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

TL;DR

This paper proves that the first higher Stasheff--Tamari orders are quotients of the higher Bruhat orders by demonstrating an order-preserving surjective and full map between them, linking combinatorics with integrable systems.

Contribution

It establishes the conjecture that the higher Tamari orders are quotients of the higher Bruhat orders through a new proof of the surjectivity and fullness of a key map.

Findings

The map from higher Bruhat to Stasheff--Tamari orders is surjective.

The map is full, making the orders quotients of each other.

Connects combinatorial orders with integrable systems literature.

Abstract

We prove the conjecture that the higher Tamari orders of Dimakis and M\"uller-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.