Symmetries of cross-ratios and the equation for M\"obius structures

We consider orthogonal representations $\eta_n:S_n \curvearrowright \mathbb{R}^N$ of the symmetry groups $S_n$, $n\ge 4$, with $N=n!/8$ motivated by symmetries of cross-ratios. For $n=5$ we find the decomposition of $\eta_5$ into irreducible components and show that one of the components gives the solution to the equations, which describe M\"obius structures in the class of sub-M\"obius structures. In this sense, the condition defining M\"obius structures is hidden already in symmetries of cross-ratios.