Noncrossing Partitions of Marked Surfaces: A Generalization of Kreweras' Noncrossing Partitions

Core Concepts

This paper introduces a novel definition of noncrossing partitions for marked surfaces, generalizing Kreweras' classical construction, and explores the properties of the resulting noncrossing partition lattices, including their lattice structure, rank function, and lower intervals.

Abstract

Bibliographic Information

Reading, N. (2024). Noncrossing partitions of a marked surface [Preprint]. arXiv:2212.13799v5

Research Objective:

This paper aims to generalize the concept of noncrossing partitions, originally defined for a cycle, to arbitrary marked surfaces. The author introduces two new definitions: noncrossing partitions of a marked surface and noncrossing partitions of a symmetric marked surface with double points. The study then investigates the properties of these new structures, focusing on their lattice structure, rank function, and the relationship between lower intervals and noncrossing partition lattices of related surfaces.

Methodology:

The paper employs a combinatorial and topological approach. It defines noncrossing partitions on marked surfaces using embedded blocks, which are essentially sub-surfaces with specific boundary conditions. The analysis leverages concepts like ambient isotopy, curve sets, simple connectors, and curve unions to characterize the partial order, cover relations, and rank function of the resulting noncrossing partition posets.

Key Findings:

The paper establishes that the natural partial order on noncrossing partitions of a marked surface forms a graded lattice. The rank function of this lattice is determined by the number of marked points and the Betti numbers of the union of blocks in a partition. Additionally, the study demonstrates that lower intervals in this lattice are isomorphic to products of noncrossing partition lattices of other marked surfaces. For symmetric marked surfaces with double points, the paper shows that while the poset is not always a lattice, it remains graded, and its rank function can be described using the dimensions of the kernel of a specific linear map on homology. Similar to the non-symmetric case, lower intervals in this poset are also isomorphic to products of noncrossing partition posets.

Main Conclusions:

The paper successfully generalizes the notion of noncrossing partitions to marked surfaces, revealing a rich combinatorial structure with connections to the topology of the underlying surface. The results provide a unifying framework for understanding noncrossing partitions in various contexts, including classical and affine Coxeter groups.

Significance:

This work extends the understanding of noncrossing partitions, a fundamental object in combinatorics with applications in representation theory, algebraic geometry, and other areas. The introduction of marked surfaces as the underlying structure offers a new perspective and potentially opens avenues for further research in related fields.

Limitations and Future Research:

The paper primarily focuses on the structural properties of noncrossing partition lattices on marked surfaces. Further investigation could explore connections to other combinatorial objects, such as triangulations and cluster algebras, and delve into the representation-theoretic aspects of these new noncrossing partition lattices. Additionally, exploring the role of punctures and their relationship to double points and symmetry could be a fruitful direction for future research.

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Noncrossing partitions of a marked surface

by Nathan Readi... at arxiv.org 11-05-2024

https://arxiv.org/pdf/2212.13799.pdf

Deeper Inquiries

How does the concept of noncrossing partitions on marked surfaces relate to the theory of cluster algebras and triangulations of surfaces?

The concept of noncrossing partitions on marked surfaces is deeply intertwined with the theory of cluster algebras and triangulations of surfaces in several ways:

Shared Foundation: Both noncrossing partitions and cluster algebras, as studied in the context of surfaces, rely heavily on the foundational work of Fomin, Shapiro, and Thurston ([13, 14]). These papers establish the framework of tagged triangulations of marked surfaces, which are crucial for understanding the combinatorics of cluster algebras arising from surfaces. Noncrossing partitions, as defined by Reading, also utilize this framework, particularly the notion of arcs and their compatibility.

Combinatorial Connections: Triangulations of a marked surface (S,M) play a significant role in the construction and study of the corresponding cluster algebra. The combinatorial structure of these triangulations, particularly their flips (local transformations), is reflected in the mutation rules of the associated cluster algebra. While noncrossing partitions don't directly model mutations, they offer a different perspective on the combinatorial structure encoded by the surface. For instance, the number of arcs in a triangulation (Proposition 2.26) is directly related to the rank of the top element in the noncrossing partition lattice NC(S,M).

Potential for Representation Theory: Cluster algebras have deep connections to representation theory. In many cases, cluster algebras provide combinatorial models for certain categories of representations. Noncrossing partitions, particularly those arising from finite Coxeter groups, also have strong ties to representation theory. It is plausible that the more general notion of noncrossing partitions on marked surfaces could provide new combinatorial models for representations related to surface cluster algebras.

Could there be alternative definitions of noncrossing partitions on marked surfaces that yield different but equally interesting lattice structures?

Yes, it's certainly possible to conceive of alternative definitions of noncrossing partitions on marked surfaces that could lead to different lattice structures. Here are a few potential avenues for exploration:

Relaxing Restrictions on Embeddings: Reading's definition imposes specific conditions on how blocks are embedded in the surface. For example, distinct blocks cannot share a boundary ring. Relaxing this condition, perhaps allowing certain types of shared boundaries, could lead to a different notion of "noncrossing" and a new lattice structure.

Incorporating Punctures: The current definition focuses on marked surfaces without punctures. Allowing punctures and incorporating them into the definition of embedded blocks and their intersections could yield a richer family of noncrossing partitions. This might be particularly interesting in light of the relationship between punctures and double points/symmetry discussed in Remark 4.1 of the paper.

Generalizing "Curves": The definition relies heavily on arcs and boundary segments as the building blocks of curve sets. One could explore using other types of curves on the surface, perhaps with different intersection properties or topological constraints, leading to new notions of noncrossingness.

Weights and Decorations: Inspired by other generalizations of noncrossing partitions, one could introduce weights or decorations on the arcs or blocks. This could lead to refined lattice structures with additional combinatorial information.
Exploring these and other alternative definitions could uncover new connections between the combinatorics of surfaces, lattice theory, and other areas of mathematics.

What insights from the study of noncrossing partitions on surfaces can be applied to problems in knot theory or the study of mapping class groups?

While the paper focuses on the lattice-theoretic aspects of noncrossing partitions on surfaces, there are potential connections to knot theory and mapping class groups that deserve further investigation:

Diagrammatic Representations of Knots and Braids: Noncrossing partitions on disks are closely related to diagrams of Temperley-Lieb algebras, which have connections to knot invariants. It's conceivable that noncrossing partitions on more general surfaces could provide new diagrammatic tools for studying knots and braids on those surfaces. The embedded blocks could potentially represent parts of a knot or braid diagram, and the lattice structure might encode information about knot or braid equivalences.

Mapping Class Group Actions: The mapping class group of a surface acts naturally on the set of isotopy classes of curves on that surface. Since noncrossing partitions are defined in terms of such curves, it's natural to ask how the mapping class group acts on the set of noncrossing partitions. Understanding this action could provide insights into both the structure of the mapping class group and the properties of noncrossing partitions.

Geometric Interpretations of Noncrossing Partitions: The paper defines noncrossing partitions in a combinatorial way. Exploring geometric realizations of these partitions, perhaps by associating specific geometric objects to the blocks, could lead to connections with geometric group theory and the study of surface diffeomorphisms.

Surface Invariants from Noncrossing Partitions: The structure of the noncrossing partition lattice NC(S,M) depends on the topology of the surface (S,M). It's possible that invariants of the lattice, such as its Möbius function or its homology, could provide new topological invariants of the surface itself.
These are just a few potential directions for future research. The study of noncrossing partitions on surfaces is still in its early stages, and it holds promise for uncovering new and deep connections between combinatorics, topology, and geometry.