Teemu Lundström: f-vector inequalities for order and chain polytopes

Abstract: From a finite poset one can construct two polytopes called the order and chain polytopes. These polytopes appear naturally in various areas of combinatorics. One thing that makes these polytopes quite special is that one can easily describe both their vertices and facets as corresponding to certain parts of the underlying poset. Much of the study of order and chain polytopes thus reduces to just doing combinatorics on posets. In this talk I will go over some of the basic properties of order and chain polytopes, focusing on their f-vectors. In particular, I will discuss a conjecture made by Hibi and Li which states that the f-vector of an order polytope is coordinate-wise smaller than the f-vector of a chain polytope, both coming from the same poset. In our main theorem we show that the conjecture holds for a certain inductively built family of posets, taking a step towards proving the full conjecture. This is joint work with Ragnar Freij-Hollanti.