Noncrossing partitions, clusters and the Coxeter plane
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arXiv
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| Format: | Preprint |
| Published: |
2009
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| _version_ | 1866918497180712960 |
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| author | Reading, Nathan |
| author_facet | Reading, Nathan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0912_5028 |
| institution | arXiv |
| publishDate | 2009 |
| record_format | arxiv |
| spellingShingle | Noncrossing partitions, clusters and the Coxeter plane Reading, Nathan Combinatorics 20F55 (Primary) 05A18 (Secondary) 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. |
| title | Noncrossing partitions, clusters and the Coxeter plane |
| topic | Combinatorics 20F55 (Primary) 05A18 (Secondary) |
| url | https://arxiv.org/abs/0912.5028 |