Horizontal and vertical mutation fans

2018/05/13 by Kiyoshi Igusa, Igusa, Kiyoshi
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1805.04799

openalex publication_date 2018/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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).

Citations

• A category of noncrossing partitions
• Remarks on lengths of maximal green sequences for quivers of type A_n,1
• The No Gap Conjecture for tame hereditary algebras
• Stability conditions, \τ-tilting Theory and Maximal Green Sequences
• Maximal green sequences for cluster-tilted algebras of finite representation type

Related

• Mutation-classes of diagrams via infinite graphs
• Ptolemy diagrams and torsion pairs in m-cluster categories of type D
• Mutation of friezes
• A geometric realization of the m-cluster categories of type D_n
• Mutation of torsion pairs in triangulated categories and its geometric realization