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 the first author 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. introduction A convex polytope P ⊂ R is called integral if all of vertices of P belong to Z. Let P ⊂ R be an integral convex polytope of dimension d. Given integers n = 1, 2, . . . , we define the function i(P, n) as follows: i(P, n) := ∣ ∣(nP ∩ Z) ∣ ∣ , where nP = {nα | α ∈ P}. We call i(P, n) the Ehrhart polynomial of P. It is known that i(P, n) is a polynomial in n of degree d with i(P, 0) = 1 (see [3]). Next, we introduce some classes of Fano polytopes. Let P ⊂ R be an integral convex polytope of dimension d. • We say that P is a Fano polytope if the origin of R is the unique integer point belonging to the interior of P. • A Fano polytope is called Gorenstein if its dual polytope is integral. (Recall that the dual polytope P of a Fano polytope P is the convex polytope which consists of those x ∈ R such that 〈x, y〉 ≤ 1 for all y ∈ P, where 〈x, y〉 is the usual inner product of R.) 2010 Mathematics Subject Classification. 13P10, 52B20.