The Weak Order of Coxeter Systems and Combinatorial Properties of Descent Sets

In this paper, we study the weak order of Coxeter systems and the combinatorial properties of descent sets. There are three main results: (1) Given a Coxeter system (W,S), some word v ∈W , some subset A ⊆ S disjoint fromDR(v), and some w ∈WA, we proved that DR(vw) ⊆ DR(v) ∪ A. (2) We obtained an explicit map for A ∪ B to dominate B in the case when A,B are commuting disjoint sets, with B finite. (3) We proved that for finite Coxeter systems (W,S), with subsets A,B ⊆ S, if A dominates B, then B ⊆ A. In particular, the third result is a generalization of a proposition in [K. Nyman, E. Swartz, Inequalities for the h-vectors and flag h-vectors of geometric lattices, Discrete Comput. Geom. 32 (2004) 533-548], while the second result gives a partial answer to one of the problems posed in [E. Swartz, g-elements, finite buildings and higher Cohen-Macaulay connectivity, J. Combin. Theory Ser. A 113 (2006) 1305-1320]. Also, this paper develops the theory of sequences of braid moves, boundary pairs, and tagging letters in reduced expressions for the general Coxeter system (W,S). A side application of inversion tables also yield an explicit formula of a reduced expression for all words in Coxeter systems of type An. All these results are new.