Frederic Chapoton, Sergey Fomin, Andrei Zelevinsky
In hep-th/0111053, a complete simplicial fan was associated to an arbitrary finite root system. It was conjectured that this fan is the normal fan of a simple convex polytope (a generalized associahedron of the corresponding type). Here we prove this conjecture by explicitly exhibiting a family of such polytopal realizations.
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A Polytopal Realization of Higher-Categorical Associahedra
本文给出了高范畴结合面的多面体实现，其正规扇基于速度扇修改并由Astra模型发现。
Polytopal realizations of non-crystallographic associahedra
本文通过折叠技术证明非单链及非晶体学根系对应的广义关联多面体可由单链广义关联多面体的截面实现。
The Syntax Of Polytopal Projections: From Permutohedra To Associahedra
本文通过自由非对称非单位操作域的等式演算，语法化地解释了从置换体到结合体的Tonks投影和Loday-Ronco映射，并证明了Tonks顶点映射保持弱Bruhat序到Tamari序的序关系。
Given any finite graph, we offer a simple realization of the graph-associahedron polytope using integer coordinates.
Brick polytopes of spherical subword complexes and generalized associahedra
将砖多胞形推广至有限Coxeter群的球面子词复形，为簇复形提供多胞形实现，并给出广义associahedra的顶点描述、Minkowski分解及交换矩阵组合描述。
We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $Δ_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan …
A~$k$-associahedron is a simplicial complex whose facets, called~$k$-triangulations, are the inclusion maximal sets of diagonals of a convex polygon where no~$k+1$ diagonals mutually cross. Such complexes are conjectured for about a decade to have realizations as convex polytopes, and therefore as complete simplicial fans. Apart from four one-parameter families including simplices, cyclic polytopes and classical associahedra, only two instances of multiassociahedra have been geometrically realized so far. This paper reports on conjectural realizations for all~$2$-associahedra, obtained by heur…
Given any connected poset $P$, we give a simple realization of Galashin's poset associahedron $\mathscr{A}(P)$ as a convex polytope in $\mathbb{R}^P.$ The realization is inspired by the description of $\mathscr{A}(P)$ as a compactification of the configuration space of order-preserving maps $P \to \mathbb{R}.$ In addition, we give an analogous realization for Galashin's affine poset cyclohedra.
Geometric realizations of $ν$-associahedra via brick polyhedra
本文利用砖多面体实现了$\ u$-associahedron的几何构造，其边图为$\ u$-Tamari格，并给出了优雅的顶点坐标描述。
This paper introduces a new method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also known as Stasheff polytopes. We show that it satisfies an easy-to-state cellular formula. For the first time, we endow a family of realizations of the associahedra (the Loday realizations) with a topological and cellular operad structure; it is shown to be compatible with the diagonal maps.
Polytopal realizations of finite type $\mathbf{g}$-vector fans
本文证明了任意有限型 $\mathbf{g}$-向量扇（无论是否无环）均为多面体，并通过构造万有结合多面体 $\mathsf{Asso}_{\mathrm{un}}(Γ)$，使得任何该型的 $\mathbf{g}$-向量扇都是其适当投影的法扇。
Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
本文证明了有限型簇代数中网格突变是 $\mathbf{g}$-向量的最小关系，并由此推导出 $\mathbf{g}$-向量扇的多面体实现空间为单纯锥，且可描述为正象限与仿射空间的交集；同时推广到 gentle 代数的非亲吻扇，并证明了相关范畴的代数结果。
This thesis explores two specific topics of discrete geometry, the multitriangulations and the polytopal realizations of products, whose connection is the problem of finding polytopal realizations of a given combinatorial structure. A k-triangulation is a maximal set of chords of the convex n-gon such that no k+1 of them mutually cross. We propose a combinatorial and geometric study of multitriangulations based on their stars, which play the same role as triangles of triangulations. This study leads to interpret multitriangulations by duality as pseudoline arrangements with contact points cove…
We give a geometric realization of the polyhedra governed by the structure of associative algebras with co-inner products, or more precisely, governed by directed planar trees. Our explicit realization of these polyhedra, which include the associahedra in a special case, shows in particular that these polyhedra are homeomorphic to balls. We also calculate the number of vertices of the lowest generalized associahedra, giving appropriate generalizations of the Catalan numbers.
We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most $n$. We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope.
Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two different ways of constructing exponential families of realizations of the n-dimensional associahedron with normal vectors in {0,1,-1}^n, generalizing the constructions of Loday (2004) and Chapoton-Fomin-Zelevinsky (2002). We classify the associahedra obtained by these constructions modulo linear equivalence of their normal fans and show, in particular, that the only realization that can be obtained with both methods is the Chapoton-Fomin-Zelevinsky (2002) associahedron. For the Hohlweg-Lange associahedra our classification…
The second author introduced 2-associahedra as a tool for investigating functoriality properties of Fukaya categories, and he conjectured that they could be realized as face posets of convex polytopes. We introduce a family of posets called categorical $n$-associahedra, which naturally extend the second author's 2-associahedra and the classical associahedra. Categorical $n$-associahedra give a combinatorial model for the poset of strata of a compactified real moduli space of a tree arrangement of affine coordinate subspaces. We construct a family of complete polyhedral fans, called velocity fa…
An associahedron is a polytope whose vertices correspond to the triangulations of a convex polygon and whose edges correspond to flips between them. A particularly elegant realization of the associahedron, due to S. Shnider and S. Sternberg and popularized by J.-L. Loday, has been generalized in two directions: on the one hand by A. Postnikov to obtain a realization of the graph associahedra of M. Carr and S. Devadoss, and on the other hand by C. Hohlweg and C. Lange to obtain multiple realizations of the associahedron parametrized by a sequence of signs. The goal of this paper is to unify and…
We investigate a family of polytopes introduced by E.M.\ Feichtner, A.\ Postnikov and B.\ Sturmfels, which were named nestohedra. The vertices of these polytopes may intuitively be understood as constructions of hypergraphs. Limit cases in this family of polytopes are, on the one end, simplices, and, on the other end, permutohedra. In between, as notable members one finds associahedra and cyclohedra. The polytopes in this family are investigated here both as abstract polytopes and as realized in Euclidean spaces of all finite dimensions. The later realizations are inspired by J.D.\ Stasheff's …
Associahedra via spines
本文通过引入三角剖分的脊柱（对偶树加标记和定向），扩展了结合多面体的经典理解，为Hohlweg-Lange构造提供新视角，并简化了相关证明。
Given a finite Coxeter system $(W,S)$ and a Coxeter element $c$, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan $F_c$, settling a conjecture of Reading that this is possible. We call this polytope the $c$-generalized associahedron. Our approach generalizes Loday's realization of the associahedron (a type $A$ $c$-generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the $c$-singleton cones, the con…
M. Carr and S. Devadoss introduced in [7] the notion of tubing on a finite simple graph $Γ$, in the context of configuration spaces on the Hilbert plane. To any finite simple graph $Γ$ they associated a finite partially ordered set, whose elements are the tubings of $Γ$ and whose geometric realization is a convex polytope ${\mathcal K}Γ$, the graph-associahedron. For the complete graphs they recovered permutahedra, for linear graphs they got Stasheff's associahedra, while for simple graph they obtained the standard simplexes. The goal of the present work is to give an \emph{algebraic} descript…
Motivated by the theory of cluster algebras, F. Chapoton, S. Fomin and A. Zelevinsky associated to each finite type root system a simple convex polytope called \emph{generalized associahedron}. They provided an explicit realization of this polytope associated with a bipartite orientation of the corresponding Dynkin diagram. In the first part of this paper, using the parametrization of cluster variables by their $g$-vectors explicitly computed by S.-W. Yang and A. Zelevinsky, we generalize the original construction to any orientation. In the second part we show that our construction agrees with…
Small covers of graph-associahedra and realization of cycles
本文研究图结合多面体小覆盖的普适实现性质，证明连通图对应的小覆盖或其二重定向覆盖为 URC-流形，并给出 Stasheff 结合多面体的更简单例子。
Compatibility fans for graphical nested complexes
本文受有限型簇复形类比启发，将Fomin-Zelevinsky相容性扇构造推广至图形嵌套复形，定义管间相容度，证明其相容向量支撑完整单纯扇，实现图嵌套复形，并恢复路径情形的加泰罗尼亚多个结合多面体实现。
Polytopal Bier spheres and nonrealizable central symmetries
该研究给出中心对称Bier球无法作为中心对称多面体边界的判据，得到新的不可几何实现的单纯多面体族，还证明顶点数≤12的Bier球均为多面体。
We prove, for an arbitrary finite root system, the periodicity conjecture of Al.B.Zamolodchikov concerning Y-systems, a particular class of functional relations arising in the theory of thermodynamic Bethe ansatz. Algebraically, Y-systems can be viewed as families of rational functions defined by certain birational recurrences formulated in terms of the underlying root system. In the course of proving periodicity, we obtain explicit formulas for all these rational functions, which turn out to always be Laurent polynomials. In a closely related development, we introduce and study a family of si…
Let $(W,S)$ be a finite Coxeter system acting by reflections on an $\mathbb R$-Euclidean space with simple roots $Δ=\{\a_s | s\in S\}$ of the same length and fundamental weights $Δ^*=\{v_s | s\in S\}$. We set $M(e)=\sum_{s\in S}κ_s v_s$, $κ_s>0$, and for $w\in W$ we set $M(w)=w(M(e))$. The permutahedron $Perm(W)$ is the convex hull of the set $\{M(w) | w\in W\}$. Given a Coxeter element $c\in W$, we have defined in a previous work a generalized associahedron $Asso_c(W)$ whose normal fan is the corresponding $c$-Cambrian fan $F_c$ defined by N. Reading. By construction, $Asso_c(W)$ is obtained …
We define Bernstein-Gelfand-Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the "truncated simple reflections" on the set of almost positive roots $Φ_{\ge -1}$ associated to a finite dimensional semisimple Lie algebra. Combining with the tilting theory in cluster categories developed in [4], we give a unified interpretation via quiver representations for the generalized associahedra associated to the root systems of all Dynkin types (a simply-laced or non-simply-laced). This confirms th…
An important problem that commonly arises in areas such as internet traffic-flow analysis, phylogenetics and electrical circuit design, is to find a representation of any given metric $D$ on a finite set by an edge-weighted graph, such that the total edge length of the graph is minimum over all such graphs. Such a graph is called an optimal realization and finding such realizations is known to be NP-hard. Recently Varone presented a heuristic greedy algorithm for computing optimal realizations. Here we present an alternative heuristic that exploits the relationship between realizations of the …
Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes
本研究将平面N=4超杨-米尔斯的格拉斯曼S矩阵表述推广至CEGM提出的广义双伴随标量振幅，证明m^(3)_n的(n-5)维留数对应完整m^(2)_n偏振幅，提出k≥4情形的推广方案，并预言结合多面体的新闵可夫斯基和实现。
A Bier sphere $Bier(K) = K\ast_ΔK^\circ$, defined as the deleted join of a simplicial complex and its Alexander dual $K^\circ$, is a purely combinatorial object (abstract simplicial complex). Here we study a hidden geometry of Bier spheres by describing their natural geometric realizations, compute their volume, describe an effective criterion for their polytopality, and associate to $K$ a natural fan $Fan(K)$, related to the Braid fan. Along the way we establish a connection of Bier spheres of maximal volume with recent generalizations of the classical Van Kampen-Flores theorem and clarify th…
Realizations of multiassociahedra via bipartite rigidity
本文利用二部图刚性矩阵作为向量配置，实现了多重结合多面体在特定参数下的多胞形与扇构造，给出了刚性拟阵的代数解释，并证明了部分大参数情形不可实现。
Realizations of multiassociahedra via rigidity
本文借助刚性理论与矩曲线构造，证明若干多关联复形可实现为多面体或单纯扇，并给出刚性实现的不可能性边界。
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