Multiple polylogarithm values at roots of unity

TL;DR

This paper investigates the linear relations among multiple polylogarithm values at roots of unity, revealing limitations of standard relations and identifying new relations using symmetry and motivic methods.

Contribution

It demonstrates that standard relations are incomplete for certain levels and weights, and introduces new relations derived from symmetry and motivic fundamental group techniques.

Findings

Standard relations do not account for all relations at certain levels and weights.

Octahedral symmetry helps find missing relations at level 4, weight 3 or 4.

Motivic fundamental group methods provide results for prime power levels.

Abstract

For any positive integer $N$ let $\mu_N$ be the group of the $N$th roots of unity. In this note we shall study the $\Q$-linear relations among values of multiple polylogarithms evaluated at $\mmu_N$. We show that the standard relations considered by Racinet do not provide all the possible relations in the following cases: (i) level N=4, weight $w=3$ or 4, and (ii) $w=2$, $7<N<50$, and $N$ is a power of 2 or 3, or $N$ has at least two prime factors. We further find some (presumably all) of the missing relations in (i) by using the octahedral symmetry of $\P^1-(\{0,\infty\}\cup \mu_4)$. We also prove some other results when $N=p$ or $N=p^2$ ($p$ prime $\ge 5$) by using the motivic fundamental group of $\P^1-(\{0,\infty\}\cup\mu_N)$.