GORENSTEIN FANO POLYTOPES ARISING FROM ORDER POLYTOPES AND CHAIN POLYTOPES

Richard Stanley introduced the order polytope 𝒪(P) and the chain polytope 𝒞(P) arising from a finite partially ordered set P, and showed that the Ehrhart polynomial of 𝒪(P) is equal to that of 𝒞(P). In addition, the unimodular equivalence problem of 𝒪(P) and 𝒞(P) was studied by Hibi and Nan Li. In the present paper, three integral convex polytopes Γ(𝒪(P),−𝒪(Q)), Γ(𝒪(P),−𝒞(Q)) and Γ(𝒞(P),−𝒞(Q)), where P and Q are partially ordered sets with |P|=|Q|, will be studied. First, it will be shown that the Ehrhart polynomial of Γ(𝒪(P),−𝒞(Q)) coincides with that of Γ(𝒞(P),−𝒞(Q)). Furthermore, when P and Q possess a common linear extension, it will be proved that these three convex polytopes have the same Ehrhart polynomial. Second, the problem of characterizing partially ordered sets P and Q for which Γ(𝒪(P),−𝒪(Q)) or Γ(𝒪(P),−𝒞(Q)) or Γ(𝒞(P),−𝒞(Q)) is a smooth Fano polytope will be solved. Finally, when these three polytopes are smooth Fano polytopes, the unimodular equivalence problem of these three polytopes will be discussed.