Higher-dimensional combinatorics in representation theory

Williams, Nicholas (2022) Higher-dimensional combinatorics in representation theory. PhD thesis, Universität Kõln.

Abstract

The main result of this thesis is that the two higher Stasheff-Tamari orders are equal, as was originally conjectured by Edelman and Reiner in 1996. These are two orders on the set of triangulations of a cyclic polytope - the first introduced by Kapranov and Voevodsky, and the second introduced by Edelman and Reiner. Our first step in proving the conjecture is to give new combinatorial interpretations of the higher Stasheff-Tamari orders which make them easier to compare. As a necessary prequel to these combinatorial interpretations, we characterise triangulations of (2d + 1)-dimensional cyclic polytopes in terms of their d-simplices. The proof itself is then by induction on the number of vertices of the cyclic polytope. As a technical tool for this proof, we develop a theory for expanding triangulations of cyclic polytopes at any vertex, which is of independent interest. We apply our results in representation theory of algebras, building on the work of Oppermann and Thomas, who show how triangulations of even-dimensional cyclic polytopes arise in the representation theory of the higher Auslander algebras of type A. Indeed, triangulations of even-dimensional cyclic polytopes are in bijection with both tilting modules and cluster-tilting objects. We choose to work in the slightly different framework of d-silting complexes, which we show are also in bijection with triangulations of even-dimensional cyclic polytopes. We show that the higher Stasheff-Tamari orders in even dimensions correspond to natural orders on d-silting complexes, which originally arose in the work of Riedtmann and Schofield concerning partial orders on tilting modules. This algebraic interpretation of the even-dimensional orders allows us to show that odd-dimensional triangulations correspond to equivalence classes of d-maximal green sequences, which we introduce as the higher-dimensional versions of classical maximal green sequences. We are then able to interpret the higher Stasheff-Tamari orders on equivalence classes of d-maximal green sequences. The orders obtained are very natural, but have not been studied before. The equivalence of the higher Stasheff-Tamari orders shows that these algebraic orders are equal for the higher Auslander algebras of type A. We prove a pair of results on mutation, one on mutating cluster-tilting objects in higher cluster categories and the other on mutating triangulations of even-dimensional cyclic polytopes. The criterion for mutating triangulations works by associating quivers to the triangulations. These quivers originate from the cluster-tilting objects which correspond to the triangulations. We further use these quivers to characterise 2d-dimensional triangulations which do not possess any interior (d + 1)-simplices. Finally, another open question we resolve comes from Dimakis and Mueller-Hoissen. These authors introduce orders known as 'the higher Tamari orders' in the context of studying KP solitons. We show that, as conjectured, these are indeed the same posets as the higher Stasheff-Tamari orders. Since the higher Tamari orders are explicitly defined as a quotient of the higher Bruhat orders, this provides a quotient map from the higher Bruhat orders to the higher Stasheff-Tamari orders. Indeed to make this precise, we develop some new theory concerning quotient posets.