Björner, Anders; Wachs, Michelle L. Generalized quotients in Coxeter groups. (English) Zbl 0659.05007 Trans. Am. Math. Soc. 308, No. 1, 1-37 (1988). 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 Cited in 5 ReviewsCited in 56 Documents MSC: 05A99 Enumerative combinatorics 06F99 Ordered structures 20B99 Permutation groups Keywords:coset representatives; Coxeter group; Bruhat order; weak order; symmetric group; Young tableaux; rooted forest PDFBibTeX XMLCite \textit{A. Björner} and \textit{M. L. Wachs}, Trans. Am. Math. Soc. 308, No. 1, 1--37 (1988; Zbl 0659.05007) Full Text: DOI