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Séminaire Lotharingien de Combinatoire, B63b (2010), 32 pp.

# Nathan Reading

# 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.

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.**

d4cl.ps

d5cl.ps

e6cl.ps

e7cl.ps

e8cl.ps

f4cl.ps

h3cl.ps

h4cl.ps

e6nc.ps

f4nc.ps

h3nc.ps

h4nc.ps

e6cl_alt.ps

e7cl_alt.ps

e8cl_alt.ps

f4cl_alt.ps

h4cl_alt.ps