Multiple polylogarithms and the Steinberg module

TL;DR

The paper establishes a deep link between weight-$n$ and depth-$d$ multiple polylogarithms on a torus and the Steinberg module, showing that all such polylogarithms can be expressed in terms of the single ladder function $\mathrm{Li}_{n-d+1,1,\dots,1}$. It builds a comprehensive Hopf-algebra and VB-module framework, introduces the truncated symbol map $\mathrm{ST}$, and proves a precise isomorphism $\mathrm{gr}_d^{\mathcal{D}}\mathbb{L}_n(\mathrm{T}_d) \cong \mathrm{St}(V)\otimes\mathrm{St}(V)\otimes\mathrm{S}^{n-d}V$, with explicit realization on generators. The work also provides a simple proof of the Bykovskiĭ theorem, connects polylogarithms to Milnor $K$-theory, and discusses dualities between polylogarithms and iterated integrals via a Koszul-dual Steinberg framework, yielding interpretations of Rognes and CFP conjectures. These results illuminate depth-reduction phenomena, deepen the motivic understanding of polylogarithms on tori, and offer new algebraic tools for studying polylogarithmic relations and their arithmetic implications.

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 Bykovskiĭ theorem, explain the duality between multiple polylogarithms and iterated integrals, and provide a polylogarithmic interpretation of the conjectures of Rognes and Church-Farb-Putman.

Theorems & Definitions (150)

• Theorem 1
• Theorem 2
• Conjecture 3
• Lemma 4
• proof
• Proposition 5
• proof
• Corollary 6
• Theorem 7
• proof