parameters.

Assume now that the hypothesis of Theorem \[thm:13\], are satisfied, that is let ${\underline}{\ell}={\underline}{p}^{\,\sharp}$ be a weight vector, with ${\underline}{p}$ pairwise coprime, and let $p:=p_0p_1\ldots p_n$. On the generators of ${\mathcal{O}}(L_q(p;{\underline}{\ell}))$ the representation above gives: $$\begin{aligned}
x_i{\left|\smash[t]{\vec{k}}\right>} &=q^{2(k_1+\ldots+k_i)}\big(1-q^{2k_{i+1}}\big){\left|\smash[t]{\vec{k}}\right>} \;, & \textup{for} \;\; 0\leq i<n, \\
x_n{\left|\smash[t]{\vec{k}}\right>} &=q^{2(k_1+\ldots+k_n)}{\left|\smash[t]{\vec{k}}\right>} \;, \\
\zeta_i{\left|\smash[t]{\vec{k}}\right>} &=q^{p_i(k_1+\ldots+k_i)}\sqrt{ {\genfrac{ \{ }{ \} }{0pt}{1}{k_{i+1}+p_i}{k_{i+1}}_{\!q}} }{\left|\smash[t]{\vec{k}+p_i\vec{e}_{i+1}}\right>} \;, & \textup{for} \;\; 0\leq i<n, \\
\zeta_n{\left|\smash[t]{\vec{k}}\right>} &=q^{p_n(k_1+\ldots+k_n)}{\left|\smash[t]{\vec{k}}\right>} \;,\end{aligned}$$ where, for $0\leq k<m$, $${\genfrac{ \{ }{ \} }{0pt}{}{m}{k}_{\!q}} := (1 - q^{2k+2})(1 - q^{2k+4})\ldots (1 - q^{2m})$$ is just a shorthand notation for the $q$-shifted factorial $(q^{2k+2};q^2)_{m-k-1}$.

This representation breaks into irreducible components for ${\mathcal{O}}(L_q(p;{\underline}{\ell}))$. To see this, we relabel the basis vectors as follows. For all $i=1,\ldots,n$ let $$k_i=p_{i-1}(m_i-m_{i-1})+r_{i-1}\, ,$$ where $\vec{m}=(m_1,\ldots,m_n)\in{\mathbb{N}}^n$ satisfies $0\leq m_1\leq m_2\leq\ldots\leq m_n$, we set $m_0:=0$, and $r_i\in\{0,\ldots,p_i-1\}$ are the remainders. The inverse transformation is then $$\label{eq:inverse}
m_i=\sum_{j=1}^i\frac{k_j-r_{j-1}}{p_{j-1}} \;.$$ Basis vectors will be renamed accordingly ${\left|\smash[t]{\vec{m};{\boldsymbol}{r}}\right>}$ and the representation breaks into the irreducible sub-representations of ${\mathcal{O}}(L_q(p;{\underline}{\ell}))$ given in the next proposition.

\[prop:A\] Fix a vector ${\boldsymbol}{r}=(r_0,\ldots,r_{n-1}) \in{\mathbb{N}}^n$ with constraints on the components $$\begin{gathered}
\quad 0 \leq r_i<p_i\;, \quad \textup{for} \,  \, \, i=0,\ldots,n-1 \, .\end{gathered}$$ Let ${\mathcal{H}}_{{\boldsymbol}{r}}$ be the Hilbert space with orthonormal basis ${\left|\smash[t]{\vec{m};{\boldsymbol}{r}}\right>}$ and 