Triangulations of Cyclic Polytopes and higher Bruhat Orders

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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.

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