Dihedral symmetries of multiple logarithms

TL;DR

This paper explores the symmetries of multiple logarithms under dihedral group actions, generalizing combinatorial methods to establish new relations applicable to iterated integrals over any field.

Contribution

It introduces a generalized combinatorial framework for multiple logarithms with dihedral symmetries, extending existing polygon representations to broader algebraic contexts.

Findings

Identifies dihedral symmetry relations in multiple logarithms

Extends combinatorial methods to iterated integrals over arbitrary fields

Provides a unified approach to symmetries in multiple logarithm functions

Abstract

This paper finds relationships between multiple logarithms with a dihedral group action on the arguments. I generalize the combinatorics developed in Gangl, Goncharov and Levin's R-deco polygon representation of multiple logarithms to find these relations. By writing multiple logarithms as iterated integrals, my arguments are valid for iterated integrals as over an arbitrary field.