On webs, polylogarithms and cluster algebras

TL;DR

This paper explores webs linked to cluster algebras and polylogarithms, introducing a new class of webs with maximal rank and demonstrating their connection to classical polylogarithmic functional equations.

Contribution

It introduces the concept of AMP webs, shows their relevance to cluster algebras and polylogarithms, and connects classical polylogarithmic equations to cluster structures.

Findings

Many classical polylogarithmic functional equations are of cluster type.

The paper defines and studies AMP webs with maximal rank.

Results and conjectures about cluster webs related to Dynkin diagrams.

Abstract

In this text, we investigate webs which can be associated to cluster algebras from the point of view of the abelian functional equations these webs carry, focusing on the polylogarithmic ones. We introduce a general notion of webs whose rank is `As Maximal as Possible' (AMP) and show that many webs associated either to polylogarithmic functional equations or to cluster algebras are of this type. In particular, we prove a few results and state some conjectures about cluster webs associated to (pairs of) Dynkin diagrams. Along the way, we show that many of the classical functional equations satisfied by low-order polylogarithms (such as Spence-Kummer's equation of the trilogarithm or the tetralogarithmic one of Kummer) are of cluster type.