摘要翻译：
推广了Propp、Fomin和Zelevinsky、Speyer、Fock和Goncharov研究的八面体递推和Fomin和Zelevinsky、Carroll和Speyer研究的三维立方体递推。这种递推的状态由带菱形的多边形的倾角来索引，而递推中的变量由这些倾角的顶点来索引。我们通过执行基本的翻转从递归的一个状态旅行到另一个状态。我们证明了递推的值与我们执行翻转的顺序无关；这个证明涉及关于菱形镶嵌的非平凡组合结果，这些结果可能是独立感兴趣的。然后我们证明了多维立方体递推呈现出洛朗现象--任意变量在其它变量中都是由一个洛朗多项式给出的。我们认识到多维立方体递推的一个特例，给出了各向同性Grassmannian IG(n-1,2n)的显式方程。最后，我们描述了多维立方体递推的一个热带版本，并证明了与热带八面体递推一样，它传播某些线性不等式。
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英文标题：
《The Multidimensional Cube Recurrence》
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作者：
Andre Henriques and David E. Speyer
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最新提交年份：
2009
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分类信息：

一级分类：Mathematics        数学
二级分类：Combinatorics        组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the three-dimensional cube recurrence studied by Fomin and Zelevinsky, and Carroll and Speyer. The states of this recurrence are indexed by tilings of a polygon with rhombi, and the variables in the recurrence are indexed by vertices of these tilings. We travel from one state of the recurrence to another by performing elementary flips. We show that the values of the recurrence are independent of the order in which we perform the flips; this proof involves nontrivial combinatorial results about rhombus tilings which may be of independent interest. We then show that the multidimensional cube recurrence exhibits the Laurent phenomenon -- any variable is given by a Laurent polynomial in the other variables. We recognize a special case of the multidimensional cube recurrence as giving explicit equations for the isotropic Grassmannians IG(n-1,2n). Finally, we describe a tropical version of the multidimensional cube recurrence and show that, like the tropical octahedron recurrence, it propagates certain linear inequalities.
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PDF链接：
https://arxiv.org/pdf/0708.2478