Minimal Paths between Maximal Chains in Finite Rank Semimodular Lattices

Abstract

We study paths between maximal chains, or “flags,” in finite rank semimodular lattices. Two flags are adjacent if they differ on at most one rank. A path is a sequence of flags in which consecutive flags are adjacent. We study the union of all flags on at least one minimum length path connecting two flags in the lattice. This is a subposet of the original lattice. If the lattice is modular, the subposet is equal to the sublattice generated by the flags. It is a distributive lattice which is determined by the “Jordan-Hölder permutation” between the flags. The minimal paths correspond to all reduced decompositions of this permutation. In a semimodular lattice, the subposet is not uniquely determined by the Jordan-Hölder permutation for the flags. However, it is a join sublattice of the distributive lattice corresponding to this permutation. It is semimodular, unlike the lattice generated by the two flags, which may not be ranked. The minimal paths correspond to some reduced decompositions of the permutation, though not necessarily all. We classify the possible lattices which can arise in this way, and characterize all possibilities for the set of shortest paths between two flags in a semimodular lattice.