# Polytopes and Hyperplane Arrangements

### Richard Ehrenborg

### Mathematics Department

Cornell

### Monday, February 23, 1998

4:00 PM

### Room 102, Bradley Hall

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.**