April 10, 2025 — Padova

A transport factorization primer

Davide Dal Martello, University of Padova

Abstract

We introduce a higher Teichmüller machinery, hereafter the “transport factorization (TF)”, as a tool for representation theory. Given an input surface \(\mathbb{S}\), such machinery indeed delivers a coordinatization of the representation space \(\mathrm{Hom}(\pi^1(\mathbb{S}),\mathrm{SL}_n)\). More importantly, TF admits a natural noncommutative version delivering “hands-on” representations of certain higher Hecke algebras. These properties will be gently described via the case study of the four-punctured sphere, stressing its remarkable connection with Painlevé \(VI\) and the DAHA of type \(C^\vee C_1\).