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

TL;DR

This paper establishes duality and reflection formulas for complex and $ ext{l}$-adic multiple polylogarithms, using algebraic relations from associators and symmetry of the projective line minus three points.

Contribution

It introduces new duality and reflection formulas for multiple polylogarithms and provides algebraic proofs based on associator relations and symmetry considerations.

Findings

Derived duality and reflection formulas for multiple polylogarithms

Provided algebraic proofs using associator relations

Linked formulas to symmetry of the projective line minus three points

Abstract

In the present paper, we derive formulas of complex and $\ell$-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.