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Noncrossing partitions, clusters and the Coxeter plane. (English) Zbl 1208.05008

Summary: 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.

MSC:

05A18 Partitions of sets
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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