Symmetries of weight 6 multiple polylogarithms and Goncharov's Depth Conjecture

TL;DR

This paper proves symmetries of certain weight 6, depth 3 multiple polylogarithms, confirming a case of Goncharov's Depth Conjecture and advancing understanding of polylogarithm symmetries.

Contribution

It establishes the higher Zagier symmetry part of the weight 6, depth 3, reduction conjecture, confirming Goncharov's Depth Conjecture for this case.

Findings

Proves 6-fold anharmonic symmetries of specific weight 6 polylogarithms.

Confirms Goncharov's Depth Conjecture for weight 6, depth 3.

Complements prior work to fully establish the conjecture in this case.

Abstract

We prove that the weight 6, depth 3, multiple polylogarithm $ \mathrm{Li}_{4,1,1}((xyz)^{-1}, x, y) $, or rather its more natural `divergent' incarnation $ \mathrm{Li}_{3;1,1,1}(x,y,z) $, satisfies the 6-fold anharmonic symmetries of the dilogarithm $ \mathrm{Li}_2 $, $ \lambda \mapsto 1-\lambda $ and $ \lambda \mapsto \lambda^{-1} $, in each of $x$, $y$ and $z$ independently, modulo terms of depth $ \leq2 $. This establishes the `higher Zagier' part of the weight 6, depth 3, reduction conjectured by Matveiakin and Rudenko. Together with their proof of the `higher Gangl' part of the weight 6, depth 3, reduction (which is formulated modulo the `higher Zagier' part), we establish Goncharov's Depth Conjecture in the case of weight 6, depth 3.