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The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuß-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuß-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras.
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This content will become publicly available on October 1, 2025
Rational noncrossing Coxeter–Catalan combinatorics
Abstract We solve two open problems in Coxeter–Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type‐uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter–Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.
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- PAR ID:
- 10574705
- Publisher / Repository:
- London Mathematical Society
- Date Published:
- Journal Name:
- Proceedings of the London Mathematical Society
- Volume:
- 129
- Issue:
- 4
- ISSN:
- 0024-6115
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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