The positive Dressian equals the positive tropical Grassmannian
The positive Dressian equals the positive tropical Grassmannian
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DOI：
10.1090/btran/67
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发表时间：
2020-03
期刊：
影响因子：
--
通讯作者：
David E. Speyer;L. Williams
中科院分区：
文献类型：
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作者：
David E. Speyer;L. Williams
德雷斯型和热带格拉斯曼型参数化抽象和可实现的热带线性空间;但一般来说，德雷斯型要比热带格拉斯曼型大得多。这两种空间都有自然的正概念--正的德雷斯空间和正的热带格拉斯曼空间（我们大约在15年前介绍过）--所以很自然地要问这两种正空间如何比较。在本文中，我们证明了正Dressian等于正热带格拉斯曼。利用超单形的正Dressian和正则正拟阵细分之间的联系，我们使用我们的结果给出了da Silva 1987年猜想的一个新的“热带”证明（2017年由Ardila-Rincon-威廉姆斯首次证明），即所有正定向拟阵都是可实现的。我们还表明，最好的正规正胚细分的超单纯形由系列平行拟阵多面体，并实现平等的施派尔的f-向量定理。最后，我们给出了一个例子的正剖分的超单形是不规则的，并作出连接的理论热带超平面安排。
The Dressian and the tropical Grassmannian parameterize abstract and realizable tropical linear spaces; but in general, the Dressian is much larger than the tropical Grassmannian. There are natural positive notions of both of these spaces -- the positive Dressian, and the positive tropical Grassmannian (which we introduced roughly fifteen years ago) -- so it is natural to ask how these two positive spaces compare. In this paper we show that the positive Dressian equals the positive tropical Grassmannian. Using the connection between the positive Dressian and regular positroidal subdivisions of the hypersimplex, we use our result to give a new "tropical" proof of da Silva's 1987 conjecture (first proved in 2017 by Ardila-Rincon-Williams) that all positively oriented matroids are realizable. We also show that the finest regular positroidal subdivisions of the hypersimplex consist of series-parallel matroid polytopes, and achieve equality in Speyer's f-vector theorem. Finally we give an example of a positroidal subdivision of the hypersimplex which is not regular, and make a connection to the theory of tropical hyperplane arrangements.