scholarly journals Noncrossing partitions and representations of quivers

2009 ◽  
Vol 145 (6) ◽  
pp. 1533-1562 ◽  
Author(s):  
Colin Ingalls ◽  
Hugh Thomas
Keyword(s):  
Coxeter Group ◽  
Stability Condition ◽  
Quiver Representations ◽  
Finitely Generated ◽  
Cluster Category ◽  
Noncrossing Partitions ◽  
Finite Coxeter Group ◽  
Group Form ◽  
Cluster Tilting ◽  
Tilting Objects

AbstractWe 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.

scholarly journals Lifting to Cluster-tilting Objects in Higher Cluster Categories

2012 ◽  
Vol 19 (04) ◽  
pp. 707-712
Author(s):  
Pin Liu
Keyword(s):  
Positive Integer ◽  
Cluster Category ◽  
Tilting Modules ◽  
Endomorphism Rings ◽  
Tilted Algebras ◽  
Cluster Categories ◽  
Cluster Tilting ◽  
Tilting Objects

Let d > 1 be a positive integer. In this note, we consider the d-cluster-tilted algebras, i.e., algebras which appear as endomorphism rings of d-cluster-tilting objects in higher cluster categories (d-cluster categories). We show that tilting modules over such algebras lift to d-cluster-tilting objects in the corresponding higher cluster category.

scholarly journals Density of g-Vector Cones From Triangulated Surfaces

2020 ◽  
Vol 2020 (21) ◽  
pp. 8081-8119
Author(s):  
Toshiya Yurikusa
Keyword(s):  
Asymptotic Behavior ◽  
Half Space ◽  
Closed Surface ◽  
Cluster Algebras ◽  
Connected Components ◽  
Cluster Category ◽  
Triangulated Surfaces ◽  
Cluster Tilting ◽  
Tilting Objects ◽  
Marked Surface

Abstract We study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$ except for a closed surface with exactly one puncture, in which case it is equal to the half space of a certain explicit hyperplane in $\mathbb{R}^n$. Our main ingredients are laminations on $(S,M)$, their shear coordinates, and their asymptotic behavior under Dehn twists. As an application, if $(S,M)$ is not a closed surface with exactly one puncture, the exchange graph of cluster tilting objects in the corresponding cluster category is connected. If $(S,M)$ is a closed surface with exactly one puncture, it has precisely two connected components.

Finite Repetitive Generalized Cluster Complexes and d-Cluster Categories

2013 ◽  
Vol 20 (01) ◽  
pp. 123-140
Author(s):  
Teng Zou ◽  
Bin Zhu
Keyword(s):  
Simplicial Complex ◽  
Root System ◽  
Cluster Complex ◽  
Cluster Complexes ◽  
Endomorphism Algebras ◽  
Tilted Algebras ◽  
Cluster Categories ◽  
Hereditary Algebras ◽  
Cluster Tilting ◽  
Tilting Objects

For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of d-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) d-cluster tilted algebras. Moreover, we prove that the generalized d-cluster categories of hereditary algebras of finite representation type provide a category model for the n-repetitive generalized cluster complexes.

scholarly journals Donaldson–Thomas invariants versus intersection cohomology of quiver moduli

2019 ◽  
Vol 2019 (754) ◽  
pp. 143-178 ◽  
Author(s):  
Sven Meinhardt ◽  
Markus Reineke
Keyword(s):  
Moduli Space ◽  
Stability Condition ◽  
Intersection Cohomology ◽  
Quiver Representations ◽  
Compactly Supported ◽  
Generic Stability ◽  
Relative Version ◽  
Zero Potential ◽  
Stable Locus

Abstract The main result of this paper is the statement that the Hodge theoretic Donaldson–Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson–Thomas “function” to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson–Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.

scholarly journals Quotient Isomorphism Invariants of a Finitely Generated Coxeter Group

2008 ◽  
Author(s):  
Michael Mihalik ◽  
John Ratcliffe ◽  
Steven Tschantzk
Keyword(s):  
Coxeter Group ◽  
Finitely Generated

scholarly journals Combinatorics of Exceptional Sequences in Type A

10.37236/6251 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Alexander Garver ◽  
Kiyoshi Igusa ◽  
Jacob P. Matherne ◽  
Jonah Ostroff
Keyword(s):  
Dynkin Diagram ◽  
Type A ◽  
Linear Extensions ◽  
Quiver Representations ◽  
Noncrossing Partitions ◽  
Underlying Graph ◽  
Exceptional Sequences

Exceptional sequences are certain sequences of quiver representations.  We introduce a class of objects called strand diagrams and use these to classify exceptional sequences of representations of a quiver whose underlying graph is a type $\mathbb{A}_n$ Dynkin diagram. We also use variations of these objects to classify $c$-matrices of such quivers, to interpret exceptional sequences as linear extensions of explicitly constructed posets, and to give a simple bijection between exceptional sequences and certain saturated chains in the lattice of noncrossing partitions. 

scholarly journals Some remarks on the algebraic structure of the finite Coxeter groupF4

1999 ◽  
Vol 22 (1) ◽  
pp. 81-84
Author(s):  
Muhammad A. Albar ◽  
Norah Al-Saleh
Keyword(s):  
Coxeter Group ◽  
Algebraic Structure ◽  
Finite Coxeter Group

We consider in this paper the algebraic structure and some properties of the finite Coxeter groupF4.

The polytopes of the H 3 group with 60 vertices and their orbit decompositions

2019 ◽  
Vol 75 (3) ◽  
pp. 541-550
Author(s):  
Emmanuel Bourret ◽  
Zofia Grabowiecka
Keyword(s):  
Coxeter Group ◽  
Dynkin Diagram ◽  
Geometrical Structure ◽  
Three Dimensions ◽  
Icosahedral Group ◽  
Finite Coxeter Group

The goal of this article is to compare the geometrical structure of polytopes with 60 vertices, generated by the finite Coxeter group H 3, i.e. an icosahedral group in three dimensions. The method of decorating a Coxeter–Dynkin diagram is used to easily read the structure of the reflection-generated polytopes. The decomposition of the vertices of the polytopes into a sum of orbits of subgroups of H 3 is given and presented as a `pancake structure'.

scholarly journals Generalized Descent Algebras

2007 ◽  
Vol 50 (4) ◽  
pp. 535-546
Author(s):  
Christophe Hohlweg
Keyword(s):  
Group Algebra ◽  
Coxeter Group ◽  
Descent Algebra ◽  
Finite Coxeter Group ◽  
Solomon Descent Algebra ◽  
Descent Algebras

AbstractIf A is a subset of the set of reflections of a finite Coxeter group W, we define a sub-ℤ-module of the group algebra ℤW. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if W is of type B, the Mantaci–Reutenauer algebra.

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