Geodesic
Lattices of parabolic subgroups in connection with hyperplane arrangements
Journal of Algebraic Combinatorics, Tome 9 (1999) no. 1, pp. 5-24.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let W be a Coxeter group acting as a matrix group by way of the dual of the geometric representation. Let L be the lattice of intersections of all reflecting hyperplanes associated with the reflections in this representation. We show that L is isomorphic to the lattice consisting of all parabolic subgroups of W. We use this correspondence to find all W for which L is supersolvable. In particular, we show that the only infinite Coxeter group for which L is supersolvable is the infinite dihedral group. Also, we show how this isomorphism gives an embedding of L into the partition lattice whenever W is of type An, Bn or Dn. In addition, we give several results concerning non-broken circuit bases (NBC bases) when W is finite. We show that L is supersolvable if and only if all NBC bases are obtainable by a certain specific combinatorial procedure, and we use the lattice of parabolic subgroups to identify a natural subcollection of the collection of all NBC bases.
Keywords: hyperplane arrangement, lattice, Coxeter group 
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     title = {Lattices of parabolic subgroups in connection with hyperplane arrangements},
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Barcelo, Hélène; Ihrig, Edwin. Lattices of parabolic subgroups in connection with hyperplane arrangements. Journal of Algebraic Combinatorics, Tome 9 (1999) no. 1, pp. 5-24. http://geodesic.mathdoc.fr/item/JAC_1999__9_1_a3/