Noncrossing partitions and representations of quivers

By Colin Ingalls, Hugh Thomas. Compositio Mathematica, 2006

254 citations recorded by the source corpus.

Abstract We situate the noncrossing partitions associated with 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 with 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 that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e0084c387d50f6c3c35ad4484fbacfc99b6475b

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Citations in the corpus

Rowmotion in slow motion (2017, background): “yofﬁnite-dimensional algebras: namely, as the lattices of torsion classes for suitably chosen algebras. The lattices that can be obtained in this way include the Cambrian lattices of simplylaced type [IT09], but also many more. (We will show in Corollary 1.5 that these extremal lattices are actually”
SIGNED EXCEPTIONAL SEQUENCES AND THE CLUSTER MORPHISM CATEGORY (2017, background): “ be a ﬁnite dimensional hereditary algebra over any ﬁeld. Then the cluster morphism category G(Λ) of Λ is deﬁned to be the category whose objects are the ﬁnitely generated wide subcategories of mod-Λ [InTh], (Section 1.1 below). Such a subcategory A⊆mod-Λ is hereditary and abelian and has a cluster ”
Ordered Exchange Graphs (2013, methodology): “Further, for an acyclic quiver, Speyer and Thomas give a combinatorial description of which collections of roots form c-matrices, see [98, Theorem 1.4].”
Torsion classes, wide subcategories and localisations (2015, background): “Passing to non-commutative rings, it was shown in [17] that over a finite-dimensional hereditary algebra A, the functorially finite wide subcategories of mod(A) are in bijection with the functorially finite torsion classes of the same category.”
Stratifying triangulated categories (2009, methodology): “The localizing subcategories of D(kQ) are parameterized by the non-crossing partitions associated to Q; this can be deduced from work of Ingalls and Thomas [20].”
$\tau$-tilting finite algebras, bricks and $g$-vectors (2015, methodology): “Algebras having finitely many isomorphism classes of basic tilting modules have been successfully investigated, see for example Riedtmann-Schofield [23], Unger [26] and Ingalls-Thomas [16].”
A survey on maximal green sequences (2019, background): “It is made completely explicit by Ingalls–Thomas in [70] (in the acyclic case).”
Exceptional sequences and clusters (2009, background): “ion holds if and only if (E 1,···,En) is a complete exceptional sequence. We note that these results are known in the ﬁnite case [2], [4] and have been extended to the aﬃne case by Ingalls and Thomas [13]. Our new proof is type independent and includes these previous results as corollaries. (We note”
Representations of Algebras and Related Topics (2011, unknown): “6 of [32] to this problem.”
Reduction of τ-Tilting Modules and Torsion Pairs (2013, background): “Support tilting A-modules are defined analogously, see [18].”
Cluster algebras and derived categories (2012, background): “• combinatorics and in particular the study of polyhedra like the Stasheff associahedra [21] [22] [42] [43] [71] [90] [102] [104] [103] [130] .”