In the last few decades, combinatorial families exhibiting noncrossing or cluster phenomena have proven useful in understanding and connecting mathematical objects arising in seemingly unrelated branches of mathematics and theoretical physics.
These phenomena can be modeled in the context of Coxeter groups and play an important role in algebraic combinatorics. In finite type, such families are enumerated by generalized Catalan numbers.
In this thesis, we consider the extension of this theory to parabolic quotients of Coxeter groups. We outline the history, present the basic definitions and constructions, and provide a number of conjectures and research challenges arising in this context. We then solve these questions in linear type A and exhibit surprising connections of this theory to certain Hopf algebras and to the theory of diagonal harmonics. We end this thesis by proposing related directions for future research.:Chapter 0. Prologue
Noncrossing partitions
Triangulations
Stack-sortable permutations
Dyck paths
Chapter 1. Preliminaries
1.1. Posets and lattices
1.1.1. A notion of order
1.1.2. Diagrams and labelings
1.1.3. Duality and multichains
1.1.4. Zeta polynomial and Möbius function
1.1.5. Lattices
1.1.6. Distributivity
1.1.7. Semidistributivity
1.1.8. Trimness
1.1.9. Congruence-uniformity
1.1.10. The core label order
1.2. Coxeter groups
1.2.1. Coxeter systems
1.2.2. The geometric representation
1.2.3. Ordering a Coxeter group
1.2.4. Orienting a Coxeter group
Chapter 2. Cataland
2.1. Catalan numbers
2.2. Aligned elements
2.2.1. Cambrian lattices
2.3. Noncrossing partitions
2.4. Clusters
2.5. Nonnesting partitions
2.5.1. v-Tamari lattices
2.6. Chapoton Triangles
Chapter 3. Parabolic Cataland: Origins
3.1. Parabolic quotients of Coxeter groups
3.2. Parabolic aligned elements
3.3. Parabolic noncrossing partitions
3.4. Parabolic clusters
3.5. Parabolic nonnesting partitions
3.6. Parabolic Chapoton triangles
Chapter 4. Parabolic Cataland: Linear type A
4.1. Definitions
4.1.1. Parabolic quotients of the symmetric group
4.1.2. The longest α-permutation
4.1.3. The root poset of S_α and α-Dyck paths
4.1.4. c-clusters for S_α
4.1.5. c-aligned elements for S_α
4.1.6. c-noncrossing partitions for S_α
4.1.7. α-trees
4.2. Bijections
4.2.1. Noncrossing α-partitions and (α, 231)-avoiding permutations
4.2.2. Noncrossing α-partitions and α-Dyck paths
4.2.3. α-trees and (α, 231)-avoiding permutations
4.2.4. α-trees and noncrossing α-partitions
4.2.5. α-trees and α-Dyck paths
4.3. Posets
4.3.1. The weak order on S_α(231)
4.3.2. The rotation order on Dyck(α)
4.3.3. The core label order of Tam(α)
4.4. Chapoton triangles
4.5. Applications
4.5.1. A Hopf algebra on pipe dreams
4.5.2. A zeta map from diagonal harmonics
Chapter 5. Epilogue
5.1. Arbitrary type A
5.2. Linear type B
5.3. (α, m)-Tamari lattices
5.4. Parabolic multiclusters
Chapter A. Data
A.1. Parabolic Catalan numbers in rank 3
A.2. Parabolic Catalan numbers in rank 4
A.3. Answers to Research Challenge 3.3.4 in rank 4 / Kombinatorische Familien, die nichtkreuzende oder Cluster-Phänomene aufweisen, haben sich in den letzten Jahrzehnten als wichtiges Werkzeug für das Verständnis und die Verbindung mathematischer Objekte aus scheinbar unverbundenen Teilgebieten der Mathematik und der theoretischen Physik erwiesen.
Diese Phänomene können im Zusammenhang mit Coxeter-Gruppen modelliert werden, und spielen eine wichtige Rolle in der algebraischen Kombinatorik. Im endlichen Fall werden derartige kombinatorische Familien von verallgemeinerten Catalanzahlen abgezählt.
In dieser Schrift betrachten wir eine Erweiterung dieser Theorie auf parabolische Quotienten von Coxeter-Gruppen. Wir stellen die historische Entwicklung und die grundlegenden Definitionen und Konstruktionen dar und präsentieren eine Reihe von Vermutungen und Forschungsfragen, die in diesem Zusammenhang entstehen. Anschließend lösen wir diese Fragen im sogenannten 'linearen Typ A' und decken überraschende Zusammenhänge dieser Theorie zu bestimmten Hopf-Algebren und zur Theorie der diagonal-harmonischen Polynome auf. Am Ende dieser Schrift schlagen wir weiterführende Forschungsrichtungen vor.:Chapter 0. Prologue
Noncrossing partitions
Triangulations
Stack-sortable permutations
Dyck paths
Chapter 1. Preliminaries
1.1. Posets and lattices
1.1.1. A notion of order
1.1.2. Diagrams and labelings
1.1.3. Duality and multichains
1.1.4. Zeta polynomial and Möbius function
1.1.5. Lattices
1.1.6. Distributivity
1.1.7. Semidistributivity
1.1.8. Trimness
1.1.9. Congruence-uniformity
1.1.10. The core label order
1.2. Coxeter groups
1.2.1. Coxeter systems
1.2.2. The geometric representation
1.2.3. Ordering a Coxeter group
1.2.4. Orienting a Coxeter group
Chapter 2. Cataland
2.1. Catalan numbers
2.2. Aligned elements
2.2.1. Cambrian lattices
2.3. Noncrossing partitions
2.4. Clusters
2.5. Nonnesting partitions
2.5.1. v-Tamari lattices
2.6. Chapoton Triangles
Chapter 3. Parabolic Cataland: Origins
3.1. Parabolic quotients of Coxeter groups
3.2. Parabolic aligned elements
3.3. Parabolic noncrossing partitions
3.4. Parabolic clusters
3.5. Parabolic nonnesting partitions
3.6. Parabolic Chapoton triangles
Chapter 4. Parabolic Cataland: Linear type A
4.1. Definitions
4.1.1. Parabolic quotients of the symmetric group
4.1.2. The longest α-permutation
4.1.3. The root poset of S_α and α-Dyck paths
4.1.4. c-clusters for S_α
4.1.5. c-aligned elements for S_α
4.1.6. c-noncrossing partitions for S_α
4.1.7. α-trees
4.2. Bijections
4.2.1. Noncrossing α-partitions and (α, 231)-avoiding permutations
4.2.2. Noncrossing α-partitions and α-Dyck paths
4.2.3. α-trees and (α, 231)-avoiding permutations
4.2.4. α-trees and noncrossing α-partitions
4.2.5. α-trees and α-Dyck paths
4.3. Posets
4.3.1. The weak order on S_α(231)
4.3.2. The rotation order on Dyck(α)
4.3.3. The core label order of Tam(α)
4.4. Chapoton triangles
4.5. Applications
4.5.1. A Hopf algebra on pipe dreams
4.5.2. A zeta map from diagonal harmonics
Chapter 5. Epilogue
5.1. Arbitrary type A
5.2. Linear type B
5.3. (α, m)-Tamari lattices
5.4. Parabolic multiclusters
Chapter A. Data
A.1. Parabolic Catalan numbers in rank 3
A.2. Parabolic Catalan numbers in rank 4
A.3. Answers to Research Challenge 3.3.4 in rank 4
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:76309 |
| Date | 15 October 2021 |
| Creators | Mühle, Henri |
| Contributors | Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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