Characterize When the Noncrossing Partition Poset is a Lattice in Rank ≥ 4

Characterize, for Coxeter groups W of rank at least 4 and Coxeter elements w, precisely when the noncrossing partition poset NC(W, w) is a lattice, i.e., when every pair of elements has a unique minimal upper bound and a unique maximal lower bound.

Background

The lattice property of NC(W, w) is classical in spherical cases and has been shown for certain affine families (~A_n and ~C_n) and all rank-three cases. McCammond proved that the lattice property fails in the remaining affine cases outside this list.

Despite these advances, a complete characterization of the lattice property in rank ≥ 4 is not known. Resolving this would refine understanding of the combinatorial structure of NC(W, w) and impact applications such as Garside theory.