Lattices of Parabolic Subgroups in Connection with Hyperplane Arrangements
Hélène Barcelo
and Edwin Ihrig
DOI: 10.1023/A:1018607830027
Abstract
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.
Pages: 5–24
Keywords: hyperplane arrangement; lattice; Coxeter group
Full Text: PDF
References
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2. H. Barcelo and A. Goupil, “Non-broken circuits of reflection groups and factorization in Dn,” Israel Journal of Mathematics 91(1-3) (1995), 285-306.
3. A. Bj\ddot orner and G. Ziegler, “Broken circuit complexes: Factorizations and generalizations,” JCT (B) 51 (1991), 96-126.
4. R.W. Carter, “Conjugacy classes in the Weyl group,” Compositio Mathematica 25 (1972), 1-59.
5. H.S.M. Coxeter, “Regular Complex-Polytopes,” 2nd edition, Cambridge University Press, 1991.
6. James E. Humphreys, “Reflection Groups and Coxeter Groups,” Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, 1990.
7. P. Orlik and H. Terao, “Arrangements of hyperplanes,” Series of Comprehensive Studies in Math. 300, Springer- Verlag, Berlin, Heidelberg, 1992.
8. R. Stanley, “Modular elements of geometric lattices,” Algebra Universalis 1 (1971), 214-217.
9. R. Stanley, “Supersolvable lattices,” Algebra Universalis 2 (1972), 197-217.
10. H. Terao, “Modular elements of lattices and topological fibration,” Advances in Mathematics 62(2) (1986), 135-154.