Horizontal and vertical mutation fans

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).