the paper, we restrict ourselves to symmetric collective states, i.e., those invariant under permutation of the internal state of any two particles. For spin-$\frac12$ particles, this corresponds to the maximal total angular momentum manifold, and our representation coincides with the standard approach.

In the general case, the symmetric subspace is spanned by the occupation number states $$\begin{gathered}
  \label{eq:N-body-basis}
  {|{n_0,m_1,l_2,\ldots}\rangle} \equiv
  \frac1{\sqrt{n!\, m!\, l! \ldots}}
  \sum_{\text{perm}}
  {|{\phi_0}\rangle}_1 \cdots {|{\phi_0}\rangle}_n
  \\ \times
  {|{\phi_1}\rangle}_{n+1} \cdots {|{\phi_1}\rangle}_{n+m}
  \; {|{\phi_2}\rangle}_{n+m+1} \cdots,\end{gathered}$$ which means that $n$ atoms are in the internal state ${|{\phi_0}\rangle}$, $m$ in ${|{\phi_1}\rangle}$, $l$ in ${|{\phi_2}\rangle}$, etc., and they sum up to $N = n + m + l + \ldots$ atoms. It is easy to see that the collective operators $$\begin{gathered}
  \label{eq:Sigma}
  \hat \Sigma_{\alpha\beta} \equiv \sum_{j=1}^N
  {|{\phi_\alpha}\rangle}_j {}_j{\langle{\phi_\beta}|},
  \qquad
  (\alpha,\beta = 0,1,\ldots,d-1)\end{gathered}$$ keep the symmetry of the sub-Hilbert space in consideration. Furthermore, we observe the following properties, $$\begin{aligned}
  \hat \Sigma_{\alpha\beta} {|{n_\alpha, m_\beta, \ldots}\rangle}
  &= \sqrt{(n+1)m}
  \nonumber\\&\quad\times
  {|{(n+1)_\alpha, (m-1)_\beta, \ldots}\rangle},
  \label{eq:Sigma-step}
  \\
  \hat \Sigma_{\alpha\alpha} {|{n_\alpha, m_\beta, \ldots}\rangle}
  &= n \; {|{n_\alpha, m_\beta, \ldots}\rangle},
  \label{eq:Sigma-number}\end{aligned}$$ and the commutator identity, $$\begin{aligned}
  \big[ \hat\Sigma_{\alpha\gamma}, \hat\Sigma_{\gamma\beta} \big]
  &= \hat \Sigma_{\alpha\beta}
  - \delta_{\alpha\beta} \hat \Sigma_{\gamma\gamma}.
  \label{eq:Sigma-commute}\end{aligned}$$ All other commutators, which can not be brought on the form $\big[ 