Noncrossing partitions, clusters and the Coxeter plane

Details Noncrossing partitions, clusters and the Coxeter plane Noncrossing partitions, clusters and the Coxeter plane

2009-12-26

arxiv.org/abs/0912.5028v2

Mathematics

Combinatorics

Very minor changes, as suggested by the referee. This is essentially the final version, which will appear in Sem. Lothar. Comb

Scientific paper

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.

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