Minimal paths between maximal chains in finite rank semimodular lattices (Q1383820)

The author studies relationships between maximal chains (``flags'') in a finite rank semimodular lattice, namely, the union of all flags on at least one minimum length path connecting two flags. He generalizes to semimodular lattices the Jordan-Hölder permutation and results concerning the sublattice generated by two flags. He relates minimal paths between two flags in a semimodular lattice to reduced decompositions of the Jordan-Hölder permutation between the flags. Many results of the paper use a comparison of the subposet of all points on at least one reduced \(X\)-\(Y\) path to the sublattice generated by two flags \(X\) and \(Y\) in a semimodular lattice and labeling functions with respect to flags. Finally, the author classifies the sets of reduced decompositions of a permutation which can correspond to a set of reduced \(X\)-\(Y\) paths in some semimodular lattice.