Red sizes of quivers

TL;DR

This paper generalizes the concept of red numbers for quivers in cluster algebras, providing a complete classification for finite mutation types and proposing conjectures for broader cases.

Contribution

It introduces unrestricted red numbers, extends existing notions, and classifies these for all finite mutation type quivers.

Findings

Complete determination of red and unrestricted red numbers for finite mutation type quivers.

Introduction of the concept of unrestricted red numbers.

Proposals of conjectures on red number values in general.

Abstract

In this article, we will expand on the notions of maximal green and reddening sequences for quivers associated to cluster algebras. The existence of these sequences has been studied for a variety of applications related to Fomin and Zelevinsky's cluster algebras. Ahmad and Li considered a numerical measure of how close a quiver is to admitting a maximal green sequence called a red number. In this paper we generalized this notion to what we call unrestricted red numbers which are related to reddening sequences. In addition to establishing this more general framework we completely determine the red numbers and unrestricted red numbers for all finite mutation type quivers. Furthermore, we give conjectures on the possible values of red numbers and unrestricted red numbers in general.