  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.
