the tango, we'd be much better at it too. It was very tough to continue to dance after that. She had a little dog and we would walk him every night. The dog would be sniffing around the hedges and we would be in the middle of the street practicing the cha-cha or samba underneath the lights."
Thankfully, he never allowed the grief to corrupt his passion and he is as active a dancer today as he ever was, with a trip to the home of the Tango, Buenos Aires, hopefully on the horizon next year.
"You get to meet people, it's very social. I didn't think about it when I was younger because I was active anyway but it's important to do it now that I'm older because if not you'd sink into a seat and it's a spiral to oblivion.
"I love the music. It gives you a buzz and when you're moving right in time with it… it's hard to describe it. It's a feeling of happiness."
In conversation with Ronan Morrissey
Follow @WaterfordNS
By rmorrissey
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More Well!
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In this week's printed edition
Déise camogie players feel disrespected by fixture change
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The Rams travelled to top of the division Tamworth Phoenix, to face a side that were unbeaten and had only conceded 18 points in their previous 3 games.
The 'Nix had the first drive of the game and were perfect on their execution as with only 4 plays they moved the ball 40+ yards and scored the first touchdown of the day. On the backfoot from the off, the Rams struggled to get anything going on offence and the teams traded punts for a few drives before Tamworth went on another long drive which was finished off with a long run through a lot of missed tackles by the Rams to give the home side a two score advantage.
The Rams started to move the ball a little better on the next drive but once again needed to punt. This time though the Phoenix would fumble the ball and the Rams would get another drive almost straight away. Peter Yianni stepping in at RB for the game (and would win Offensive MVP for his performance) moved the ball successfully on a couple of plays but the Rams would once again have to punt. Tamworth would block this one though and, with good field position, must have thought they would extend their lead even further but they would turnover the ball for the second drive running as James Hudson came up with the interception.
Yianni once again made some yards but as the second quarter began, the Rams were forced to punt again and a 'Nix punt followed soon after. The Rams started deep in their own half and again resorting to punting, the Rams benefited from a roughing penalty on the home side and would get another set of downs. They couldn't take advantage though and on third down a high snap was recovered by QB Mark Gaze with Tamworth defenders seeing a chance for an extra TD.
This time the punt was returned to just outside the endzone by the Phoenix returner and on the next play, they would finish the job with a pass to a Receiver sat on the floor. Nothing much would happen in the rest of the half, except for another punt from the Rams that was muffed by the returner that gave the Rams a long-awaited chance in the Phoenix half. The ball was picked off though but Tamworth couldn't take advantage.
After half time the Phoenix pressure continued and they scored a passing touchdown into the corner of the endzone after a long run from a pass earlier in the drive. The lead would grow steadily throughout the second half as turnovers started to creep into the Rams' play. An interception got the 'Nix to the one yard line and it was run in for another TD.
Straight after that the ball was fumbled on a run play and the Tamworth defence ran the ball straight in and the game was effectively over as it petered with only a Safety added on for the home side as QB Dan Berry was adjudged (debatably) to have intentionally grounded the ball in the endzone.
The Rams will have plenty of work to do as they go on the road to face the Lancashire Wolverines on 14th June, having been defeated by that very team 50-0 at home.


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Brighter Futures, Praxis
We want to be at the forefront of new approaches to funding - working in partnership with other funders and organisations to increase our reach and make more of a difference.
We have worked with a range of partners to develop new funds that focus on particular parts of the country as well as in specific interest areas or to tackle specific issues. You can find a few examples of these below. We also list our funding partnerships in our annual report.
Live Work Fund
The Live Work Fund is a partnership between Jerwood Arts, Wolfson Foundation, Esmée Fairbairn Foundation and The Linbury Trust to support early-career artists, creatives and producers to enable them to adapt and thrive through the challenges of COVID-19.
Act for Change Fund
The Act for Change Fund is a £3.6 million partnership for organisations supporting young people working for change. It is a joint initiative between Paul Hamlyn Foundation and Esmée, in partnership with the National Lottery Community Fund.
Nesta Arts & Culture Impact Fund
Managed by Nesta, the Arts & Culture Impact Fund (and its predecessor, the Arts Impact Fund), brings together public, private and charitable funding to demonstrate the potential and impact of social investment in the arts.
Museums Association Collections Fund
The Collections Fund provides funding to a range of projects that demonstrate the significance, distinctiveness and power of collections to people.
Environment Agency, Defra and Esmée Fairbairn Foundation collaboration
A collaborative pilot with the Environment Agency and Defra to support environmental projects in creating funding models which could be replicated by other projects up and down the country.
LocalMotion is a joint initiative by six funders to support communities in a more radical, joined-up way.
Creative Civic Change
Creative Civic Change is an experimental funding programme designed to fund and test ways in which culture and creativity can support community change. It was set up as a funding partnership between Local Trust, National Lottery Community Fund, Calouste Gulbenkian Foundation and Esmée Fairbairn Foundation.


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Appellate Procedure / Zoning Validity Challenges
Appeal of Chester County Outdoor, LLC, 64 A.3d 1148 (Pa. Commw. Ct. 2013).
Appeal of Chester Cty. Outdoor, LLC, 167 A.3d 280 (Pa. Commw. Ct. 2017), aff'd. Appeal of Chester Cty. Outdoor, LLC, 2018 Pa. LEXIS 545.
Where a proposal is not appended to a validity challenge of a zoning ordinance, the one-year time limitation to file for a building permit under Section 10916(g) of the Pennsylvania Municipalities Planning Code (MPC) is not applicable. Further, a trial court does not have jurisdiction to grant site specific relief if the developer did not first file an appropriate application with the local municipality. Finally, while a developer must first submit its request for site specific relief to a municipality, the trial court is the ultimate decision maker and has broad discretion as to its review of the record and additional evidence.
Case Details:
Developer successfully challenged Township's zoning ordinance in a claim that billboards were unlawfully excluded. Developer did not append a proposal or plan to the application for the validity challenge. Developer did not seek a building permit within one year of winning its validity challenge. During this process, the Township approved its own curative amendment. Three years after winning its validity challenge, Developer sought site-specific relief and the Township's Zoning Hearing Board (ZHB) appointed a special master and denied the requested relief. Deferring to the ZHB's findings of fact and conclusions of law, the trial court denied Developer's appeal from the ZHB without reviewing the evidence de novo. In addition, the trial court held Developer's permit application was untimely. Developer appealed to the Commonwealth Court.
The Commonwealth Court held Developer was not untimely in filing its request for a building permit. Section 916(g) of the MPC states, in part: "[w]here the proposal appended to the . . . validity challenge is approved but does not require further application under any subdivision or land development ordinance, the developer shall have one year within which to file for a building permit." Since Developer did not append a proposal to its validity challenge, Section 916(g) was inapplicable. Section 916(g) applies only where a proposal is appended to a validity challenge.
In addition, the Court held Developer's arguments regarding jurisdiction to determine whether site-specific relief is warranted failed. Per the Court, trial courts are not authorized to approve site-specific relief unless the relief originally was sought from the municipality. Indeed, a trial court lacks jurisdiction to "review plans or applications that had never been submitted to the appropriate authority for consideration." Accordingly, Developer could not have sought site-specific relief directly from the trial court.
Finally, when a trial court has jurisdiction to review a request for site-specific relief, the trial court is required to "conduct a de novo review of the evidence." After conducting its de novo review, the trial court is permitted to accept a zoning hearing board's findings as its own. It also may hold a hearing and take additional evidence. Following its review, the trial court must grant site-specific relief, unless the municipality proves the challenger's (Developer) proposal is incompatible with "the materiality of certain unchallenged, pre-existing, and generally applicable provisions of the Ordinance." In addition, where a zoning ordinance has been amended after a validity challenge was filed, the trial court may not apply the new adopted provisions. Further, if the trial court determines the proposed site-specific relief is improper, it must "consider alternative sites and/or alternative configurations for the proposed [use] and fashion some form of site-specific relief." In Developer's case, the trial court failed to comply with all of the procedures and requirements above. Accordingly, the Commonwealth Court remanded the case for the trial court to properly review the matter.
External Link to Full Case
Date of Decision: 7/28/17
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Pennypacker v. Ferguson Township, Pa. Commw. (2017)
Davis-Haas v. Exeter Township Zoning Hearing Board, Pa. Commw. (2017)


I've been labelled a *nix fanboy! I win!
@jsnover nothing irks me more than *nix fanboys slamming CMD. It feels like wilfull ignorance of #PowerShell. I love *nix, but not bash!
A brief summary of my daily computing environment: I use an android phone currently, iOS iPads, Osx on laptops, win8 on a desktop machine. I develop software components that target most of the above, as well as various flavors of linux for appliances and cloud hosted environments. On a less frequent basis I use a surface RT that I bought at launch. Each os has its pros and cons, each machine has its merits.
I have no emotional attachment to any os because deep down they aren't all that differentiated. And I can't afford the luxury of irrational attachment — I need to target users wherever they are, and I need to use tools that are the most efficient. My own usage, and that of most users, is inherently spread across many vendors and many oses, and I don't see that changing. Multiplatform and interoperability is just a fact of life.
So when I said wanted ssh, I don't want something better/different than ssh. I want exactly ssh because I need to connect to a variety of environments and ssh is the only reasonable common denominator. Powershell remoting may be super awesome and wonderful, but it doesn't solve any problem I have, and just introduces a new non-interoperable tool and protocol with its attendant complexity. Not at all helpful — doesn't save me time, just creates complexity. I am not saying that Powershell remoting is bad, it just isn't useful in an inherently multiplatform environment. If your environment is Windows only, then by all means embrace Powershell if that works for you.
And when I say I want easy copy/paste, I don't want copy/paste with some completely different keyboard and mouse interface. Ctrl-C/Ctrl-V and Cmd-C/Cmd-V have become second nature to me, I have a lot of muscle memory built up around those. Giving me copy/paste using some other keyboard interface or mouse interface is not very helpful (nor is command window text selection that works on blocks instead of lines). I don't need or value innovation in the copy/paste interface, it doesn't make me more productive.
So for those reasons, for me, OSX is substantially better than Windows as a development tool today. Which frustrates me because these are stupid little reasons, and there is no reason why Windows couldn't add these features, it would only make Windows more valuable, and would make my life better. And frustrates me because I still have some affinity for Microsoft, and these seem like such petty features over which to alienate developers. If I was in charge, I'd make sure Windows was excellent at modern cross-platform devops, as well as having Windows-centric features. I can't see the negative in this.
« Why is OSX terminal so much better than Windows command prompts?


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Awaiting the translation of my poem ' Write to me the words' by you in Tamil. It's part of the deal, don't forget. Have an awesome Tuesday !