Generalized associahedra via quiver representations

By B. Marsh, Markus Reineke, Andrei Zelevinsky. 2002

224 citations recorded by the source corpus.

We provide a quiver-theoretic interpretation of certain smooth complete simplicial fans associated to arbitrary finite root systems in recent work of S. Fomin and A. Zelevinsky. The main properties of these fans then become easy consequences of the known facts about tilting modules due to K. Bongartz, D. Happel and C. M. Ringel.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5eea97b778ded18c0d990ba2b282c3e2ee442c97

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Citations in the corpus

Cambrian Lattices (2004, unknown): “Since their construction in [21] starts with the same combinatorial data as ours, it would be interesting to know if their fans coincide with the Cambrian fans.”
Mutation of cluster-tilting objects and potentials (2008, methodology): “There is, motivated by this theory via [ MRZ ], a mutation of cluster-tilting objects in cluster categories, and more generally Homfinite triangulated 2-Calabi-Yau (2-CY for short) categories over an algebraically closed field K [BMRRT, IY].”
Tilting theory and cluster combinatorics (2004, methodology): “In [MRZ], it was shown that the combinatorics of A could be obtained from the category of decorated representations of a quiver Q with underlying graph ∆.”
A survey on maximal green sequences (2019, background): “vertices are the seeds obtained from pQ,xq by iterated mutation and whose edges correspond to mutations. By deﬁnition, the edges of the exchange graph are unoriented. However, it was noticed early on [88] that there is a natural partial order on seeds whose minimal inclusions correspond to edges of ”
Categorification of Skew-symmetrizable Cluster Algebras (2009, background): “More recently, Derksen et al. [18, 19] have obtained a far-reaching generalization of [ 53 ].”
Acyclic Calabi–Yau categories (2006, background): “It was the discovery of Marsh, Reineke and Zelevinsky [MRZ03] that they are closely connected to quiver representations.”
Clusters and seeds in acyclic cluster algebras (2005, methodology): “These were introduced to model some of the ingredients in the theory of cluster algebras in a module theoretic/categorical way, inspired by [ MRZ ], with the hope of obtaining feedback to the theory of cluster algebras.”
Cluster-tilted algebras (2004, background): “Motivated by [ MRZ ], tilting theory in a particular factor category of the bounded derived category D b (mod H) of the finitely generated modules over a finite dimensional hereditary algebra H over a field k was investigated in [BMRRT], along with the relationship to tilting theory for hereditary a”
Subword complexes, cluster complexes, and generalized multi-associahedra (2011, background): “This simplicial complex along with the generalized associahedron has become the object of intensive studies and generalizations in various contexts in mathematics, see, for instance, [12, 27, 40, 48].”
On cluster algebras with coefficients and 2-Calabi-Yau categories (2007, background): “…we pursue the representation-theoretic approach to Fomin-Zelevinsky’s cluster algebras [19], [20], [8], [21] developed by Marsh-Reineke-Zelevinsky [35], Buan-Marsh-Reineke-Reiten-Todorov [6], [7], Caldero-Chapoton and Caldero-Keller [9], [10], [11], Geiss-Leclerc-Schr¨oer [25], [26], [23],…”
Cluster algebras as Hall algebras of quiver representations (2004, background): “The implicit question behind all articles [ MRZ03 , BMR+, CCS04] dealing with”
A categorification of non-crossing partitions (2013, background): “One connection between noncrossing partitions and representations of quivers has already been pointed out by Ingalls and Thomas [34]; it arises from the categorification of cluster algebras [20] via cluster categories [9, 40].”