Distributive coset graphs of finite Coxeter groups

Götz Pfeiffer, Gerhard Röhrle
2003 Journal of group theroy  
Let W be a finite Coxeter group, W J a parabolic subgroup of W and X J the set of distinguished coset representatives of W J in W equipped with the induced weak Bruhat ordering of W. All instances when X J is a distributive lattice are known. In this note we present a new short conceptual proof of this result. s −→ y whenever y = xs and l(y) = l(x) + 1 for x, y ∈ X J and s ∈ S. Thus, Γ J is the labelled Hasse diagram of the poset (X J , ). Note that Γ ∅ is the Cayley graph of W with respect to
more » ... W with respect to the generating set S. Remark. In terms of the coset graph Γ J , a Mackey decomposition of X J with respect to K ⊆ S ([6, (2.1.9)]) can be visualised by omitting the edges not labelled
doi:10.1515/jgth.2003.021 fatcat:mxnm7bgrhjektpfz7qu76k4isu