Reddening sequences and mutation of infinite quivers

TL;DR

This paper develops a combinatorial framework for mutation of infinite quivers, extending the concept of reddening sequences from finite to infinite cases, and unifies topological and combinatorial approaches in cluster algebra theory.

Contribution

It introduces a new combinatorial approach to infinite quiver mutation, generalizing existing surface-based results and extending reddening sequences to infinite settings.

Findings

Defined mutation for infinite quivers as limits of finite quivers

Extended reddening sequences to infinite quivers

Unified topological and combinatorial perspectives in cluster algebras

Abstract

Cluster algebras, introduced by Fomin and Zelevinsky through the process of quiver mutation, have become central objects in modern algebra and geometry, linking combinatorial constructions with diverse mathematical domains such as Teichmuller theory, total positivity, and even theoretical physics. Building on foundational work by Fomin, Shapiro, and Thurston connecting cluster algebras to triangulated surfaces, recent research has extended mutation theory to infinite settings, including the infinity-gon and more general marked surfaces. In this paper, we develop a purely combinatorial framework for mutation of infinite quivers, independent of but compatible with these topological constructions. By formalizing infinite quivers as limits of embedded finite quivers, we establish a consistent definition of mutation that generalizes prior surface-based results. We then apply this framework to extend the notion of reddening sequences, special mutation sequences with significant algebraic consequences, from the finite to the infinite setting. Our approach not only unifies previous topological and combinatorial perspectives but also provides a technical foundation for further generalizations of cluster algebra theory in the infinite case.