The first higher Stasheff-Tamari orders are quotients of the higher Bruhat orders
The electronic journal of combinatorics, Tome 30 (2023) no. 1
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.
@article{10_37236_10877,
author = {Nicholas J. Williams},
title = {The first higher {Stasheff-Tamari} orders are quotients of the higher {Bruhat} orders},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {1},
doi = {10.37236/10877},
zbl = {1515.06006},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10877/}
}
Nicholas J. Williams. The first higher Stasheff-Tamari orders are quotients of the higher Bruhat orders. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/10877
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