g-Elements, finite buildings and higher Cohen-Macaulay connectivity
نویسنده
چکیده
Chari proved that if ∆ is a (d − 1)-dimensional simplicial complex with a convex ear decomposition, then h0 ≤ · · · ≤ hbd/2c [7]. Nyman and Swartz raised the problem of whether or not the corresponding g-vector is an M -vector [18]. This is proved to be true by showing that the set of pairs (ω,Θ), where Θ is a l.s.o.p. for k[∆], the face ring of ∆, and ω is a g-element for k[∆]/Θ, is nonempty whenever the characteristic of k is zero. Finite buildings have a convex ear decomposition. These decompositions point to inequalities of the flag h-vector of such spaces similar in spirit to those examined in [18] for order complexes of geometric lattices. This also leads to connections between higher Cohen-Macaulay connectivity and conditions which insure that h0 < · · · < hi for a predetermined i. One of the most basic combinatorial invariants of a (finite) simplicial complex is its f -vector, or equivalently, its h-vector. In order to analyze h-vectors of matroid independence complexes Chari introduced the notion of a convex ear decomposition [7]. He showed that (d − 1)-dimensional complexes which have such a decomposition satisfy hi ≤ hd−i and hi ≤ hi+1 for all i ≤ bd/2c. In addition, he proved that independence complexes of matroids have a PS-ear decomposition, a special type of convex ear decomposition. Spaces with a PSear decomposition satisfy the additional condition that their g-vector, (g0, g1, . . . , gdd/2e), where gi = hi − hi−1, is an M-vector [27]. Our main result, Theorem 3.9, is that this holds for all spaces with a convex ear decomposition. In section 2 we introduce a convex ear decomposition for finite buildings. In addition to the enumerative conclusions above, this will allow an analysis of the flag h-vector of such complexes. We end with an examination of a connection between higher Cohen-Macaulay connectivity and increasing h-vectors suggested by finite buildings. Throughout, ∆ is a finite (d − 1)-dimensional abstract simplicial complex with vertex set V, |V | = n. A maximal face of ∆ is a facet. Partially supported by NSF grant DMS-0245623. 1
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عنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 113 شماره
صفحات -
تاریخ انتشار 2006