Duality-reflection formulas of multiple polylogarithms and their $\mathrm{\ell }$-adic Galois analogues

In the present paper, we derive formulas of complex and $\mathrm{\ell }$-adic multiple polylogarithms, which have two aspects: a duality in terms of indexes and a reflection in terms of variables. We provide an algebraic proof of these formulas by using algebraic relations between associators arising from the $S_3$-symmetry of the projective line minus three points.