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Generalized quotients in Coxeter groups. (English) Zbl 0659.05007
For (W,S) a Coxeter group, sets of the form $$W/V=\{w\in W| l(wv)=l(w)+l(v)\quad for\quad all\quad v\in V\},\quad V\subseteq W,$$ are called generalized quotients. It is shown that they have rich combinatorial structure under Bruhat order and weak order. It is proved that Bruhat intervals in W/V are lexicographically shellable and that the Möbius function on W/V under Bruhat order takes values in $$\{$$- 1,0,1$$\}$$. Further, W/V is always a complete meet-semilattice and a convex order ideal as a subset of W under weak order.
The latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics; e.g. the set of standard Young tableaux of a fixed shape or the set of linear extensions of a rooted forest are interpreted and studied as generalized quotients.
Reviewer: M.Demlová

##### MSC:
 05A99 Enumerative combinatorics 06F99 Ordered structures 20B99 Permutation groups
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