Symmetries of double ratios and an equation for Möbius structures

Abstract:

Orthogonal representations $\eta _n\colon S_n\curvearrowright \mathbb {R}^N$ of the symmetric groups $S_n$, $n\ge 4$, with $N=n!/8$, emerging from symmetries of double ratios are treated. For $n=5$, the representation $\eta _5$ is decomposed into irreducible components and it is shown that a certain component yields a solution of the equations that describe the Möbius structures in the class of sub-Möbius structures. In this sense, a condition determining the Möbius structures is implicit already in symmetries of double ratios.