Noncrossing partitions, clusters and the Coxeter plane

Abstract

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 small W-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 H_3 and I_2(m) and less simple criteria can be found for compatibility in types E_6, F_4 and H_4. Our construction also explains why simple combinatorial models are elusive in the larger exceptional types.Comment: Very minor changes, as suggested by the referee. This is essentially the final version, which will appear in Sem. Lothar. Combin. 32 pages. About 12 of the pages are taken up by 29 figure