Redundancy in string cone inequalities and multiplicities in potential functions on cluster varieties

TL;DR

This paper investigates the inequalities defining string cones in cluster varieties using potential functions, proposing criteria for minimal inequality sets and conjecturing their equivalence.

Contribution

It introduces a criterion for potential functions to yield minimal inequalities and conjectures an equivalence related to string cone inequalities in cluster varieties.

Findings

Proposed a necessary criterion for potential functions to produce minimal inequalities.

Conjectured an equivalence between potential functions and string cone inequalities.

Analyzed inequalities in the context of reduced double Bruhat cells.

Abstract

We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential function to provide a minimal set of inequalities via tropicalization and conjecture an equivalence.