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).

23 pages, 10 figures, presented at Workshop on Cluster Algebras and Related Topics held at the Chern Institute of Mathematics, Nankai University, China, July 10-13, 2017, accepted for publication in SCIENCE CHINA Mathematics, v2: minor corrections

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