2022/12/16 by Nicholas J. Williams, Williams, Nicholas J. · 1 citation
Mathematics · #52B12 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Primary 05E10 #secondary 06A07
paper · pdf · doi:10.48550/arxiv.2212.08422
openalex publication_date 2022/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the relationship discovered by Oppermann and Thomas between triangulations of cyclic polytopes and the higher Auslander algebras of type A, denoted And, is an incredibly rich one. The higher Stasheff--Tamari orders are two orders on triangulations of cyclic polytopes, conjectured to be equivalent, defined in the 1990s by Kapranov and Voevodsky, and Edelman and Reiner. We first show that these orders correspond in even dimensions to natural orders on tilting modules defined by Riedtmann and Schofield and studied by Happel and Unger. This result allows us to show that triangulations of odd-dimensional cyclic polytopes are in bijection with equivalence classes of d-maximal green sequences of And, which we introduce as a higher-dimensional generalisation of the original maximal green sequences of Keller. We further interpret the higher Stasheff--Tamari orders in odd dimensions, where they correspond to natural orders on equivalences classes of d-maximal green sequences. The conjecture that these two partial orders on equivalence classes of d-maximal green sequences are equal amounts to an oriented version of the ``no-gap'' conjecture of Brüstle, Dupont, and Perotin. A corollary of our results is that this conjecture holds for An, and that here the set of equivalence classes of (1-)maximal green sequences is a lattice.