From Permutahedron to Associahedron

Polytopal realizations of generalized associahedra

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.

有限型 $\mathbf{g}$-向量扇的多面体实现

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。

Coxeter群W的Cambrian格及其与广义associahedron的联系

Cambrian Lattices

本文定义了Coxeter群W的Cambrian格，通过弱序的商格定义，并探讨其与广义associahedron的联系，通过纤维多面体构造给出了类型A和B的格的组合实现，并证明其与广义associahedron的正常扇的同构性。

Fan realizations for some 2-associahedra

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…

寒武纪扇形

Cambrian fans

本文研究了Coxeter群W的c-Cambrian扇形，证明其与广义associahedron的正常扇形在组合上同构，并探讨了c-Cambrian扇形与c-簇的线性同构关系。

The tropical totally positive Grassmannian

Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). The theory of total positivity is a natural generalization of the study of matrices with all minors positive. In this paper we introduce the totally positive part of the tropicalization of an arbitrary affine variety, an object which has the structure of a polyhedral fan. We then investigate the case of the Grassmannian, denoting the resulting fan \Trop^+ Gr_{k,n}. We show that \Trop^+ Gr_{2,n} is the Stanley-Pitman fan, which is combinatorially the fan dual to the (type A_{n-3}) associahedron, and that \Trop^+ G…

图嵌套复形的相容性扇

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。

多重三角剖分与热带Pfaffian

Multitriangulations and tropical Pfaffians

本文研究了$k$-结合多面体与Pfaffian簇的联系，通过刻画Gröbner锥证明$k$-三角剖分为代数拟阵的基，并揭示了其热带几何嵌入及与簇代数$\mathbf{g}$-向量扇的同构关系。

ABHY 结合多面体与有限型簇代数的 $F$-多项式的牛顿多面体

ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras

本文将ABHY结合多面体新构造推广至任意单标Dynkin类型的广义结合多面体，并发现该构造同样可生成对应簇代数所有 $F$-多项式的牛顿多面体，同时证明了g-向量扇相关环簇的nef锥具有单纯性。

Cluster algebras II: Finite type classification

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin. The combinatorial structure behind a cluster algebra of finite type is captured by its cluster…

有限型簇代数的关联多面体与 $\mathbf{g}$-向量的最小关系

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。