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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Convex-Ear Decompositions and the Flag h-Vector. | Convex-Ear Decompositions and the Flag h-Vector Jay Schweig University of Kansas Kansas U.S.A. j schweig@math.ku.edu Submitted Jun 10 2010 Accepted Dec 13 2010 Published Jan 5 2011 Mathematics Subject Classification 05E45 06F30 52B22 Abstract We prove a theorem allowing us to find convex-ear decompositions for rank-selected subposets of posets that are unions of Boolean sublattices in a coherent fashion. We then apply this theorem to geometric lattices and face posets of shellable complexes obtaining new inequalities for their h-vectors. Finally we use the latter decomposition to give a new interpretation to inequalities satisfied by the flag h-vectors of face posets of Cohen-Macaulay complexes. 1 Introduction The f-vector of a finite simplicial complex A which counts the number of faces of the complex in each dimension is arguably its most fundamental invariant. The h-vector of A is the image of its f-vector under an invertible transformation. Somewhat surprisingly properties of a complex s f-vector are sometimes better expressed through its h-vector. A good example of this phenomenon are the Dehn-Sommerville relations see for instance 17 which state that the h-vector of a simplicial polytope boundary is symmetric. The main complexes we study in this paper are all order complexes namely complexes whose simplices correspond to chains in posets. Since a poset and its order complex hold the same information we often refer to them interchangeably. E.g. we may speak of the facets or h-vector of a poset or to a chain in an order complex. Convex-ear decompositions were first introduced by Chari in 3 . Heuristically a complex admits a convex-ear decomposition if it is a union of simplicial polytope boundaries which fit together coherently see Definition 2.3 . Suppose a d 1 -dimensional complex A admits such a decomposition. In 3 Chari shows that the h-vector ho hl . hd of A satisfies for i d 2 hi hi 1 and hi hd-i. In 15 Swartz shows that A is 2-CM and that h0 h1 h0 h2 h1
