Polytopal Realizations of Generalized Associahedra

By Frédéric Chapoton, Sergey Fomin, Andrei Zelevinsky. Canadian mathematical bulletin, 2002

236 citations recorded by the source corpus.

Abstract We prove polytopality of the generalized associahedra introduced in [5].

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

View Paper: https://www.semanticscholar.org/paper/36b797960b45d4abd00cc769a09b902accb8bd6a

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In Algebra and Number Theory

1. B. Keller
2. I. Reiten
3. Bernard Leclerc
4. Osamu Iyama
5. Aslak Bakke Buan
6. C. Geiss
7. Ralf Schiffler
8. J. Schröer

In Discrete Mathematics and Combinatorics

1. B. Keller
2. I. Reiten
3. Bernard Leclerc
4. Aslak Bakke Buan
5. Nathan Reading
6. C. Geiss
7. J. Schröer
8. Laurent Demonet

In Geometry and Topology

1. B. Keller
2. I. Reiten
3. Bernard Leclerc
4. Osamu Iyama
5. Aslak Bakke Buan
6. C. Geiss
7. J. Schröer
8. Ralf Schiffler

Citations in the corpus

• Cluster algebras II: Finite type classification (2002, methodology): “See [11, Section 3.5] and [ 7, Section 4.2 ] for details.”
• Enumerative properties of generalized associahedra (2004, unknown): “These fans are associated with convex polytopes called generalized associahedra [8] and have been shown to be related to classical combinatorial objects such as triangulations, noncrossing and nonnesting partitions and Catalan numbers.”
• Realization of the Stasheff polytope (2002, background): “It is a convex polytope of dimension n with one vertex for each planar binary tree with n + 1 leaves.”
• Root systems and generalized associahedra (2005, methodology): “nt polynomial, in agreement with Theorem 4.4. 4.3. Cluster complexes and generalized associahedra This section is based on [19, 21], except for the last statement in Theorem 4.11, which was proved in [13]. It can be shown that in a given cluster algebra of finite type, each seed is uniquely determine”
• Positivity and canonical bases in rank 2 cluster algebras of finite and affine types (2003, background): “• A new family of convex polytopes (generalized associahedra) including as special cases Stasheff’s associahedron, and Bott-Taubes cyclohedron [8, 2].”
• Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “These are the vertices of the exchange graph, which is isomorphic to the third Stasheff associahedron [120] [ 35 ]: /.-, ()*+ 2”
• Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups (2006, background): “However, the only known geometric realization of ∆(W ) as a convex polytope (due to Chapoton, Fomin and Zelevinsky [43]) does not obviously extend to the noncrystallographic case, and it is an open problem to give a geometric realization of ∆(W ) when W is noncrystallographic.”
• Cluster complexes via semi-invariants (2007, background): “One can show this triangulation is Poincar´e dual to the Cluster Associahedron of [3].”
• Cambrian Lattices (2004, unknown): “In this section we provide background information on Coxeter groups and the weak order, including a repetition in greater depth of a few concepts mentioned in the introduction.”
• Cluster structures from 2-Calabi–Yau categories with loops (2008, background): “A simplicial complex was associated in [17] to any finite root system, and it was conjectured there and later proved in [13] that these simplicial complexes are the face complexes of certain polytopes which were called generalised associahedra.”
• Nested complexes and their polyhedral realizations (2005, background): “The argument is completely parallel to that in [2]; in particular, it leads to a specific realization of Π which can be shown to be the same as the realization given in [9].”