A Symmetrized Basis for Transitions in the Heisenberg Model

The spin-S Heisenberg model has 2S+1 states on each site, for which there are (2S+1)$^{2}$ possible transitions between these states. For N sites there are (2S+1)$^{N}$ states and (2S+1)$^{2N}$ transitions between states. This rapid increase in the number of transitions with sites appears to limit calculations to just a few sites. However for transitions induced by spin-spin interactions, we construct a symmetrized basis which only grows as 2$^{N-3}$, making possible computations for much larger systems.