Noncrossing Partitions of an Annulus: A Planar Model for Intervals in the Absolute Order on Affine Coxeter Groups of Types A and C

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This paper introduces a novel planar model for visualizing intervals in the absolute order on affine Coxeter groups of types A and C, utilizing noncrossing partitions of an annulus and their relation to the Coxeter plane.

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Bibliographic Information:

Brestensky, L. G., & Reading, N. (2024). Noncrossing partitions of an annulus. arXiv preprint arXiv:2212.14151v4.

Research Objective:

This paper aims to develop planar models for visualizing intervals in the absolute order on affine Coxeter groups of types A and C, extending the existing planar models for finite Coxeter groups.

Methodology:

The authors utilize the concept of projecting a small orbit onto the Coxeter plane, a technique previously used for finite Coxeter groups. They adapt this method to affine Coxeter groups by considering an orbit in the vector space V and projecting it onto the Coxeter plane in the dual space V*.

Key Findings:

• In type eA, the resulting model consists of noncrossing partitions of an annulus, where elements of the interval [1, c]T are distinguished by the absence of dangling annular blocks.
• In type eC, the model utilizes symmetric noncrossing partitions of an annulus, which can be equivalently represented as noncrossing partitions of a two-orbifold disk.
• The paper demonstrates that the combinatorial structure of these planar diagrams suggests a natural way of factoring translations in type eA, leading to a lattice completion of the interval [1, c]T.

Main Conclusions:

The planar models presented provide a visual and intuitive way to understand the structure of intervals in the absolute order on affine Coxeter groups of types A and C. The factorization of translations in type eA, inspired by the planar model, offers a new perspective on the work of McCammond and Sulway on lattice completions of these intervals.

Significance:

This research extends the understanding of noncrossing partitions and their connections to Coxeter groups, providing valuable tools for studying Artin groups and related algebraic structures. The planar models offer a powerful visual aid for researchers in combinatorics and related fields.

Limitations and Future Research:

The paper focuses on affine Coxeter groups of types A and C. Future research could explore similar planar models for other affine types, such as eD and eB, as indicated by the authors. Additionally, investigating the implications of these models for the representation theory of affine Coxeter groups could be a fruitful avenue for further exploration.

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Noncrossing partitions of an annulus

by Laura G. Bre... om arxiv.org 11-05-2024

https://arxiv.org/pdf/2212.14151.pdf

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How can the planar models presented in this paper be generalized to other Coxeter groups beyond the classical finite and affine types?

While the paper focuses on planar models for classical finite and affine types, generalizing to other Coxeter groups presents exciting challenges and possibilities:
1. Beyond Planar Diagrams: The inherent limitation of planar diagrams is their inability to represent structures beyond two dimensions. For Coxeter groups with more complex geometry, we need to explore higher-dimensional analogs. Some potential avenues include:
* **Higher-Dimensional Representations:** Instead of projecting onto a plane, investigate projections onto suitable higher-dimensional subspaces that capture the group's structure.
* **Combinatorial Complexes:** Explore representations using combinatorial objects like simplicial complexes or cell complexes, which can encode higher-dimensional relations.

2. Identifying Suitable Projections: The success in classical types relies on projecting a small orbit onto the Coxeter plane. Generalization requires:
* **Finding Analogous Subspaces:** Identify subspaces in the reflection representation or its dual that play a role analogous to the Coxeter plane. These subspaces should reflect the Coxeter element's action and the group's structure.
* **Characterizing "Small" Orbits:** Define a notion of "small" orbits for the specific Coxeter group, ensuring they capture enough information for a meaningful projection.

3. Interpreting the Projections: Once a projection is established, the next challenge lies in interpreting the resulting diagrams:
* **Defining Noncrossing Partitions:** Formulate a suitable definition of "noncrossing partitions" in the context of the chosen projection and the specific Coxeter group.
* **Connecting to Absolute Order:** Establish a clear connection between the combinatorial properties of the diagrams and the algebraic structure of the absolute order in the Coxeter group.

4. Exploring Exceptional Types: The exceptional Coxeter groups (E6, E7, E8, F4, H3, H4) pose unique challenges due to their intricate structure. Investigating these cases might require novel approaches and combinatorial constructions tailored to their specific properties.
5. Leveraging Existing Tools: Existing tools from Coxeter group theory and related fields can aid in this generalization:
* **Root Systems and Reflection Groups:** Utilize the properties of root systems and reflection groups to guide the choice of projections and the interpretation of diagrams.
* **Geometric Group Theory:** Employ techniques from geometric group theory, such as studying the action of the Coxeter group on its Cayley graph or other geometric spaces.

Could alternative factorization schemes for translations in type eA, different from the one suggested by the planar model, lead to different lattice completions with distinct properties?

Yes, alternative factorization schemes for translations in type eA could indeed lead to different lattice completions with distinct properties. While the planar model suggests a natural factorization, it's not unique. Here's why:

Non-Uniqueness of Factorization: The fundamental reason is that the translations form an infinite cyclic subgroup, and there are infinitely many ways to factor an element within an infinite cyclic group.
Impact on Lattice Properties: Different factorizations would lead to different generating sets for the larger group containing the affine Coxeter group. This can affect:

Lattice Presentation: The relations defining the lattice would change, potentially altering its combinatorial and algebraic properties.
Garside Structure: The existence and properties of a Garside structure for the associated Artin group could be affected. Different Garside structures can lead to different normal forms for elements, impacting algorithms and computations.
Geometric Realizations: The geometric realization of the lattice, if it exists, might be different, leading to different insights into the group's structure.
Exploring Alternative Schemes:

Varying the "Break Points": The planar model suggests breaking an annular block into two dangling annular blocks. One could explore breaking it into more than two blocks or choosing different "break points" on the annulus.
Introducing New Generators: Instead of just factoring existing translations, one could introduce entirely new generators and relations that interact with the translations in a controlled manner.
Analyzing the Consequences:

Lattice Isomorphisms: Investigate whether different factorization schemes lead to isomorphic lattices. Even if the lattices are isomorphic, their presentations and associated Garside structures might differ.
Artin Group Properties: Study how different factorizations impact the properties of the associated Artin groups, such as their word problem, cohomology, and connections to geometric structures.

What are the implications of these planar models for the study of the associated Artin groups, particularly in the context of their geometric and topological properties?

The planar models presented in the paper have significant implications for studying the associated Artin groups, offering valuable insights into their geometric and topological properties:
1. Geometric Realizations:

Visualizing the Group: Planar models provide a concrete and visual way to represent elements and relations in the Artin group. This visualization can aid in understanding the group's structure and developing geometric intuition.
Constructing Complexes: The lattice structure of noncrossing partitions, revealed by the planar models, suggests connections to cell complexes. These complexes can provide geometric realizations of the Artin group, where cells correspond to elements and attaching maps reflect group relations.
2. Normal Forms and Algorithms:

Dual Presentations: The lattice structure of [1,c]T, captured by the planar models, is crucial for constructing dual presentations of Artin groups. These presentations use the lattice elements as generators and provide alternative ways to represent and manipulate group elements.
Solving the Word Problem: Dual presentations, arising from the lattice structure, can lead to efficient algorithms for solving the word problem in Artin groups. This has implications for computational group theory and applications in areas like cryptography.
3. Topological Invariants:

Cohomology: The geometric realizations of Artin groups, inspired by planar models, can be used to compute topological invariants like cohomology groups. These invariants provide insights into the group's structure and its connections to other geometric and topological objects.
K-Theory: The lattice structure of noncrossing partitions has connections to K-theory, a branch of algebraic topology. Exploring these connections could lead to new invariants and a deeper understanding of Artin groups.
4. Generalizations and Connections:

Surface Groups: The planar models, particularly the annular case, resemble diagrams used to study surface groups. This suggests deeper connections between Artin groups and surface groups, potentially leading to new insights and generalizations.
Cluster Algebras: Noncrossing partitions and their planar representations are closely related to cluster algebras, a vibrant area of current research in algebra and geometry. The planar models could provide a bridge between the study of Artin groups and cluster algebras.
5. Open Questions and Future Directions:

Extending to Other Types: Exploring whether similar planar models exist for other types of Artin groups beyond the classical cases is an active area of research.
Geometric Properties of Complexes: Investigating the geometric properties of complexes arising from planar models, such as their curvature, isometry groups, and connections to other geometric structures, is an exciting avenue for future research.