Permutation symmetry of the scattering equations

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Closed formulas for tree amplitudes of $n$-particle scatterings of gluon, graviton, and massless scalar particles have been proposed by Cachazo, He, and Yuan. They depend on $(n\ensuremath{-}3)$ quantities ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ which satisfy a set of coupled scattering equations, with momentum dot products as input coefficients. These equations are known to have $(n\ensuremath{-}3)!$ solutions; hence, each ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ is believed to satisfy a single polynomial equation of degree $(n\ensuremath{-}3)!$. In this article, we derive the transformation properties of ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ under momentum permutation and verify them with known solutions at low $n$, and with exact solutions at any $n$ for special momentum configurations. For momentum configurations not invariant under a certain momentum permutation, new solutions can be obtained for the permuted configuration from these symmetry relations. These symmetry relations for ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ lead to symmetry relations for the $(n\ensuremath{-}3)!+1$ coefficients of the single-variable polynomials, whose correctness are checked with the known cases at low $n$. The extent to which the coefficient symmetry relations can determine the coefficients is discussed.