Multiplicative Inequalities In Cluster Algebras Of Finite Type

TL;DR

This paper characterizes the cone of bounded Laurent monomials in cluster variables of finite type, showing the extremal rays are the u-variables and exploring factorization properties in various Grassmannians.

Contribution

It describes the cone of bounded Laurent monomials in finite type cluster algebras and identifies the extremal rays as u-variables, with applications to Grassmannians.

Findings

Extreme rays of the cone are the u-variables.

All bounded ratios are bounded by 1.

Factorization into primitive ratios holds in certain Grassmannians.

Abstract

Generalizing the notion of a multiplicative inequality among minors of a totally positive matrix, we describe, over full rank cluster algebras of finite type, the cone of Laurent monomials in cluster variables that are bounded as a real-valued function on the positive locus of the cluster variety. We prove that the extreme rays of this cone are the u-variables of the cluster algebra. Using this description, we prove that all bounded ratios are bounded by 1 and give a sufficient condition for all such ratios to be subtraction free. This allows us to show in Gr(2, n), Gr(3, 6), Gr(3, 7), Gr(3, 8) that every bounded Laurent monomial in Pl\"ucker coordinates factors into a positive integer combination of so-called primitive ratios. In Gr(4, 8) this factorization does not exists, but we provide the full list of extreme rays of the cone of bounded Laurent monomials in Pl\"ucker coordinates.