On the approximate periodicity of sequences attached to noncrystallographic root systems

TL;DR

This paper explores the behavior of sequences related to noncrystallographic root systems, demonstrating approximate periodicity and classifying mutation classes for types H3 and H4, advancing understanding in cluster algebra structures.

Contribution

It introduces approximately periodic sequences for noncrystallographic root systems and characterizes matrix mutation classes for types H3 and H4.

Findings

Constructed approximately periodic sequences for rank 2 noncrystallographic root systems.

Described matrix mutation classes for types H3 and H4.

Extended cluster algebra mutation theory to noncrystallographic cases.

Abstract

We study Fomin-Zelevinsky's mutation rule in the context of noncrystallographic root systems. In particular, we construct approximately periodic sequences of real numbers for the noncrystallographic root systems of rank 2 by adjusting the exchange relation for cluster algebras. Moreover, we describe matrix mutation classes for type H3 and H4.