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Noncrossing partitions and representations of quivers

Part of: Algebraic combinatorics Representation theory of rings and algebras

Published online by Cambridge University Press:  03 December 2009

Colin Ingalls  and
Hugh Thomas
Colin Ingalls
Affiliation:
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada (email: cingalls@unb.ca)
Hugh Thomas
Affiliation:
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada (email: hthomas@unb.ca)
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Abstract

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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.

Keywords

noncrossing partitionsDynkin quiversreflections groupscluster categoryquiver representationstorsion classwide subcategoriesCambrian latticesemistable subcategories

MSC classification

Secondary: 16G20: Representations of quivers and partially ordered sets 05E15: Combinatorial aspects of groups and algebras
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 6 , November 2009 , pp. 1533 - 1562
Copyright
Copyright © Foundation Compositio Mathematica 2009

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