158 research outputs found

    Cambrian triangulations and their tropical realizations

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    This paper develops a Cambrian extension of the work of C. Ceballos, A. Padrol and C. Sarmiento on ν\nu-Tamari lattices and their tropical realizations. For any signature ε{±}n\varepsilon \in \{\pm\}^n, we consider a family of ε\varepsilon-trees in bijection with the triangulations of the ε\varepsilon-polygon. These ε\varepsilon-trees define a flag regular triangulation Tε\mathcal{T}^\varepsilon of the subpolytope conv{(ei,ej)0i<jn+1}\operatorname{conv} \{(\mathbf{e}_{i_\bullet}, \mathbf{e}_{j_\circ}) \, | \, 0 \le i_\bullet < j_\circ \le n+1 \} of the product of simplices {0,,n}×{1,,(n+1)}\triangle_{\{0_\bullet, \dots, n_\bullet\}} \times \triangle_{\{1_\circ, \dots, (n+1)_\circ\}}. The oriented dual graph of the triangulation Tε\mathcal{T}^\varepsilon is the Hasse diagram of the (type AA) ε\varepsilon-Cambrian lattice of N. Reading. For any I{0,,n}I_\bullet \subseteq \{0_\bullet, \dots, n_\bullet\} and J{1,,(n+1)}J_\circ \subseteq \{1_\circ, \dots, (n+1)_\circ\}, we consider the restriction TI,Jε\mathcal{T}^\varepsilon_{I_\bullet, J_\circ} of the triangulation Tε\mathcal{T}^\varepsilon to the face I×J\triangle_{I_\bullet} \times \triangle_{J_\circ}. Its dual graph is naturally interpreted as the increasing flip graph on certain (ε,I,J)(\varepsilon, I_\bullet, J_\circ)-trees, which is shown to be a lattice generalizing in particular the ν\nu-Tamari lattices in the Cambrian setting. Finally, we present an alternative geometric realization of TI,Jε\mathcal{T}^\varepsilon_{I_\bullet, J_\circ} as a polyhedral complex induced by a tropical hyperplane arrangement.Comment: 16 pages, 11 figure

    Which nestohedra are removahedra?

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    A removahedron is a polytope obtained by deleting inequalities from the facet description of the classical permutahedron. Relevant examples range from the associahedra to the permutahedron itself, which raises the natural question to characterize which nestohedra can be realized as removahedra. In this note, we show that the nested complex of any connected building set closed under intersection can be realized as a removahedron. We present two different complementary proofs: one based on the building trees and the nested fan, and the other based on Minkowski sums of dilated faces of the standard simplex. In general, this closure condition is sufficient but not necessary to obtain removahedra. However, we show that it is also necessary to obtain removahedra from graphical building sets, and that it is equivalent to the corresponding graph being chordful (i.e. any cycle induces a clique).Comment: 13 pages, 4 figures; Version 2: new Remark 2

    The greedy flip tree of a subword complex

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    We describe a canonical spanning tree of the ridge graph of a subword complex on a finite Coxeter group. It is based on properties of greedy facets in subword complexes, defined and studied in this paper. Searching this tree yields an enumeration scheme for the facets of the subword complex. This algorithm extends the greedy flip algorithm for pointed pseudotriangulations of points or convex bodies in the plane.Comment: 14 pages, 10 figures; various corrections (in particular deletion of Section 4 which contained a serious mistake pointed out by an anonymous referee). This paper is subsumed by our joint results with Christian Stump on "EL-labelings and canonical spanning trees for subword complexes" (http://arxiv.org/abs/1210.1435) and will therefore not be publishe

    Brick polytopes, lattice quotients, and Hopf algebras

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    This paper is motivated by the interplay between the Tamari lattice, J.-L. Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf algebra on binary trees. We show that these constructions extend in the world of acyclic kk-triangulations, which were already considered as the vertices of V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural surjection from the permutations to the acyclic kk-triangulations. We show that the fibers of this surjection are the classes of the congruence k\equiv^k on Sn\mathfrak{S}_n defined as the transitive closure of the rewriting rule UacV1b1VkbkWkUcaV1b1VkbkWU ac V_1 b_1 \cdots V_k b_k W \equiv^k U ca V_1 b_1 \cdots V_k b_k W for letters a<b1,,bk<ca < b_1, \dots, b_k < c and words U,V1,,Vk,WU, V_1, \dots, V_k, W on [n][n]. We then show that the increasing flip order on kk-triangulations is the lattice quotient of the weak order by this congruence. Moreover, we use this surjection to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on permutations, indexed by acyclic kk-triangulations, and to describe the product and coproduct in this algebra and its dual in term of combinatorial operations on acyclic kk-triangulations. Finally, we extend our results in three directions, describing a Cambrian, a tuple, and a Schr\"oder version of these constructions.Comment: 59 pages, 32 figure

    Geometric realizations of the accordion complex of a dissection

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    Consider 2n2n points on the unit circle and a reference dissection D\mathrm{D}_\circ of the convex hull of the odd points. The accordion complex of D\mathrm{D}_\circ is the simplicial complex of non-crossing subsets of the diagonals with even endpoints that cross a connected subset of diagonals of D\mathrm{D}_\circ. In particular, this complex is an associahedron when D\mathrm{D}_\circ is a triangulation and a Stokes complex when D\mathrm{D}_\circ is a quadrangulation. In this paper, we provide geometric realizations (by polytopes and fans) of the accordion complex of any reference dissection D\mathrm{D}_\circ, generalizing known constructions arising from cluster algebras.Comment: 25 pages, 10 figures; Version 3: minor correction

    The brick polytope of a sorting network

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    The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of flip graphs on pseudoline arrangements with contacts supported by a given sorting network. In this paper, we construct the brick polytope of a sorting network, obtained as the convex hull of the brick vectors associated to each pseudoline arrangement supported by the network. We combinatorially characterize the vertices of this polytope, describe its faces, and decompose it as a Minkowski sum of matroid polytopes. Our brick polytopes include Hohlweg and Lange's many realizations of the associahedron, which arise as brick polytopes for certain well-chosen sorting networks. We furthermore discuss the brick polytopes of sorting networks supporting pseudoline arrangements which correspond to multitriangulations of convex polygons: our polytopes only realize subgraphs of the flip graphs on multitriangulations and they cannot appear as projections of a hypothetical multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization of our results to spherical subword complexes on finite Coxeter groups (http://arxiv.org/abs/1111.3349

    The weak order on Weyl posets

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    We define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice induced by antisymmetric closed subsets of roots is again a lattice. We then study further subposets of this lattice which naturally correspond to the elements, the intervals and the faces of the permutahedron and the generalized associahedra of the corresponding Weyl group. These results extend to arbitrary finite crystallographic root systems the recent results of G. Chatel, V. Pilaud and V. Pons on the weak order on posets and its induced subposets.Comment: 23 pages, 5 figure

    Graph properties of graph associahedra

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    A graph associahedron is a simple polytope whose face lattice encodes the nested structure of the connected subgraphs of a given graph. In this paper, we study certain graph properties of the 1-skeleta of graph associahedra, such as their diameter and their Hamiltonicity. Our results extend known results for the classical associahedra (path associahedra) and permutahedra (complete graph associahedra). We also discuss partial extensions to the family of nestohedra.Comment: 26 pages, 20 figures. Version 2: final version with minor correction

    Multitriangulations, pseudotriangulations and primitive sorting networks

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    We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.Comment: 60 pages, 40 figures; minor corrections and improvements of presentatio

    Enumerating topological (nk)(n_k)-configurations

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    An (nk)(n_k)-configuration is a set of nn points and nn lines in the projective plane such that their point-line incidence graph is kk-regular. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. We provide an algorithm for generating, for given nn and kk, all topological (nk)(n_k)-configurations up to combinatorial isomorphism, without enumerating first all combinatorial (nk)(n_k)-configurations. We apply this algorithm to confirm efficiently a former result on topological (184)(18_4)-configurations, from which we obtain a new geometric (184)(18_4)-configuration. Preliminary results on (194)(19_4)-configurations are also briefly reported.Comment: 18 pages, 11 figure
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