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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Shellable and Cohen-Macaulay partially ordered sets


Author: Anders Björner
Journal: Trans. Amer. Math. Soc. 260 (1980), 159-183
MSC: Primary 06A10; Secondary 13H10, 52A25
DOI: https://doi.org/10.1090/S0002-9947-1980-0570784-2
MathSciNet review: 570784
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Abstract: In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to R. Stanley, is generalized to posets and shown to imply shellability, while Stanley's main theorem on the Jordan-HÖlder sequences of such labelings remains valid. Further, we show a number of ways in which shellable posets can be constructed from other shellable posets and complexes. These results give rise to several new examples of Cohen-Macaulay posets. For instance, the lattice of subgroups of a finite group G is Cohen-Macaulay (in fact shellable) if and only if G is supersolvable. Finally, it is shown that all the higher order complexes of a finite planar distributive lattice are shellable.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0570784-2
Keywords: Shellable complex, Cohen-Macaulay complex, poset, lattice, Möbius function, admissible lattice, supersolvable group, barycentric subdivision, convex polytope, locally semimodular poset, Hirsch conjecture, planar distributive lattice
Article copyright: © Copyright 1980 American Mathematical Society