The First Higher Stasheff-Tamari Orders are Quotients of the Higher Bruhat Orders
The First Higher Stasheff-Tamari Orders are Quotients of the Higher Bruhat Orders
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DOI：
10.37236/10877
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发表时间：
2020-12
期刊：
影响因子：
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通讯作者：
Nicholas J. Williams
中科院分区：
文献类型：
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作者：
Nicholas J. Williams
我们证明了Dimakis和m<s:1> ller- hoissen的高阶Tamari阶与第一高阶Stasheff-Tamari阶一致的猜想。为此，我们证明了高阶Tamari阶可以被看作是从高阶Bruhat阶到第一阶Stasheff-Tamari阶的保序映射的像。这张图是通过取一个循环带栖体的一个立方的第一个横截面来定义的。我们提供了一个新的证明，证明了这个映射是满射的，并进一步证明了这个映射是满的，这包含了前面提到的猜想。我们解释了满射和满的保序映射如何对应于偏序集的商。我们的结果将第一高阶Stasheff-Tamari阶与高阶Tamari阶在可积系统中的作用的文献联系起来。
We prove the conjecture that the higher Tamari orders of Dimakis and Müller-Hoissen coincide with the first higher Stasheff-Tamari orders. To this end, we show that the higher Tamari orders may be conceived as the image of an order-preserving map from the higher Bruhat orders to the first higher Stasheff-Tamari orders. This map is defined by taking the first cross-section of a cubillage of a cyclic zonotope. We provide a new proof that this map is surjective and show further that the map is full, which entails the aforementioned conjecture. We explain how order-preserving maps which are surjective and full correspond to quotients of posets. Our results connect the first higher Stasheff-Tamari orders with the literature on the role of the higher Tamari orders in integrable systems.