置换的非循环系与单形的细混合细分
Acyclic systems of permutations and fine mixed subdivisions of simplices
摘要：
对一个大小为n的（d-1）-单形T进行细混合细分，在T的边上产生一个[n]的${d \choose 2}$置换系统，并在T内产生n个单位（d-1）-单形的集合。哪些置换系统和哪些单形集合是这样产生的？Ardila和Billey的扩展单纯形猜想给出了第二个问题的答案。我们提出并证明了第一个问题的答案，无环系统猜想。证明了T的置换系决定T的单形的集合。这建立了非循环系统猜想，作为证明扩展单纯形猜想的第一步。我们使用这种方法来证明这两个代数n=3在任意维。
英文摘要：
A fine mixed subdivision of a (d-1)-simplex T of size n gives rise to a system of ${d \choose 2}$ permutations of [n] on the edges of T, and to a collection of n unit (d-1)-simplices inside T. Which systems of permutations and which collections of simplices arise in this way? The Spread Out Simplices Conjecture of Ardila and Billey proposes an answer to the second question. We propose and give evidence for an answer to the first question, the Acyclic System Conjecture. We prove that the system of permutations of T determines the collection of simplices of T. This establishes the Acyclic System Conjecture as a first step towards proving the Spread Out Simplices Conjecture. We use this approach to prove both conjectures for n=3 in arbitrary dimension.
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