Symmetries of many-body systems imply distance-dependent potentials

Considering interatomic potential $U(\mathbf{q})$ where $\mathbf{q}=[{\mathbf{q}}_1,{\mathbf{q}}_2,\dots ,{\mathbf{q}}_N]\in {\mathbb{R}}^{3N}$ is a vector describing positions, ${\mathbf{q}}_i\in {\mathbb{R}}^3$, it is shown that $U$ can be defined as a function of the interatomic distance variables $r_{ij}=\mathrm{\mid }{\mathbf{q}}_i-{\mathbf{q}}_j\mathrm{\mid }$, provided that the potential $U$ satisfies some symmetry assumptions. Moreover, the potential $U$ can be defined as a function of a proper subset of the distance variables $r_{ij}$, provided that $N>5$, with the number of distance variables used scaling linearly with the number of atoms, $N$.