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

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 thehigher 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.