Generalized exchange operators for a system of spin-1 particles

The irreps $(SU(2),{\cal H},U)$ of SU(2) of dimension $(2S+1)^N$, i.e. operators acting on the space ${\cal H}={\cal H}_N={\bf C}^{(2S+1)^N}$ of $N$ identical particles with spin $S$, are described by Clebsch-Gordan decomposition into inequivalent irreps. In the special case $S=1/2$, Dirac \cite{Dir1} discovered that there is another rep given by $({\cal S}(N),{\cal H},V)$ where ${\cal S}(N)$ is the permutation group, Thus, the standard ``linear'' Hamiltonian, or Heisenberg interaction Hamiltonian $H_0=\sum_{1\leq i\leq N}\vec S_i\cdot\vec S_j$, where $\vec \sigma_i=2\vec S_i$ is the vector of Pauli matrices, can be interpreted as the sum of the ``Exchange Operators'' $P_{ij}$ between particles $i$ and $j$. Schr\"odinger \cite{Sch} generalized to higher spin numbers $S$ the Exchange Operator $P_{ij}=P_S(\vec S_i\cdot \vec S_j)$ as a polynomial of degree $2S$ in $\vec S_i\cdot \vec S_j$. This we call the $P$-representation. There is another rep induced by the one particle permutation of states operators $\widetilde Q_\alpha$, which we call the $Q$-rep. Our main purpose is to write some physical Hamiltonians for a few particles in the $P$- or $Q$-rep and compute their spectrum. The simplest case where there are as many particles as available states for the spin operator along the $z$-axis, i.e. $N=2S+1=3$, see Weyl \cite{Wey} or Hamermesh \cite{Ham}. Finally, we consider the relationship between permutations and rotation invariance when $S=1/2$ and $S=1$.