Discrete Morse functions from lexicographic orders

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 with $\hat {0}$ and $\hat {1}$ 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 $\Pi _n/S_{\lambda }$ of partitions of a set $\{ 1^{\lambda _1 },\dots ,k^{\lambda _k }\}$ with repetition is homotopy equivalent to a wedge of spheres of top dimension when $\lambda$ is a hook-shaped partition; it is likely that the proof may be extended to a larger class of $\lambda$ and perhaps to all $\lambda$, despite a result of Ziegler (1986) which shows that $\Pi _n/S_{\lambda }$ is not always Cohen-Macaulay.