Noncrossing partitions and representations of quivers

Ingalls, Colin; Thomas, Hugh

doi:10.1112/s0010437x09004023

Cited by 149 publications

(176 citation statements)

References 34 publications

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“…In [19], two of the authors of the present paper showed that when Q is Dynkin, any abelian and extension-closed subcategory arises as the θ-semi-stable subcategory of a suitable choice of θ. In fact, somewhat more is known; see [19,Theorem 1.1].…”

Section: Introductionmentioning

confidence: 76%

“…In Dynkin type, the semi-stable subcategories (or equivalently the abelian and extension-closed subcategories) form a lattice. It was shown in [19] that this poset is isomorphic to the lattice of noncrossing partitions associated to the Weyl group corresponding to Q. These lattices had already been studied by combinatorialists and group theorists and, especially relevant for our purposes, they play a central role in the construction of the dual Garside structure of Bessis [3] on the corresponding Artin group, which also leads to their use in constructing Eilenberg-Mac Lane spaces for the Artin groups [4,5].…”

Section: Theorem 55 For Q a Euclidean Quiver An Abelian And Extensmentioning

confidence: 99%

“…In fact, somewhat more is known; see [19,Theorem 1.1]. For any acyclic quiver Q, the following conditions on an abelian, extensionclosed subcategory A of rep(Q) are equivalent; see Proposition 4.4.…”

Section: Introductionmentioning

confidence: 99%

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Semi-stable subcategories for Euclidean quivers

Ingalls

Paquette

Thomas

2015

Proceedings of the London Mathematical Society

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Abstract. In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of semi-stable subcategories may not be semi-stable. However, only a finite number of exceptions occur, and we give a description of these subcategories. Moreover, one can attach a simplicial fan in Q n to any acyclic quiver Q, and this simplicial fan allows one to completely determine the canonical presentation of any element in Z n . This fan has a nice description in the Dynkin and Euclidean cases: it is described using an arrangement of convex codimension-one subsets of Q n , each such subset being indexed by a real Schur root or a set of quasi-simple objects. This fan also characterizes when two different stability conditions give rise to the same semi-stable subcategory.

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Section: Introductionmentioning

confidence: 76%

Section: Theorem 55 For Q a Euclidean Quiver An Abelian And Extensmentioning

confidence: 99%

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Semi-stable subcategories for Euclidean quivers

Ingalls

Paquette

Thomas

2015

Proceedings of the London Mathematical Society

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“…These notions were already studied by Auslander and Smalø [9] in the 1980's, but we are using here the terminology adopted by Adachi, Iyama and Reiten in [2], which generalises the work of Ingalls and Thomas [52] on hereditary algebras.…”

mentioning

confidence: 99%

Ordered exchange graphs

Brüstle¹,

Yang²

2014

Advances in Representation Theory of Algebras

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Abstract. The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated with objects such as a finite-dimensional algebra or a differential graded algebra concentrated in nonpositive degrees. These constructions often come from variations of the concept of tilting, the vertices of the exchange graph being torsion pairs, t-structures, silting objects, support τ -tilting modules and so on. All these exchange graphs stemming from representation theory have the additional feature that they are the Hasse quiver of a partial order which is naturally defined for the objects. In this sense, the exchange graphs studied in this article can be considered as a generalization or as a completion of the poset of tilting modules which has been studied by Happel and Unger. The goal of this article is to axiomatize the thus obtained structure of an ordered exchange graph, to present the various constructions of ordered exchange graphs and to relate them among each other.

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“…Following Ingalls-Thomas [9], the cone S(Q) of finite stability conditions is, by definition, the set of all σ ∈ Q Q 0 for which there are finitely many, up to isomorphism, σ-stable representations.…”

Section: Introductionmentioning

confidence: 99%

Cluster fans, stability conditions, and domains of semi-invariants

Chindris

2011

Trans. Amer. Math. Soc.

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Abstract. We show that the cone of finite stability conditions of a quiver Q without oriented cycles has a fan covering given by (the dual of) the cluster fan of Q. Along the way, we give new proofs of Schofield's results (1991) on perpendicular categories. From our results, we recover Igusa-Orr-TodorovWeyman's theorem on cluster complexes and domains of semi-invariants for Dynkin quivers. For arbitrary quivers, we also give a description of the domains of semi-invariants labeled by real Schur roots in terms of quiver exceptional sets.

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Noncrossing partitions and representations of quivers

Cited by 149 publications

References 34 publications

Semi-stable subcategories for Euclidean quivers

Semi-stable subcategories for Euclidean quivers

Ordered exchange graphs

Cluster fans, stability conditions, and domains of semi-invariants

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