The F -triangle of the generalised cluster complex, Topics in discrete mathematics
The F -triangle of the generalised cluster complex, Topics in discrete mathematics
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广义簇复形的 F 三角形，离散数学主题
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发表时间：
2006
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通讯作者：
Camille Jordan
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作者：
Camille Jordan
F三角形是Fomin和阅读的广义簇复合体的精细面数。我们计算的F-三角形明确所有不可约的有限根系。此外，我们使用这些结果来部分地证明阿姆斯特朗的“M = F猜想“，该猜想预言了F-三角形和与有限根系相关联的m-可分分偏序集的Möbius函数之间的惊人关系。1.导论. Fomin和Zelevinsky在[11]中发明了簇代数，开创了一个新的令人兴奋的研究领域。[12]中有限型簇代数的分类表明有限型簇代数与有限根系之间存在一一对应。此外，对于每个有限根系Φ，Fomin和Zelevinsky [13]定义了一个对应于相关簇代数的单纯复形，簇复形φ（Φ）。这是一个在根集合Φ的子集上的单纯复形。正如他们所展示的，这种复合物具有许多显着的特性。特别地，刻面的数目由根系Φ的Catalan数给出，并且，此外，所有的面数由优美的乘积公式给出。Chapoton在[8]中发现了更显著的（最初是几何的）性质。在这篇论文中，他将面枚举细化为他所谓的“F三角形”。“他计算了所有类型的F三角形（并在[8]中部分揭示了他的发现），并观察到F三角形和与Φ相关的非交叉划分格N C（Φ）的莫比乌斯函数之间的惊人关系（见[8，猜想1]），后者是由于Bessis [4]和布雷迪和瓦特[5]。这种关系，我们将在续集中称为“F = M猜想“，最近已由Athanasiadis证明[2]。关于F-三角形的更多有趣的性质，请参见[8]。
The F-triangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the F-triangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the " M = F Conjecture " of Armstrong which predicts a surprising relation between the F-triangle and the Möbius function of his m-divisible partition poset associated to a finite root system. 1. Introduction. Fomin and Zelevinsky created a new exciting research field when they invented cluster algebras in [11]. The classification of cluster algebras of finite type from [12] says that there is a one-to-one correspondence between finite-type cluster algebras and finite root systems. Furthermore, for each finite root system Φ, Fomin and Zelevinsky [13] defined a simplicial complex corresponding to the associated cluster algebra , the cluster complex ∆(Φ). This is a simplicial complex on a subset of the set of roots Φ. As they showed, this complex has many remarkable properties. In particular, the number of facets is given by the Catalan number for the root system Φ, and, moreover, all the face numbers are given by elegant product formulae. Further remarkable (originally, conjectural) properties have been discovered by Chapoton in [8]. In this paper, he refines the face enumeration to, what he calls, the " F-triangle. " He computed the F-triangle for all types (and revealed his findings partially in [8]) and observed a surprising relationship (see [8, Conjecture 1]) between the F-triangle and the Möbius function of the non-crossing partition lattice N C(Φ) associated to Φ, the latter being due to Bessis [4] and Brady and Watt [5]. This relationship, to which we shall refer in the sequel as the " F = M Conjecture , " has been recently proved by Athanasiadis [2]. For further fascinating properties of the F-triangle see [8].