On tropical friezes associated with Dynkin diagrams

TL;DR

This paper explores tropical friezes in the context of Dynkin diagrams using triangulated categories, establishing a bijection with integer tuples and characterizing their structure, while also proving a related conjecture for cluster-additive functions.

Contribution

It characterizes tropical friezes on cluster categories of Dynkin quivers and proves a conjecture of Ringel for cluster-additive functions.

Findings

Tropical friezes correspond bijectively to integer n-tuples in Dynkin cases.

Any tropical frieze has a specific form related to cluster-tilting objects.

Proof of Ringel's conjecture for cluster-additive functions.

Abstract

Tropical friezes are the tropical analogues of Coxeter-Conway frieze patterns. In this note, we study them using triangulated categories. A tropical frieze on a 2-Calabi-Yau triangulated category $\mathcal{C}$ is a function satisfying a certain addition formula. We show that when $\mathcal{C}$ is the cluster category of a Dynkin quiver, the tropical friezes on ${\mathcal{C}}$ are in bijection with the $n$-tuples in ${\mathbb{Z}}^n$, any tropical frieze $f$ on $\mathcal{C}$ is of a special form, and there exists a cluster-tilting object such that $f$ simultaneously takes non-negative values or non-positive values on all its indecomposable direct summands. Using similar techniques, we give a proof of a conjecture of Ringel for cluster-additive functions on stable translation quivers.