TRIANGULATIONS OF CYCLIC POLYTOPES AND HIGHER BRUHAT ORDERS

Recently Edelman and Reiner suggested two poset structures S-1(n, d) a nd 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 and Reiner the bounds of S-2(n, d) were also confirmed for S-1(n, d) whenever d l ess than or equal to 5, leaving the general case as a conjecture. In t his paper their conjecture is answered in the affirmative for all d, u sing several new functorial constructions. Moreover, a structure theor em is presented, stating that the elements of S-1(n, d+1) are in one-t o-one correspondence to certain equivalence classes of maximal chains of 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-2, d-1) of Manin and Schechtman, we define an order-preserving map from B(n-2, d-1) to S-1 (n, d), thereby concretizing a result by Kapranov and Voevodsky in the theory of ordered n-categories.