Lattices of Parabolic Subgroups in Connection with Hyperplane Arrangements

Journal of Algebraic Combinatorics - Tập 9 Số 1 - Trang 5-24 - 1999
Barcelo, Hélène1, Ihrig, Edwin1
1Department of Mathematics, Arizona State University, Tempe

Tóm tắt

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.

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Journal of Algebraic Combinatorics

Tập 9 Số 1

5-24

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