Symmetries of cross-ratios and the equation for Möbius structures

Abstract

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 Möbius structures in the class of sub-Möbius structures. In this sense, the condition defining Möbius structures is hidden already in symmetries of cross-ratios.

arXiv comment13 pages