Masato Kounoike: Poset Connectivity and h-Vectors of Marked Order Polytopes

Abstract: Order polytopes, introduced by Stanley, are convex polytopes naturally associated with finite posets. In this talk, I will study their h-vectors from a poset-theoretic point of view.   The top nontrivial h-coefficient is a global alternating combination of face numbers. Nevertheless, for a finite poset P with d elements, it has the simple formula   h_d(O(P)) = (-1)^{d+1}(c(P) - 1),   where c(P) denotes the number of connected components of the cover graph of P. In particular, this coefficient vanishes whenever P is connected.   I will extend this result to marked order polytopes by introducing a combinatorial invariant that records how connected components change between consecutive marking values. I will also explain some of the ideas behind the proof, based on products, hyperplane cuts, and the face structure of marked order polytopes.   Finally, I will discuss an open question on whether the same formula extends to marked chain-order polytopes.