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

Abstract

We prove two related conjectures concerning the higher Bruhat orders and the first higher Stasheff-Tamari orders. Namely, we prove the conjecture of Danilov, Karzanov, and Koshevoy that every triangulation of a cyclic polytope arises as a vertex figure of a cubillage of a cyclic zonotope. This gives an order-preserving surjection from the higher Bruhat orders to the first higher Stasheff-Tamari orders. We then go further by showing that this map is full, which proves the conjecture of Dimakis and Müller-Hoissen that the higher Tamari orders are the same posets as the first higher Stasheff-Tamari orders. We show that order-preserving maps which are surjective and full correspond to quotients of posets. This notion of quotient posets is more general than those that have been considered previously. The equivalence of the first higher Stasheff-Tamari orders and the higher Tamari orders entails that the first higher Stasheff-Tamari orders play a role in the theories of KP solitons and polygon equations from mathematical physics. Contents 1. Introduction 1 Acknowledgements 3 2. Terminology and conventions 3 3. Background 4 3.1. Higher Bruhat orders 4 3.2. Higher Stasheff-Tamari orders 8 4. Interpretations 9 4.1. Cubillages 9 4.2. Separated collections 12 4.3. Admissible orders 13 5. Quotient maps of posets 15 6. Surjectivity 16 7. Fullness 20 References 27

The two higher Stasheff-Tamari orders are equal

“…We conclude this section by noting that it suffices to show explicitly that the two orders are equal for one of either odd dimensions or even dimensions. The other parity can then be deduced for the parity that has been shown explicitly, in a similar manner to the method used in [Wil20b]. This provides a shorter proof than giving an explicit proof for both parities.…”

“…The example of this par excellence is the application of triangulations of cyclic polytopes to define higher Segal spaces [DK19], see also [Pog17;DJW19]. In integrable systems, regular triangulations of cyclic polytopes describe the evolution of a class of solitary waves modelled by the Kadomtsev-Petviashvili equation [DM12;Wil20b], see also [Hua15;KK21;GPW19]. The amplituhedron of Arkani-Hamed and Trnka [AT14] is a cyclic polytope for particular values of its parameters.…”