University of Georgia
Mathematics Department Colloquium
Richard Ehrenborg
Royal Institute of Technology
Monday, February 14, 2000
Inequalities for the cd-index
Abstract:
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.
Surprisingly, there is no full description of the
linear inequalities holding between the flag
f-vector entries.
The cd-index, conjectured by Fine and proved by
Bayer and Klapper, gives an explicit basis for
the subspace spanned by the generalized
Dehn-Sommerville relations. It offers an efficient
way to encode the flag f-vector of a polytope and
it is emerging as an important tool to understand
the flag f-vector.
We prove an inequality involving the cd-indices of
a convex polytope P, a face F of the polytope and the
link P/F. As a consequence we settle a conjecture of
Stanley that the cd-index of d-dimensional polytopes
is minimized on the d-dimensional simplex. Moreover,
we present an upper bound theorem for the cd-index
of polytopes.
Lastly, we will discuss the current state of
inequalities for flag vectors of 4-dimensional
polytopes.