MANY NON-EQUIVALENT REALIZATIONS OF THE ASSOCIAHEDRON
MANY NON-EQUIVALENT REALIZATIONS OF THE ASSOCIAHEDRON
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DOI：
10.1007/s00493-014-2959-9
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发表时间：
2015-10-01
期刊：
影响因子：
1.1
通讯作者：
Ziegler, Guenter M.
中科院分区：
文献类型：
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作者：
Ceballos, Cesar;Santos, Francisco;Ziegler, Guenter M.
Hohlweg 和 Lange (2007) 以及 Santos (2004，未发表) 发现了两种不同的方法来构造具有 {0, +/- 1}(n) 中的法向量的 n 维关联面体的指数族实现，概括了 Loday (2004) 和 Chapoton-Fomin-Zelevinsky (2002) 的构造。我们对通过这些结构获得的正常扇形模线性等价的关联面体进行分类，并特别表明，使用这两种方法可以获得的唯一实现是 Chapoton-Fomin-Zelevinsky (2002) 关联面体。对于 Hohlweg-Lange 关联面体，我们的分类先验地比正常扇形的等距分类更粗，通过伯杰龙-霍尔韦格-朗格-托马斯 (2009)。然而，两者产生相同的类。因此，我们得到两个 Hohlweg-Lange 联面体具有线性等价的法向扇形，当且仅当它们是等距的。桑托斯构造，它产生了一个更大的联面体族，在这里首次出现在印刷品中。除了详细描述它之外，我们还将它与 C 簇复合体和 A 型簇代数中的分母扇形联系起来。关联面体的第三个经典构造，作为凸 n 边形的次多胞形（Gelfand-Kapranov-Zelevinsky，1990），被证明永远不会产生与其他两种构造中任何一个构造线性等效的法向扇形。
Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two different ways of constructing exponential families of realizations of the n-dimensional associahedron with normal vectors in {0, +/- 1}(n),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 is a priori coarser than the classification up to isometry of normal fans, by Bergeron-Hohlweg-Lange-Thomas (2009). However, both yield the same classes. As a consequence, we get that two Hohlweg-Lange associahedra have linearly equivalent normal fans if and only if they are isometric.The Santos construction, which produces an even larger family of associahedra, appears here in print for the first time. Apart of describing it in detail we relate it with the c-cluster complexes and the denominator fans in cluster algebras of type A.A third classical construction of the associahedron, as the secondary polytope of a convex n-gon (Gelfand-Kapranov-Zelevinsky, 1990), is shown to never produce a normal fan linearly equivalent to any of the other two constructions.