Polyhedral parametrizations of canonical bases & cluster duality

TL;DR

This paper connects potential and decoration functions on double Bruhat cells to canonical bases, providing explicit polyhedral parametrizations and linking tropicalizations with classical parametrizations.

Contribution

It establishes a relation between potential and decoration functions and explicitly identifies polyhedral parametrizations with classical string and Lusztig parametrizations.

Findings

Explicit identification of polyhedral parametrizations with classical bases

Connection between tropicalizations and classical parametrizations

Unified framework for potential and decoration functions

Abstract

We establish the relation of the potential function constructed by Gross-Hacking-Keel-Kontsevich's and Berenstein-Kazhdan's decoration function on the open double Bruhat cell in the base affine space $G/\mathcal{N}$ of a simple, simply connected, simply laced algebraic group $G$. As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on $G/\mathcal{N}$ arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations.