Abstract. 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 with0 and1 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 Möbius function may be computed as the alternating sum of Morse numbers.The above construction enables us to prove that the poset Πn/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 λ is a hook-shaped partition; it is likely that the proof may be extended to a larger class of λ and perhaps to all λ, despite a result of Ziegler (1986) which shows that Πn/S λ is not always Cohen-Macaulay.

Fomin and Shapiro conjectured that the link of the identity in the Bruhat stratification of the totally nonnegative real part of the unipotent radical of a Borel subgroup in a semisimple, simply connected algebraic group defined and split over R is a regular CW complex homeomorphic to a ball. The main result of this paper is a proof of this conjecture. This completes the solution of the question of Bernstein of identifying regular CW complexes arising naturally from representation theory having the (lower) intervals of Bruhat order as their closure posets. A key ingredient is a new criterion for determining whether a finite CW complex is regular with respect to a choice of characteristic maps; it most naturally applies to images of maps from regular CW complexes and is based on an interplay of combinatorics of the closure poset with codimension one topology.

This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group S n on the cohomology of the configuration space of n ordered points in R d . This cohomology is known to vanish outside of dimensions divisible by d − 1; it is shown here that the S n -representation on the i(d − 1) st cohomology stabilizes sharply at n = 3i (resp. n = 3i + 1) when d is odd (resp. even).The result comes from analyzing S n -representations known to control the cohomology: the Whitney homology of set partition lattices for d even, and the higher Lie representations for d odd. A similar analysis shows that the homology of any rank-selected subposet in the partition lattice stabilizes by n ≥ 4i, where i is the maximum rank selected.Further properties of the Whitney homology and more refined stability statements for S n -isotypic components are also proven, including conjectures of J. Wiltshire-Gordon.2010 Mathematics Subject Classification. 55R80, 05E45, 05E05, 20C30, 06A07.

In 1998, Forman introduced discrete Morse theory as a tool for studying CW complexes by producing smaller, simpler-to-understand complexes of critical cells with the same homotopy types as the original complexes. This paper addresses two questions: (1) under what conditions may several gradient paths in a discrete Morse function simultaneously be reversed to cancel several pairs of critical cells, to further collapse the complex, and (2) which gradient paths are individually reversible in lexicographic discrete Morse functions on poset order complexes. The latter follows from a correspondence between gradient paths and lexicographically first reduced expressions for permutations. As an application, a new partial order on the symmetric group recently introduced by Remmel is proven to be Cohen-Macaulay.