Thumbnail Image

Triangulations of Cyclic Polytopes and higher Bruhat Orders

Rambau, Jörg

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

Recently Edelman & Reiner} suggested two poset structures S}1(n,d) and S}2(n,d) on the set of all triangulations of the cyclic d-polytope C(n,d) with n vertices. Both posets are generalizations of the well-studied Tamari lattice. While S}2(n,d) is bounded by definition, the same is not obvious for S}1(n,d). In the paper by Edelman & Reiner} the bounds of S}2(n,d) were also confirmed for S}1(n,d) whenever d \le 5, leaving the general case as a conjecture. In this paper their conjecture is answered in the affirmative for all~d, using several new functorial constructions. Moreover, a structure theorem is presented, stating that the elements of S}1(n,d+1) are in one-to-one correspondence to certain equivalence classes of maximal chains in S}1(n,d). In order to clarify the connection between S}1(n,d) and the higher Bruhat order B}(n-2,d-1) of Manin & Schechtman}, we define an order-preserving map from B}(n-2,d-1) to S}1(n,d), thereby concretizing a result by Kapranov & Voevodsky} in the theory of ordered n-categories.