Repository: DepositOnce – institutional repository for research data and publications of TU Berlin https://depositonce.tu-berlin.de
TY - RPRT
AU - Rambau, Jörg
PY - 1996
TI - Triangulations of Cyclic Polytopes and higher Bruhat Orders
DO - 10.14279/depositonce-14367
UR - http://dx.doi.org/10.14279/depositonce-14367
PB - Technische Universität Berlin
LA - en
AB - 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.
KW - bruhat order
KW - cyclic polytopes
KW - edelman reiner
KW - several new functorial construction
KW - maximal chain
KW - similar method
KW - structure theorem
KW - well-studied tamari lattice
KW - kapranov voevod-sky
KW - general case
KW - order-preserving map
KW - one-to-one correspondence
KW - manin schechtman
KW - ordered n-categories
KW - certain equivalence class
KW - poset structure
KW - cyclic d-polytope
ER -