h-VECTORS OF GORENSTEIN* SIMPLICIAL POSETS

A simplicial poset P (also called a boolean poset and a poset of boolean type) is a finite poset with a smallest element 0̂ such that every interval [0̂, y] for y ∈ P is a boolean algebra, i.e., [0̂, y] is isomorphic to the set of all subsets of a finite set, ordered by inclusion. The set of all faces of a (finite) simplicial complex with empty set added forms a simplicial poset ordered by inclusion, where the empty set is the smallest element. Such a simplicial poset is called the face poset of a simplicial complex, and two simplicial complexes are isomorphic if and only if their face posets are isomorphic. Therefore, a simplicial poset can be thought of as a generalization of a simplicial complex. Although a simplicial poset P is not necessarily the face poset of a simplicial complex, it is always the face poset of a CW-complex Γ(P ). In fact, to each y ∈ P\{0̂} = P , we assign a (geometrical) simplex whose face poset is [0̂, y] and glue those geometrical simplices according to the order relation in P . Then we get the CW-complex Γ(P ) such that all the attaching maps are inclusions. For instance, if two simplicies of a same dimension are identified on their boundaries via the identity map, then it is not a simplicial complex but a CW-complex obtained from a simplicial poset. The CW-complex Γ(P ) has a well-defined barycentric subdivision which is isomorphic to the order complex ∆(P ) of the poset P . Here ∆(P ) is a simplicial complex on the vertex set P whose faces are the chains of P . We say that y ∈ P has rank i if the interval [0̂, y] is isomorphic to the boolean algebra of rank i (in other words, the face poset of an (i−1)-simplex), and the rank of P is defined to be the maximum of ranks of all elements in P . Let d = rankP . In exact analogy to simplicial complexes, the f -vector of the simplicial poset P , (f0, f1, . . . , fd−1), is defined by fi = fi(P ) = ♯{y ∈ P | rank y = i+ 1} and the h-vector of P , (h0, h1, . . . , hd), is defined by the following identity: