Chains of modular elements and shellability

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:119

Article

Woodroofe, Russ

关键词: Shellable;    Modular;    Lattice;    Left modular;    Discrete Morse theory;    Subgroup lattice;   

DOI  :  10.1016/j.jcta.2012.03.009

来源: Elsevier

【 摘 要 】

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. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free