the constraints $$\label{eq:mconstr}
0\leq m_1\leq m_2\leq\ldots\leq m_k\;,\qquad\quad m_{k+1}>m_{k+2}>\ldots>m_n\geq 0 \;,$$ the former condition being empty if $k=0$, and the latter one being empty if $k=n$. For every $0\leq k\leq n$, a representation is defined as follows (all the representations are on the same Hilbert space). Firstly, for $0\leq i<k\leq n$, denote by $\vec{e}_{ik}\in\{0,1\}^n$ the array $$\vec{e}_{ik}:= 
(\,\,\stackrel{i\;{\mathrm}{times}}{\overbrace{0,0,\ldots,0}}\,,
\stackrel{k-i\;{\mathrm}{times}}{\overbrace{1,1,\ldots,1}}\,,
\stackrel{n-k\;{\mathrm}{times}}{\overbrace{0,0,\ldots,0}}) \;.$$ Then, we set $\pi^{(n)}_k(x_i)=\pi^{(n)}_k(\zeta_i)=0$ if $i>k$, while the remaining generators are given by: $$\begin{aligned}
\pi^{(n)}_k(x_i){\left|\smash[t]{\vec{m}}\right>} &=
               q^{2\sum_{j=0}^{i-1}r_j}q^{2\sum_{j=1}^ip_{j-1}(m_j-m_{j-1})}
               \big(1-q^{2r_i}q^{2p_i(m_{i+1}-m_i)}\big){\left|\smash[t]{\vec{m}}\right>} \, , \quad \textup{for} \;0\leq i<k, \\[5pt]
\pi^{(n)}_k(x_k){\left|\smash[t]{\vec{m}}\right>} &=
               q^{2\sum_{j=0}^{k-1}r_j}q^{2\sum_{j=1}^kp_{j-1}(m_j-m_{j-1})}
               {\left|\smash[t]{\vec{m}}\right>} \;, \\
\pi^{(n)}_k(\zeta_i){\left|\smash[t]{\vec{m}}\right>} &=
               q^{p_i\sum_{j=0}^{i-1}r_j}q^{\sum_{j=1}^ip_{j-1}(m_j-m_{j-1})}
               \sqrt{ {\genfrac{ \{ }{ \} }{0pt}{}{ p_i(m_{i+1}-m_i+1) }{ p_i(m_{i+1}-m_i) }_{\!q}} }{\left|\smash[t]{\vec{m}+\vec{e}_{ik}}\right>} \;, \\
                & \hspace{11cm} \textup{for} \;0\leq i<k, \\[-5pt]
\pi^{(n)}_k(\zeta_k){\left|\smash[t]{\vec{m}}\right>} &=
               q^{p_k\sum_{j=0}^{k-1}r_j}q^{\sum_{j=1}^kp_{j-1}(m_j-m_{j-1})}
               {\left|\smash[t]{\vec{m}}\right>} \;,\end{aligned}$$

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(with $m_0:=0$) on the subspace ${\mathcal}{V}^n_k$ and they are zero on the orthogonal subspace.

The representation $\pi^{(n)}_k$, when restricted to ${\mathcal}{V}^n_k$, is the direct sum of several copies of the irreducible representation for ${\mathcal{O}}(L_q(p_0\ldots p_k;\ell_0,\ldots,\ell_k))$ given in Proposition \[prop:A\], and pulled back to ${\mathcal{O}}(L_q(p;\ell_0,\ldots,\ell_n))$: one copy of the representation for each value of the additional labels $m_{k+1},\ldots,m_n$.

Lemma 2 of [@DL10] still holds (and we do not repeat the proof here):

\[le:pl\] The spaces $\mathcal{V}^n_k$ are such that $\mathcal{V}^n_j\perp\mathcal{V}^n_k$ if $|j-k|>1$, 