Multiple polylogarithm values at roots of unity

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 $\P^1-(\{0,\infty\}\cup\mu_N)$.