The two higher Stasheff–Tamari orders are equal

Abstract

The set of triangulations of a cyclic polytope possesses two a priori different partial orders, known as the higher Stasheff–Tamari orders. The first of these orders was introduced by Kapranov and Voevodsky, while the second order was introduced by Edelman and Reiner, who in 1996 also conjectured the two to coincide. In this paper, we prove their conjecture, thereby substantially increasing our understanding of these orders. This result also has ramifications in the representation theory of algebras, as established in previous work of the author. Indeed, it means that the two corresponding orders on tilting modules, $d$-silting complexes, and their maximal chains are equal for the higher Auslander algebras of type $A$.