×

Orderings of Coxeter groups. (English) Zbl 0594.20029

Combinatorics and algebra, Proc. Conf., Boulder/Colo. 1983, Contemp. Math. 34, 175-195 (1984).
[For the entire collection see Zbl 0546.00008.]
If (W,S) is a Coxeter group, the length \(\ell (W)\) of \(w\in W\) is the least k such that \(w=s_ 1s_ 2...s_ k\), all \(s_ i\in S\). Such a minimal length expression \(s_ 1s_ 2...s_ k\) for w is called a reduced decomposition. Let \(T=\{wsw^{-1}|\) \(w\in W\), \(s\in S\}\). If there exist \(t_ 1,t_ 2,...,t_ k\in T\) such that \(\ell (ut_ 1t_ 2...t_ i)=\ell (u)+i\) for \(1\leq i\leq k\), \(ut_ 1t_ 2...t_ k=w\) and u,w\(\in W\), it is said that u precedes w in the strong ordering, written \(u\leq w\). Similarly it is said to u precedes w in the weak ordering, written \(u\preccurlyeq w\), if there exist \(s_ 1,s_ 2,...,s_ k\in S\) such that \(\ell (us_ 1...s_ i)=\ell (u)+i\) for \(1\leq i\leq k\) and \(us_ 1s_ 2...s_ k=w\). Clearly, \(u\preccurlyeq w\) implies \(u\leq w\). Both these orderings are naturally suggested by associated geometric structures. In this paper, the author reports some discoveries about the general structure of the two partial orderings of Coxeter groups.
Reviewer: K.Otsuka

MSC:

20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups

Citations:

Zbl 0546.00008