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

We consider orthogonal representations $\eta_n:S_n \curvearrowright $\mathbb{R}$^N$ofthesymmetrygroups$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.