On optimizing discrete Morse functions

TL;DR

This paper advances discrete Morse theory by identifying conditions for reversing multiple gradient paths to simplify complexes and applying these methods to symmetric group posets, providing new proofs and insights.

Contribution

It introduces conditions for reversing multiple gradient paths in discrete Morse functions and applies these techniques to symmetric group posets and Cohen-Macaulayness.

Findings

Conditions for reversing multiple gradient paths established

New proof of homotopy type for intervals in the weak order

Application to Cohen-Macaulayness of a new partial order

Abstract

Forman introduced discrete Morse theory as a tool for studying CW complexes by essentially collapsing them onto smaller, simpler-to-understand complexes of critical cells in [Fo]. Chari reformulated discrete Morse theory for regular cell complexes in terms of acyclic matchings on face posets in [Ch]. 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) how to use lexicographically first reduced expressions for permutations (in the sense of [Ed]) to make (1) practical for poset order complexes. Applications include Cohen-Macaulayness of a new partial order, recently introduced by Remmel, on the symmetric group (by refinement on the underlying partitions into cycles) as well as a simple new proof of the homotopy type for intervals in the weak order for the symmetric group. Additional applications appear in [HW].