Multiple polylogarithms and the Steinberg module

TL;DR

This paper links multiple polylogarithms on a torus to the Steinberg module, providing new insights into their structure, duality, and related conjectures, with implications for understanding their algebraic and geometric properties.

Contribution

It establishes a novel connection between multiple polylogarithms and the Steinberg module, offering a unified expression and new proofs of existing theorems.

Findings

Expressed multiple polylogarithms via a single function $ ext{Li}_{n-d+1,1,\

Provided a simple proof of the Bykovski theorem.

Offered a polylogarithmic interpretation of conjectures by Rognes and Church-Farb-Putman.

Abstract

We establish a connection between multiple polylogarithms on a torus and the Steinberg module of $\mathbb{Q}$, and show that multiple polylogarithms of depth $d$ and weight $n$ can be expressed via a single function $\mathrm{Li}_{n-d+1,1,\dots,1}(x_1,x_2,\dots,x_d)$. Using this connection, we give a simple proof of the Bykovski\u{\i} theorem, explain the duality between multiple polylogarithms and iterated integrals, and provide a polylogarithmic interpretation of the conjectures of Rognes and Church-Farb-Putman.