may significantly reduce the dimension of the problem. Rotational and spin-ordering symmetries are present in all the three spin models. As a typical example, let us consider a hexagonal lattice ($p=6$) and its particular base configuration of spins on the lattice polygon $\{\sigma^{*}\} = \{{\uparrow}{\downarrow}{\uparrow}{\uparrow}{\downarrow}{\downarrow}\}$. Rotational symmetry requires that the tensor elements corresponding to the set of configurations $\{{\downarrow}{\uparrow}{\downarrow}{\uparrow}{\uparrow}{\downarrow}\}$, $\{{\downarrow}{\downarrow}{\uparrow}{\downarrow}{\uparrow}{\uparrow}\}$, $\{{\uparrow}{\downarrow}{\downarrow}{\uparrow}{\downarrow}{\uparrow}\}$, $\{{\uparrow}{\uparrow}{\downarrow}{\downarrow}{\uparrow}{\downarrow}\}$, $\{{\downarrow}{\uparrow}{\uparrow}{\downarrow}{\downarrow}{\uparrow}\}$ are identical to $W_{p=6}(\{\sigma^{*}\})$. Next, let us consider a spin-ordering operation, which reverses the order of the polygon spins. In particular, if the spins are labelled clockwise, the operation reorders them in the anti-clockwise direction. It means that the configuration $\{{\uparrow}{\downarrow}{\uparrow}{\uparrow}{\downarrow}{\downarrow}\}$ is equivalent to $\{{\downarrow}{\downarrow}{\uparrow}{\uparrow}{\downarrow}{\uparrow}\}$ by the spin-ordering symmetry and to all the rotations of the latter configuration ($\{{\uparrow}{\downarrow}{\downarrow}{\uparrow}{\uparrow}{\downarrow}\}$, $\{{\downarrow}{\uparrow}{\downarrow}{\downarrow}{\uparrow}{\uparrow}\}$, $\{{\uparrow}{\downarrow}{\uparrow}{\downarrow}{\downarrow}{\uparrow}\}$, $\{{\uparrow}{\uparrow}{\downarrow}{\uparrow}{\downarrow}{\downarrow}\}$, $\{{\downarrow}{\uparrow}{\uparrow}{\downarrow}{\uparrow}{\downarrow}\}$) by a composition of the spin-ordering and the rotational symmetry. As a result, the 12 tensor elements $W_6(\{\sigma\})$ corresponding to the configuration $\{\sigma^{*}\}$ and its 11 equivalent configurations are represented by a single variational parameter, as they share the same value.

By performing a similar analysis on the set of all $2^p$ configurations $\{\sigma\}$ we can factorize it into $N_{\rm Ising}^{(p)}$ classes of equivalence with representatives $\theta_j$, where $j \in \{1, \dots, N_{\rm Ising}^{(p)}\}$ 