The University of Michigan Combinatorics Seminar
An arrangement of hyperplanes (codimension one subspaces) in a
real vector space cuts the vector space into connected components called
regions. The adjacency graph of the regions can be turned into a poset by
directing it away from some fixed "base region". This "poset of regions"
has been studied by Bjorner, Edelman and Ziegler, who, in particular,
showed that if the arrangement is supersolvable, then for a certain choice
of base region the poset of regions is a lattice. I will sketch a proof
that the order dimension of the poset of regions of a supersolvable
arrangement is exactly the rank of the arrangement (the dimension of the
linear span of the hyperplanes' normal vectors). The weak Bruhat order on
a Coxeter group is the poset of regions of the corresponding Coxeter
arrangement, and for types A and B the arrangements are supersolvable.
So in particular, the order dimension of the weak order on Coxeter groups
of types A and B is the number of generators.