The University of Michigan Combinatorics Seminar
Winter 2002
February 1, 4:10-5:00, 3866 East Hall





Order Dimension of Supersolvable Arrangements

Nathan Reading

University of Minnesota




Abstract

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.

The talk will not require any prior knowledge of order dimension or of supersolvable hyperplane arrangements.