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On finite Coxeter lattices. (Sur les treillis de Coxeter finis.) (French) Zbl 0802.06016

Summary: A. Björner [Contemp. Math. 34, 175-195 (1984; Zbl 0594.20029)] has pointed out that the weak Bruhat order of a finite Coxeter group is a lattice. In the case of the symmetric group \(S_ n\), this result (permutohedron lattice) was proved by G. T. Guilbaud and P. Rosenstiehl [Math. Sci. Hum. 4, 9-33 (1963)].
In this paper we show that several known properties of the permutohedron lattices hold for any Coxeter lattice. Especially we show that Coxeter lattices are pseudocomplemented, so that any Coxeter lattice has a congruence whose quotient is a Boolean algebra. L. Solomon’s result [J. Algebra 41, 255-268 (1976; Zbl 0355.20007)] on a subalgebra of the group algebra concerns the same congruence.
In the case of the permutohedron, the equivalence “same first Young’s tableau” is a subequivalence of that equivalence associated with the congruence. Using the pseudocomplementation and an isomorphism property of intervals, we show that Coxeter lattices are semidistributive.
Properties that hold for every Coxeter lattice are especially interesting in the permutohedron case.

MSC:

06F15 Ordered groups
06D15 Pseudocomplemented lattices
20F65 Geometric group theory
20F60 Ordered groups (group-theoretic aspects)
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