New interpretations of the higher Stasheff--Tamari orders

TL;DR

This paper connects higher Stasheff--Tamari orders with the representation theory of higher Auslander algebras of type A, providing new combinatorial and algebraic insights into triangulations of cyclic polytopes.

Contribution

It introduces algebraic interpretations of the higher Stasheff--Tamari orders and relates triangulations to representation theory, including maximal green sequences.

Findings

Orders arise naturally in representation theory of $A_{n}^{d}$

Triangulations correspond to equivalence classes of maximal green sequences

New properties characterize $(2d+1)$-dimensional triangulations

Abstract

In 1996, Edelman and Reiner defined the two higher Stasheff--Tamari orders on triangulations of cyclic polytopes and conjectured them to coincide. We open up an algebraic angle for approaching this conjecture by showing how these orders arise naturally in the representation theory of the higher Auslander algebras of type $A$, denoted $A_{n}^{d}$. For this we give new combinatorial interpretations of the orders, making them comparable. We then translate these combinatorial interpretations into the algebraic framework. We also show how triangulations of odd-dimensional cyclic polytopes arise in the representation theory of $A_{n}^{d}$, namely as equivalence classes of maximal green sequences. We furthermore give the odd-dimensional counterpart to the known description of $2d$-dimensional triangulations as sets of non-intersecting $d$-simplices of a maximal size. This consists in a definition of two new properties which imply that a set of $d$-simplices produces a $(2d+1)$-dimensional triangulation.