Noncrossing Partitions, Clusters and the Coxeter Plane

Nathan Reading

Séminaire lotharingien de combinatoire, Tome 63 (2010)

Résumé

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.

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