From polygons and symbols to polylogarithmic functions
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.
Cite
@article{arxiv.1110.0458,
title = {From polygons and symbols to polylogarithmic functions},
author = {Claude Duhr and Herbert Gangl and John R. Rhodes},
journal= {arXiv preprint arXiv:1110.0458},
year = {2015}
}
Comments
75 pages. Mathematica files with the expression of all HPLs up to weight 4 in terms of the spanning set are included