Zonotopes associated with higher Bruhat orders Original Research Article

The higher Bruhat orders B(n,k) are combinatorially defined partial orders (and hence graphs) that “look like” the graphs of (n−k)-dimensional zonotopes—and they are for small parameters. Here we explain that this is since they contain the graphs of zonotopes of this dimension, but that in general they are not covered by these zonotopal graphs, and they are not polytopal in general. As a special case, this applies to the graph Gn of all arrangements of n pseudolines connected by flips, since this graph is the graph of the higher Bruhat order B(n,2).