On some relations between generalized associators

Let Phi be the KZ associator and Psi_N be its analogue for N-th roots of 1. We prove a hexagon relation for Psi_4. Similarly to the Broadhurst (for Psi_2) and Okuda (for Psi_4) duality relations, it relies on the "supplementary" (i.e., non-dihedral) symmetries of C^* - mu_4(C) (i.e., the octahedron group S_4). We also derive relations between Phi and Psi_2, which are analogues of equations, found by Nakamura and Schneps, satisfied by the image of the Galois group of Q in the Grothendieck-Teichm"uller group.