Recall that the f-vector enumerates the number of faces of a polytope according to dimension, that is, f_i is the number of faces of dimension i. The flag f-vector is a refinement of the f-vector which counts flags of faces in the polytope. There are linear relations between the entries of the flag f-vector known as the generalized Dehn-Sommerville relations. Hence it would be interesting to have an explicit basis for the subspace spanned by these relations.
The cd-index, conjectured by Fine and proved by Bayer and Klapper, gives such a basis. It offers an efficient way to encode the flag f-vector of a polytope. Very little is known about the cd-index of a general polytope. Recent developments based on coalgebra techniques have shown how the cd-index changes under geometric operations.
We will present an explicit method to compute the cd-index of zonotopes, a subclass of polytopes. This result gives the explicit connection, first proven to exist by Bayer and Sturmfels, between the flag vector of the lattice of regions and the flag vector of the intersection lattice of a hyperplane arrangement.
This is joint work with L. J. Billera and M. Readdy.
The talk will be accessible to a general audience.