Patricia Hersh

Discrete Morse functions from lexicographic orders

This paper shows how to construct a discrete Morse function with a relatively small number of critical cells for the order complex of any finite poset with O and I from any lexicographic order on its maximal chains. Specifically, if we attach facets according to the lexicographic order on maximal chains, then each facet contributes at most one new face which is critical, and at most one Betti number changes; facets which do not change the homotopy type also do not contribute any critical faces. Dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets. As one application, the Mobius function may be computed as the alternating sum of Morse numbers. The above construction enables us to prove that the poset Π n /S λ of partitions of a set {1 λ 1,...,k λ k} with repetition is homotopy equivalent to a wedge of spheres of top dimension when A is a hook-shaped partition; it is likely that the proof may be extended to a larger class of A and perhaps to all A, despite a result of Ziegler (1986) which shows that Π n /S λ is not always Cohen-Macaulay.

Lexicographic Shellability for Balanced Complexes

We introduce a notion of lexicographic shellability for pure, balanced boolean cell complexes, modelled after the CL-shellability criterion of Björner and Wachs (Adv. in Math.43 (1982), 87–100) for posets and its generalization by Kozlov (Ann. of Comp.1(1) (1997), 67–90) called CC-shellability. We give a lexicographic shelling for the quotient of the order complex of a Boolean algebra of rank 2n by the action of the wreath product S2 ≀ Sn of symmetric groups, and we provide a partitioning for the quotient complex Δ(Πn)/Sn.Stanley asked for a description of the symmetric group representation βS on the homology of the rank-selected partition lattice ΠnS in Stanley (J. Combin. Theory Ser. A32(2) (1982), 132–161), and in particular he asked when the multiplicity bS(n) of the trivial representation in βS is 0. One consequence of the partitioning for Δ(Πn)/Sn is a (fairly complicated) combinatorial interpretation for bS(n); another is a simple proof of Hanlons result (European J. Combin.4(2) (1983), 137–141) that b1,⋯,i(n) = 0. Using a result of Garsia and Stanton from (Adv. in Math.51(2) (1984), 107–201), we deduce from our shelling for Δ(B2n)/S2 ≀ Sn that the ring of invariants k[x1,⋯,x2n]S2 ≀ Sn is Cohen-Macaulay over any field k.

Symmetric Chain Decomposition for Cyclic Quotients of Boolean Algebras and Relation to Cyclic Crystals

The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of the

SB-labelings and posets with each interval homotopy equivalent to a sphere or a ball

\mathfrak{sl}_2

Chains of Modular Elements and Lattice Connectivity

lowering operator in the theory of crystal bases.

Chain decomposition and the flag f -vector

We introduce a new class of poset edge labelings for locally finite lattices which we call

On exact n -step domination

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From the weak Bruhat order to crystal posets

-labelings. We prove for finite lattices which admit an

A partitioning and related properties for the quotient complex Δ(Blm)/Sl≀Sm

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Two Generalizations of Posets of Shuffles

-labeling that each open interval has the homotopy type of a ball or of a sphere of some dimension. Natural examples include the weak order, the Tamari lattice, and the finite distributive lattices.