135 research outputs found
g-elements, finite buildings and higher Cohen-Macaulay connectivity
The main result is a proof that the g-vector of a simplicial complex with a
convex ear decomposition is an M-vector. This is a generalization of similar
results for matroid complexes. We also show that a finite building has a convex
ear decomposition. This leads to connections between higher Cohen-Macaulay
connectivity and increasing h-vectors.Comment: To appear in JCT A. 20 page
Antichain cutsets of strongly connected posets
Rival and Zaguia showed that the antichain cutsets of a finite Boolean
lattice are exactly the level sets. We show that a similar characterization of
antichain cutsets holds for any strongly connected poset of locally finite
height. As a corollary, we get such a characterization for semimodular
lattices, supersolvable lattices, Bruhat orders, locally shellable lattices,
and many more. We also consider a generalization to strongly connected
hypergraphs having finite edges.Comment: 12 pages; v2 contains minor fixes for publicatio
Chains of modular elements and shellability
Let L be a lattice admitting a left-modular chain of length r, not
necessarily maximal. We show that if either L is graded or the chain is
modular, then the (r-2)-skeleton of L is vertex-decomposable (hence shellable).
This proves a conjecture of Hersh. Under certain circumstances, we can find
shellings of higher skeleta. For instance, if the left-modular chain consists
of every other element of some maximum length chain, then L itself is
shellable. We apply these results to give a new characterization of finite
solvable groups in terms of the topology of subgroup lattices.
Our main tool relaxes the conditions for an EL-labeling, allowing multiple
ascending chains as long as they are lexicographically before non-ascending
chains. We extend results from the theory of EL-shellable posets to such
labelings. The shellability of certain skeleta is one such result. Another is
that a poset with such a labeling is homotopy equivalent (by discrete Morse
theory) to a cell complex with cells in correspondence to weakly descending
chains.Comment: 20 pages, 1 figure; v2 has minor fixes; v3 corrects the technical
lemma in Section 4, and improves the exposition throughou
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