An official website of the United States government
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.
more »
« less
This content will become publicly available on November 1, 2025
Charmed roots and the Kroweras complement
Abstract Although both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under theKreweras complementand nonnesting partitions under a Coxeter‐theoretically natural cyclic action we call theKroweras complement. Our equivariant bijection is the unique bijection that is both equivariant and support‐preserving, and is built using local rules depending on a new definition ofcharmed roots. Charmed roots are determined by the choice of Coxeter element — in the special case of the linear Coxeter element , we recover one of the standard bijections between noncrossing and nonnesting partitions.
more »
« less
- Award ID(s):
- 2246877
- PAR ID:
- 10611177
- Publisher / Repository:
- Journal of the London Mathematical Society
- Date Published:
- Journal Name:
- Journal of the London Mathematical Society
- Volume:
- 110
- Issue:
- 5
- ISSN:
- 0024-6107
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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.more » « less
-
null (Ed.)We enumerate factorizations of a Coxeter element in a well-generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of 2W-noncrossing partitions.more » « less
-
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.more » « less
-
Abstract We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams. Using this characterization and the recent bijection of Gao–Huang between reduced pipe dreams and reduced bumpless pipe dreams, we give a bijection between alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We also give a different bijection in the 1432- and 2143-avoiding case that preserves natural poset structures on the associated pipe dreams and bumpless pipe dreams.more » « less
