Noncrossing Partitions, Clusters and the Coxeter Plane

When W is a finite Coxeter group of classical type (A, B, or D), noncrossing partitions associated to W and compatibility of almost positive roots in the associated root system are known to be modeled by certain planar diagrams. We show how the classical-type constructions of planar diagrams arise uniformly from projections of smallW -orbits to the Coxeter plane. When the construction is applied beyond the classical cases, simple criteria are apparent for noncrossing and for compatibility for W of types H3 and I2(m) and less simple criteria can be found for compatibility in types E6, F4 and H4. Our construction also explains why simple combinatorial models are elusive in the larger exceptional types.