Abstract
The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed f-vectors and h-vectors are introduced, and the latter are shown to be nonnegative in the shellable case. Shellable complexes have the homotopy type of a wedge of spheres of various dimensions, and their Stanley-Reisner rings admit a combinatorially induced direct sum decomposition. The technique of lexicographic shellability for posets is similarly extended from pure posets (all maximal chains of the same length) to the general case. Several examples of nonpure lexicographically shellable posets are given, such as the k-equal partition lattice (the intersection lattice of the k-equal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the k-equal arrangement. It also determines the homotopy type of intervals in a Tamari lattice and in the lattice of number partitions ordered by dominance, thus strengthening some known Möbius function formulas. The extension to regular CW complexes is briefly discussed and shown to be related to the concept of lexicographic shellability.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1299-1327 |
| Number of pages | 29 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 348 |
| Issue number | 4 |
| State | Published - 1996 |
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Shellable nonpure complexes and posets. I. / Björner, Anders; Galloway, Michelle L.
In: Transactions of the American Mathematical Society, Vol. 348, No. 4, 1996, p. 1299-1327.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Shellable nonpure complexes and posets. I
AU - Björner, Anders
AU - Galloway, Michelle L
PY - 1996
Y1 - 1996
N2 - The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed f-vectors and h-vectors are introduced, and the latter are shown to be nonnegative in the shellable case. Shellable complexes have the homotopy type of a wedge of spheres of various dimensions, and their Stanley-Reisner rings admit a combinatorially induced direct sum decomposition. The technique of lexicographic shellability for posets is similarly extended from pure posets (all maximal chains of the same length) to the general case. Several examples of nonpure lexicographically shellable posets are given, such as the k-equal partition lattice (the intersection lattice of the k-equal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the k-equal arrangement. It also determines the homotopy type of intervals in a Tamari lattice and in the lattice of number partitions ordered by dominance, thus strengthening some known Möbius function formulas. The extension to regular CW complexes is briefly discussed and shown to be related to the concept of lexicographic shellability.
AB - The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed f-vectors and h-vectors are introduced, and the latter are shown to be nonnegative in the shellable case. Shellable complexes have the homotopy type of a wedge of spheres of various dimensions, and their Stanley-Reisner rings admit a combinatorially induced direct sum decomposition. The technique of lexicographic shellability for posets is similarly extended from pure posets (all maximal chains of the same length) to the general case. Several examples of nonpure lexicographically shellable posets are given, such as the k-equal partition lattice (the intersection lattice of the k-equal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the k-equal arrangement. It also determines the homotopy type of intervals in a Tamari lattice and in the lattice of number partitions ordered by dominance, thus strengthening some known Möbius function formulas. The extension to regular CW complexes is briefly discussed and shown to be related to the concept of lexicographic shellability.
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UR - http://www.scopus.com/inward/citedby.url?scp=0039581354&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:0039581354
VL - 348
SP - 1299
EP - 1327
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 4
ER -