Chains of modular elements and shellability

Details Chains of modular elements and shellability Chains of modular elements and shellability

2011-04-05

arxiv.org/abs/1104.0936v3

J. Combin. Theory Ser. A. 119 (2012), no. 6, 1315--1327

Mathematics

Combinatorics

20 pages, 1 figure; v2 has minor fixes; v3 corrects the technical lemma in Section 4, and improves the exposition throughout

Scientific paper

10.1016/j.jcta.2012.03.009

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.

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