Horizontal and vertical mutation fans

TL;DR

This paper introduces horizontal and vertical mutation fans for m-cluster categories, visual tools that generalize known diagrams for standard cluster categories to better understand mutations and c-vectors.

Contribution

It develops new diagrammatic tools called mutation fans for m-cluster categories, extending the concept of mutation fans from standard cluster theory.

Findings

Mutation fans visualize mutations in m-cluster categories.

They provide a framework for understanding c-vectors and maximal green sequences.

The diagrams generalize known structures for m=1 to higher m.

Abstract

We introduce diagrams for $m$-cluster categories which we call "horizontal" and "vertical" mutation fans. These are analogous to the mutation fans (also known as "semi-invariant pictures" or "scattering diagrams") for the standard ($m=1$) cluster case which are dual to the poset of finitely generated torsion classes. The purpose of these diagrams is to visualize mutations and analogues of maximal green sequences in the $m$-cluster category with special emphasis on the $c$-vectors (the "brick" labels).