# The Catalan combinatorics of the hereditary artin algebras

@article{Ringel2015TheCC,
title={The Catalan combinatorics of the hereditary artin algebras},
author={Claus Michael Ringel},
journal={arXiv: Representation Theory},
year={2015}
}
• C. Ringel
• Published 23 February 2015
• Mathematics
• arXiv: Representation Theory
This is a survey on the categorification of the poset of generalized non-crossing partitions, using the representation theory of a hereditary artin algebra H, looking at the set P of exceptional subcategories in mod H. This categorification is due to Ingalls and Thomas, and a subsequent paper by Igusa and Schiffler. Starting point is a refinement of the classical tilting theory for mod H, replacing torsion pairs by torsion triples, thus putting it into the realm of the stability theory of King… Expand
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#### References

SHOWING 1-10 OF 106 REFERENCES
The number of complete exceptional sequences for a Dynkin algebra
• Mathematics
• 2013
We consider Dynkin algebras, these are the hereditary artin algebras of finite representation type. The indecomposable modules for a Dynkin algebra correspond bijectively to the positive roots of aExpand
EXCEPTIONAL SEQUENCES FOR QUIVERS OF DYNKIN TYPE
Let kQ be the path algebra of a quiver Q without oriented cycles with n vertices. An indecomposable kQ-module without self-extensions is called exceptional. The braid group B n with n − 1 generatorsExpand
On a refinement of the generalized Catalan numbers for Weyl groups
Let Φ be an irreducible crystallographic root system with Weyl group W, coroot lattice Q and Coxeter number h, spanning a Euclidean space V, and let m be a positive integer. It is known that the setExpand
The braid group action on the set of exceptional sequences of a hereditary artin algebra
Let A be a hereditary artin algebra with s = s(A) simple modules. The indecomposable A-modules without self-extensions are of great importance, they may be called complete exceptional sequences.Expand
Noncrossing Partitions for the Group Dn
• Computer Science, Mathematics
• SIAM J. Discret. Math.
• 2004
A combinatorial description of this lattice in terms of noncrossing planar graphs in the case of the Coxeter group of type Dn, thus answering a question of Bessis, and a proof of a theorem valid for all root systems. Expand
Noncrossing partitions and representations of quivers
• Mathematics
• Compositio Mathematica
• 2009
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 noncrossingExpand
ad-nilpotent ideals of a Borel subalgebra: generators and duality
It was shown by Cellini and Papi that an ad-nilpotent ideal determines certain element of the affine Weyl group, and that there is a bijection between the ad-nilpotent ideals and the integral pointsExpand
The module theoretical approach to quasi-hereditary algebras
• Mathematics
• 1992
Quasi-hereditary algebras were introduced by L.Scott [S] in order to deal with highest weight categories as they arise in the representation theory of semi–simple complex Lie algebras and algebraicExpand
On algebras of finite representation type
Introduction. Since D. G. Higman proved that bounded representation type and finite representation type are equivalent for group algebras at prime characteristic, there has been a renewed interest inExpand
Perpendicular Categories with Applications to Representations and Sheaves
• Mathematics
• 1991
This paper is concerned with the omnipresence of the formation of the subcategories right (left) perpendicular to a subcategory of objects in an abelian category. We encounter these subcategories inExpand