Multiple polylogarithms at non-positive indices and combinatorics of Magnus polynomials
Kohei Kitamura
TL;DR
The paper develops a combinatorial-algebraic framework linking non-positive multiple polylogarithms to Magnus polynomials in a free associative algebra. It derives a Magnus-type representation for products of mono-indexed non-positive MPLs and a binomial-structured expansion that relates these functions to Magnus polynomials via a symbol map. It further investigates how permuting indices yields linear functionals in the kernel of the Li^- map, yielding functional equations and clarifying the underlying algebraic structure. The work connects these non-positive MPL identities to Magnus expansion theory and Na23 dualities, and outlines directions for understanding the full kernel and additional relations (KKN).
Abstract
In this paper we investigate multiple polylogarithms with non-positive multi-indices (nonpositive MPLs) from a combinatorial and algebraic viewpoint. By introducing a correspondence between non-positive multiple polylogarithms and Magnus polynomials in a free associative algebra, we obtain an explicit Magnus-type representation of products of mono-indexed non-positive MPLs. The main identity (Theorem A) expresses such a product as a single non-positive MPL indexed by a Magnus polynomial, which may be regarded as a Möbius inversion of the expansion formula due to Duchamp-Hoang Ngoc Minh-Ngo. Moreover, we study the effects of permuted indices and show that certain differences of Magnus polynomials belong to the kernel of the linear map ${\rm Li}^-_{\bullet}$ , leading to new functional equations among non-positive MPLs of the same weight and depth. These results clarify the combinatorial structure underlying non-positive MPLs and reveal a close connection with the Magnus expansion in non-commutative algebra.