On the partial trace over collective spin-degrees of freedom

We derive analytical properties for the degeneracy $\nu(N,j)$ occurring in the decomposition $\bigoplus\limits_{j}^\frac{N}{2}\nu(N,j)\mathbb C^{2j+1}$ of the state space $\mathbb C^{2\otimes N}$. We also investigate the dynamics of two qubits coupled via Ising interactions to separate spin baths, and we study the thermodynamic limit.