Many non-equivalent realizations of the associahedron

Discrete & Computational Geometry, 2007

We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the Bott-Taubes polytope) and compare them to the permutahedron of type A and B respectively.

Associahedra, Tamari Lattices and Related Structures, 2012

There are many open problems and some mysteries connected to the realizations of the associahedra as convex polytopes. In this note, we describe threeconcerning special realizations with the vertices on a sphere, the space of all possible realizations, and possible realizations of the multiassociahedron.

2021

Taking a representation-theoretic viewpoint, we construct a continuous associahedron motivated by the realization of the generalized associahedron in the physical setting. We show that our associahedron shares important properties with the generalized associahedron of type A. Our continuous associahedron is convex and manifests a cluster theory: the points which correspond to the clusters are on its boundary, and the edges that correspond to mutations are given by intersections of hyperplanes. This requires development of several methods that are continuous analogues of discrete methods. We conclude the paper by showing that there is a sequence of embeddings of type A generalized associahedra into our continuous associahedron.

2021

The hypersimplex ∆k+1,n is the image of the positive Grassmannian Gr ≥0 k+1,n under the moment map. It is a polytope of dimension n− 1 in R. Meanwhile, the amplituhedron An,k,2 is the projection of the positive Grassmannian Gr≥0 k,n into the Grassmannian Grk,k+2 under the amplituhedron map Z̃. Introduced in the context of scattering amplitudes, it is not a polytope, and has full dimension 2k inside Grk,k+2. Nevertheless, there seem to be remarkable connections between these two objects via T-duality, as was first discovered in [LPW20]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes – images of positroid cells of Gr≥0 k+1,n under the moment map – translate into sign conditions characterizing the T-dual Grasstopes – images of positroid cells of Gr≥0 k,n under the amplituhedron map. ...

2011

We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description of generalized associahedra, a Minkowski sum decomposition into Coxeter matroid polytopes, and a combinatorial description of the exchange matrix of any cluster in a finite type cluster algebra.

A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply-laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the F-polynomials of the corresponding cluster algebras.

Advances in Mathematics, 2015

We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description of generalized associahedra, a Minkowski sum decomposition into Coxeter matroid polytopes, and a combinatorial description of the exchange matrix of any cluster in a finite type cluster algebra.

2021

Grzegorz Rajchel-Mieldzioć,1 Kamil Korzekwa,2 Zbigniew Puchała,2, 3 and Karol Życzkowski1, 2 Center for Theoretical Physics, Polish Academy of Sciences, 02-668 Warszawa, Poland Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Kraków, Poland Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, 44-100 Gliwice, Poland (Dated: January 28, 2021)

Arnold mathematical journal, 2018

We show that the cyclohedron (Bott-Taubes polytope) W n arises as the polar dual of a Kantorovich-Rubinstein polytope KR(ρ), where ρ is an explicitly described quasimetric (asymmetric distance function) satisfying strict triangle inequality. From a broader perspective, this phenomenon illustrates the relationship between a nestohedron ∆ F (associated to a building set F) and its non-simple deformation ∆ F , where F is an irredundant or tight basis of F (Definition 21). Among the consequences are a new proof of a recent result of Gordon and Petrov (Arnold Math. J. 3 (2), 205-218 (2017)) about f-vectors of generic Kantorovich-Rubinstein polytopes and an extension of a theorem of Gelfand, Graev, and Postnikov, about triangulations of the type A, positive root polytopes.

Algebraic Combinatorics, 2021

The polytope subalgebra of deformations of a zonotope can be endowed with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. We explore this construction and find relations between statistics on (signed) permutations and the module structure in the case of (type B) generalized permutahedra. In type B, the module structure surprisingly reveals that any family of generators (via signed Minkowski sums) for generalized permutahedra of type B will contain at least 2 d-1 full-dimensional polytopes. We find a generating family of simplices attaining this minimum. Finally, we prove that the relations defining the polytope algebra are compatible with the Hopf monoid structure of generalized permutahedra.