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Cyclic polytopes and representation theory

We describe the relationship between triangulations of cyclic polytopes and the higher Auslander algebras of type~$A$, denoted $A_{n}^{d}$. Indeed, triangulations of even-dimensional cyclic polytopes are in bijection with certain silting complexes over these algebras. We show how two partial orders, called the `higher Stasheff--Tamari orders', on the triangulations translate to two orders on silting complexes already known to representation theorists. This then allows us to show that triangulations of odd-dimensional cyclic polytopes are in bijection with equivalences classes of $d$-maximal green sequences over $A_{n}^{d}$. By proving the 1996 conjecture of Edelman and Reiner that the two higher Stasheff--Tamari orders are equal, we obtain new results on the representation theory of these algebras.