Journal of Algebraic Combinatorics

, Volume 9, Issue 1, pp 5–24

Lattices of Parabolic Subgroups in Connection with Hyperplane Arrangements

  • Hélène Barcelo
  • Edwin Ihrig

DOI: 10.1023/A:1018607830027

Cite this article as:
Barcelo, H. & Ihrig, E. Journal of Algebraic Combinatorics (1999) 9: 5. doi:10.1023/A:1018607830027


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.

hyperplane arrangement lattice Coxeter group 

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Hélène Barcelo
    • 1
  • Edwin Ihrig
    • 1
  1. 1.Department of MathematicsArizona State UniversityTempe

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