Séminaire Lotharingien de Combinatoire, B63b (2010), 32 pp.
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 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 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.
Received: January 28, 2010.
Accepted: February 15, 2010.
The following versions are available:
Supplement: Postscript files for Coxeter-plane diagrams of parabolic subgroups and almost positive roots
These files are explained in Section 1.3 of the paper.
Here is a word of caution from that section: The files are optimized for small file size as PostScript files, by taking advantage of the fact that PostScript is a complete programming language.
It is recommended viewing the files (or at least those files associated with the larger exceptional groups) with a PostScript viewer that interprets PostScript directly, rather than converting to PDF.
File sizes in PDF will be up to about 350 times larger than the postscript files.
WARNING: If you click certain of these links in a browser that insists on converting Postscript files to PDF, your browser may be busy for a long time.