Noncrossing partitions and representations of quivers

Abstract

We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading's bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated to a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver $Q$ without oriented cycles are in natural bijection with the cluster-tilting objects in the associated cluster category. We also show these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

Fixed typos, added an example, minor improvements to the exposition

References in corpus (2)

• Cambrian fans
• Sortable elements and Cambrian lattices

Cited by in corpus (39)

• $\tau$-tilting finite algebras, bricks and $g$-vectors
• Reduction of $\tau$-tilting modules and torsion pairs
• Brick polytopes of spherical subword complexes and generalized associahedra
• Torsion classes, wide subcategories and localisations
• Lattice theory of torsion classes: Beyond $\tau$-tilting theory
• Lattice structure of torsion classes for path algebras
• Noncrossing partitions and the shard intersection order
• Introduction to $\tau$-tilting theory
• Rowmotion in slow motion
• C-sortable words as green mutation sequences
• The Jordan-Hölder property and Grothendieck monoids of exact categories
• Cambrian combinatorics on quiver representations (type A)
• Celebrating Loday's Associahedron
• Rigid modules and ICE-closed subcategories in quiver representations
• Linearity of stability conditions
• Quotient closed subcategories of quiver representations
• ICE-closed subcategories and wide $\tau$-tilting modules
• Monobrick, a uniform approach to torsion-free classes and wide subcategories
• Derived categories of representations of small categories over commutative noetherian rings
• An algebraic approach to Harder-Narasimhan filtrations
• Schur's lemma for exact categories implies abelian
• Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras
• $\tau$-perpendicular wide subcategories
• Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank
• Stability and Indecomposability of the Representations of Quivers of $A_n$-type
• Semi-stable subcategories for Euclidean quivers
• Trimness of Closed Intervals in Cambrian Semilattices
• Central support for triangulated categories
• Ungarian Markov Chains
• Ext-quivers of hearts of A-type and the orientation of associahedron
• Mutation and torsion pairs
• Dual structures on Coxeter and Artin groups of rank three
• Parabolic Tamari Lattices in Linear Type B
• From frieze patterns to cluster categories
• Thick subcategories on weighted projective curves and nilpotent representations of quivers
• Exceptional sequences and rooted labeled forests
• $c$-Birkhoff polytopes
• Bijection between positive clusters and projectively signed exceptional sequences
• Rooted Clusters for Graph LP Algebras