From polygons and symbols to polylogarithmic functions

TL;DR

This paper reviews the symbol map for simplifying multiple polylogarithms, introduces a combinatorial method for deriving symbols from polygons, and discusses ambiguities in reconstructing functions from symbols.

Contribution

It provides a systematic approach to obtaining symbols from multiple polylogarithms and explores ambiguities in reconstructing functions, advancing the mathematical understanding of polylogarithmic functions.

Findings

Presented a recipe for obtaining symbols from decorated polygons

Illustrated the construction of functions from symbols for harmonic polylogarithms up to weight four

Identified non-trivial elements in the kernel of the symbol map at arbitrary weight

Abstract

We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illustrate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of non-trivial elements in the kernel of the symbol map for arbitrary weight.