Permutation symmetry of the scattering equations

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-3) quantities σ_α which satisfy a set of coupled scattering equations, with momentum dot products as input coefficients. These equations are known to have (n-3)! solutions; hence, each σ_α is believed to satisfy a single polynomial equation of degree (n-3)!. In this article, we derive the transformation properties of σ_α 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 σ_α lead to symmetry relations for the (n-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.