Abstract

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 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). By similar methods it is proved that all triangulations of cyclic polytopes are shellable. In order to clarify the connection between S 1 (n;d) and the higher Bruhat order B(n \Gamma 2;d \Gamma 1) of MANIN & SCHECHTMAN, we define an order-preserving map from B(n \Gamma 2;d \Gamma 1) to S 1 (n;d), thereby concretizing a result by KAPRANOV & VOEVOD-SKY in the theory of ordered n-categories.

Citations