scholarly journals The Existence of a Maximal Green Sequence is not Invariant under Quiver Mutation

10.37236/5412 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Greg Muller
Keyword(s):  
Maximal Green Sequence ◽  
Quiver Mutation

This note provides a quiver which does not admit a maximal green sequence, but which is mutation-equivalent to a quiver which does admit a maximal green sequence. The proof uses the `scattering diagrams' of Gross-Hacking-Keel-Kontsevich to show that a maximal green sequence for a quiver determines a maximal green sequence for any induced subquiver.

scholarly journals Quiver Mutation Sequences and $q$-Binomial Identities

2017 ◽  
Vol 2018 (23) ◽  
pp. 7335-7358
Author(s):  
Akishi Kato ◽  
Yuma Mizuno ◽  
Yuji Terashima
Keyword(s):  
Quiver Mutation ◽  
Binomial Identities

scholarly journals Coloured quiver mutation for higher cluster categories

2009 ◽  
Vol 222 (3) ◽  
pp. 971-995 ◽  
Author(s):  
Aslak Bakke Buan ◽  
Hugh Thomas
Keyword(s):  
Quiver Mutation ◽  
Cluster Categories

scholarly journals Quiver Mutation Loops and Partition q-Series

2014 ◽  
Vol 336 (2) ◽  
pp. 811-830 ◽  
Author(s):  
Akishi Kato ◽  
Yuji Terashima
Keyword(s):  
Quiver Mutation

scholarly journals Braid groups and quiver mutation

2017 ◽  
Vol 290 (1) ◽  
pp. 77-116 ◽  
Author(s):  
Joseph Grant ◽  
Robert Marsh
Keyword(s):  
Braid Groups ◽  
Quiver Mutation

scholarly journals Acyclic Calabi–Yau categories

2008 ◽  
Vol 144 (5) ◽  
pp. 1332-1348 ◽  
Author(s):  
Bernhard Keller ◽  
Idun Reiten
Keyword(s):  
Alternative Proof ◽  
Combinatorial Proof ◽  
Closed Field ◽  
Cluster Category ◽  
Similar Characterization ◽  
Endomorphism Algebras ◽  
Orbit Category ◽  
Quiver Mutation ◽  
Cluster Tilting ◽  
Stable Categories

AbstractWe prove a structure theorem for triangulated Calabi–Yau categories: an algebraic 2-Calabi–Yau triangulated category over an algebraically closed field is a cluster category if and only if it contains a cluster-tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable category of maximal Cohen–Macaulay modules over a certain isolated singularity of dimension 3 is a cluster category. This implies the classification of the rigid Cohen–Macaulay modules first obtained by Iyama and Yoshino. As an application to the combinatorics of quiver mutation, we prove the non-acyclicity of the quivers of endomorphism algebras of cluster-tilting objects in the stable categories of representation-infinite preprojective algebras. No direct combinatorial proof is known as yet. In the appendix, Michel Van den Bergh gives an alternative proof of the main theorem by appealing to the universal property of the triangulated orbit category.

scholarly journals ORIENTED FLIP GRAPHS, NONCROSSING TREE PARTITIONS, AND REPRESENTATION THEORY OF TILING ALGEBRAS

2019 ◽  
Vol 62 (1) ◽  
pp. 147-182 ◽  
Author(s):  
ALEXANDER GARVER ◽  
THOMAS MCCONVILLE
Keyword(s):  
Representation Theory ◽  
Derived Categories ◽  
Module Category ◽  
Indecomposable Modules ◽  
Quiver Mutation ◽  
Extension Spaces ◽  
Dynkin Quiver ◽  
Torsion Pairs ◽  
Gentle Algebras

AbstractThe purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

scholarly journals Maximal green sequences for arbitrary triangulations of marked surfaces (Extended Abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Matthew R. Mills
Keyword(s):  
Boundary Component ◽  
Maximal Green Sequence ◽  
Higher Genus ◽  
International Audience ◽  
Marked Surface

International audience In general, the existence of a maximal green sequence is not mutation invariant. In this paper we show that it is in fact mutation invariant for cluster quivers associated to most marked surfaces. We develop a procedure to find maximal green sequences for cluster quivers associated to an arbitrary triangulation of closed higher genus marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with at least one boundary component has a maximal green sequence.

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