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The noncrossing partition poset associated to a Coxeter group $$W$$ and Coxeter element $$c$$ is the interval $$[1,c]_T$$ in the absolute order on $$W$$. We construct a new model of noncrossing partititions for $$W$$ of classical affine type, using planar diagrams. The model in type $$\afftype{A}$$ consists of noncrossing partitions of an annulus. In type~$$\afftype{C}$$, the model consists of symmetric noncrossing partitions of an annulus or noncrossing partitions of a disk with two orbifold points. Following the lead of McCammond and Sulway, we complete $$[1,c]_T$$ to a lattice by factoring the translations in $$[1,c]_T$$, but the combinatorics of the planar diagrams leads us to make different choices about how to factor.
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This content will become publicly available on March 31, 2026
Noncrossing Partitions of a Marked Surface
We define noncrossing partitions of a marked surface without punctures (interior marked points). We show that the natural partial order on noncrossing partitions is a graded lattice and describe its rank function topologically. Lower intervals in the lattice are isomorphic to products of noncrossing partition lattices of other surfaces. We similarly define noncrossing partitions of a symmetric marked surface with double points and prove some of the analogous results. The combination of symmetry and double points plays a role that one might have expected to be played by punctures.
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- Award ID(s):
- 2054489
- PAR ID:
- 10649439
- Publisher / Repository:
- Society for Industrial and Applied Mathematics
- Date Published:
- Journal Name:
- SIAM Journal on Discrete Mathematics
- Volume:
- 39
- Issue:
- 1
- ISSN:
- 0895-4801
- Page Range / eLocation ID:
- 231 to 266
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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