Lattices of parabolic subgroups in connection with hyperplane arrangements (Q1283501)

The authors investigate the lattice \(P_W\) of all parabolic subgroups of a Coxeter group \(W\). They first construct a natural isomorphism between \(P_W\) and the lattice \(L_W\) of intersections of reflecting hyperplanes in the dual of the geometric representation of \(W\). In particular, \(P_W\) is a geometric lattice and the authors study the modular elements of this lattice and characterize those Coxeter groups for which \(P_W\) is supersoluble. It turns out that the dihedral groups and the finite groups of type \(A_n\) and \(B_n\) are the only Coxeter groups with this property. Finally, non-broken circuit bases of \(P_W\) and \(L_W\) are studied.