Coxeter Complexes and Graph-Associahedra

By Michael Carr, Satyan L. Devadoss. 2004

240 citations recorded by the source corpus.

Abstract Given a graph Γ , we construct a simple, convex polytope, dubbed graph-associahedra , whose face poset is based on the connected subgraphs of Γ . This provides a natural generalization of the Stasheff associahedron and the Bott–Taubes cyclohedron. Moreover, we show that for any simplicial Coxeter system, the minimal blow-ups of its associated Coxeter complex has a tiling by graph-associahedra. The geometric and combinatorial properties of the complex as well as of the polyhedra are given. These spaces are natural generalizations of the Deligne–Knudsen–Mumford compactification of the real moduli space of curves.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2b9a7700eb4b28880dc4ea353b6270399a727837

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Citations in the corpus

Toric Topology (2012, methodology): “The corresponding nestohedra are called graph-associahedra; they were introduced and studied in the work of Carr and Devadoss [59], and independently in the work of Toledano Laredo [222] under the name De Concini–Procesi associahedra.”
A link surgery spectral sequence in monopole Floer homology (2009, methodology): “While the relationship between permutohedra and hypercube lattices is well-known, more generally, we construct maps using graph associahedra [5], which we couple to the combinatorics of finite product lattices of arbitrary shape.”
Matroid polytopes, nested sets and Bergman fans (2004, background): “ER AND BERND STURMFELS Acknowledgements: We are grateful to Paul Hacking for his helpful comments on a draft of this paper, and to Lauren Williams and Alex Postnikov for pointing us to the references [2, 6, 17]. The IAS/Park City Mathematics Institute (PCMI, July 2004) and the Mathematical Sciences ”
A semi-small decomposition of the Chow ring of a matroid (2020, background): “We refer to [7] and [13] for detailed discussions of graph associahedra and their realizations.”
Stellahedral geometry of matroids (2022, background): “We remark that the stellahedron Π 𝐸 is a realization of the graph associahedron of the star graph with the set of endpoints E ; see, for example, [PRW08, §10.4].”
Generalized cluster complexes and Coxeter combinatorics (2005, unknown): “ferent generalization of associahedra [ 7 , 10, 20] which produces the cyclohedron in case of an n-cycle.”
Nested complexes and their polyhedral realizations (2005, background): “More general complexes associated with arbitrary flnite graphs were introduced and studied in [1] and more recently in [10]; these papers also present a construction of associated simple convex polytopes (dubbed graph associahedra in [ 1 ], and De Concini - Procesi associahedra in [10]).”
Hopf Monoids and Generalized Permutahedra (2017, background): “There are natural bijections between tubings on a path pn of length n, valid parenthesizations of the expression x0x1 · · ·xn, and plane rooted trees with n+1 unlabeled leaves.[19] [75, Chapter 6] This bijection allows us to state Proposition 25.”
Permutohedra, Associahedra, and Beyond (2005, methodology): “, desc(8, T )} = {[1, 1], [1, 4], [3, 3], [3, 4], [1, 8], [6, 8], [7, 7], [7, 8]}.”
Convex Hull Realizations of the Multiplihedra (2007, unknown): “Thanks to XY-pic for the diagrams.”