Mathematics Colloquia and Seminars

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Description

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.