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arXiv 2012.103712020-12math.CO

The first higher Stasheff-Tamari orders are quotients of the higher Bruhat orders

Nicholas J. Williams

备注：: 42 pages (1.35x line spacing), 7 figures. v2: added references and improved notation in final two sections, along with other minor changes. v3: edited paper to reflect discovery that surjectivity was already known; changed formatting

英文摘要：

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.

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