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Cayley lattices of finite Coxeter groups are bounded. (English) Zbl 1097.06001

Given a poset \(P\) and an interval \(I\) in \(P\), an interval doubling \(P(I)\) replaces \(I\times\{0\leq1\}\) maintaining the order \(x< I\) by \(x< I\times\{0\}\) and \(x> I\) by \(x> I\times\{1\}\). If \(P\) can be obtained from \(\{0\leq 1\}\) by a finite sequence of interval doublings, then it is bounded. In this useful paper a class of lattices HH is introduced, probably of importance in its own right, whose elements are shown to be bounded. Coxeter lattices are obtained from the transitive closures of the Cayley graphs of finite Coxeter groups via the weak order for general Coxeter groups. They are shown to be members of HH, which makes HH an interesting class of lattices ipso facto and bounded. The relationship between construction and deconstruction as in HH can be observed on the Coxeter lattices with meaningful interpretations in terms of reflections and parabolic subgroups whose lattices may each be reached from the over-arching lattice by a series of interval contractions as shown. Due to the intimate connections between Coxeter groups and geometry, the notions and results introduced and obtained here also yield new interpretations via these connections for even greater interest as noted by the authors.

MSC:

06A07 Combinatorics of partially ordered sets
20F55 Reflection and Coxeter groups (group-theoretic aspects)
06B05 Structure theory of lattices
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