Generalized exchange operators for a system of spin-1 particles

The irreps $(SU(2),\mathcal{H},U)$ of SU(2) of dimension $(2S+1{)}^N$, i.e. operators acting on the space $\mathcal{H}={\mathcal{H}}_N={\mathbf{C}}^{(2S+1{)}^N}$ of $N$ identical particles with spin $S$, are described by Clebsch-Gordan decomposition into inequivalent irreps. In the special case $S=1\mathrm{/}2$, Dirac \cite{Dir1} discovered that there is another rep given by $(\mathcal{S}(N),\mathcal{H},V)$ where $\mathcal{S}(N)$ 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 $\vec σ_i=2\vec S_i$ is the vector of Pauli matrices, can be interpreted as the sum of the Exchange Operators’’ $P_{ij}$ between particles $i$ and $j$. Schrödinger \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 ${\tilde{Q}}_{\alpha }$, 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 $S=1\mathrm{/}2$ and $S=1$.