Associahedra, Tamari Lattices and Related Structures

Name: Associahedra, Tamari Lattices and Related Structures | Tamari Memorial Festschrift
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Tamari Memorial Festschrift

Folkert Müller-Hoissen Jean Marcel Pallo Jim Stasheff(Herausgeber*in)

Birkhäuser (Verlag)

1. Auflage

Erschienen am 13. Juli 2012

XX, 436 Seiten

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Inhalt

Intro
Associahedra, Tamari Latticesand Related Structures
Preface
Contents
List of Contributors
Dov Tamari (formerly Bernhard Teitler)
1 Germany
2 Palestine
3 France
4 Tamari's thesis and beyond
5 Israel
6 Via Princeton to Brazil
7 The Netherlands
8 USA
9 Tamari-Technion Affair
9.1 Freudenthal's initiative
9.2 AMS and CAFTES
10 After retirement
10.1 Mathematical activities
10.2 Back to the roots
11 Further recollections
References
On Being a Student of Dov Tamari
How I 'met' Dov Tamari
1 Introduction/History
1.1 The so-called Stasheff polytope
1.2 Giving due credit
1.3 Outline of this paper
2 Realization
3 Other polytopes
3.1 Multiplihedra
3.2 Cyclohedra and generalized associahedra
3.3 Compactified configuration spaces
4 Applications
4.1 Coderivation interpretation and curvature
4.2 Pentagonal algebras
4.3 The pentagonal algebra of open string field theory
4.4 An-structures for 4 & n & 8
5 Opportunities - missed and not
References
Dichotomy of the Addition of Natural Numbers
Introduction
1 About the formula 1 + 1 = 2
2 Splitting the integers into pieces
3 Trees and addition on trees
4 Where we show that 1 + 1 = 2 + 1 = 3
5 The integers as molecules
6 Multiplication of trees
7 Trees and polynomials
8 Realizing the associahedron
8.1 Tamari poset
8.2 Associahedron and regular pentagons
8.3 Realizations of the associahedron
9 Associahedron and the trefoil knot
References
Partial Groupoid Embeddings in Semigroups
1 Introduction
2 Definitions and preliminary results
3 Results
4 Further results
4.1 Embedding into a partially ordered semigroup
4.2 Embedding into a commutative semigroup
5 Conclusion
References
Moduli Spaces of Punctured Poincaré Disks
1 Motivation from physics
2 Particles on the Poincaré disk
3 The Fulton-McPherson compactification
4 Group actions on screens
5 Combinatorial results
References
Realizing the Associahedron: Mysteries and Questions
1 Introduction
2 Realization space
3 Vertices on a sphere
4 Realizing the multiassociahedron
References
Permutahedra and Associahedra
1 Introduction
2 Permutahedra and finite reflection groups
2.1 Finite reflection groups
2.2 Permutahedra as V -polytopes
2.3 Root systems and permutahedra as H -polytopes
2.4 Faces of permutahedra and the weak order
2.4.1 Faces and standard parabolic subgroups
2.4.2 Edges and weak order
3 Coxeter generalized associahedra
3.1 Coxeter elements and Coxeter singletons
3.2 Coxeter generalized associahedra as H -polytopes
3.3 Faces and almost positive roots
3.4 Edges and Cambrian lattices
3.5 Isometry classes
3.6 Integer coordinates
4 Further developments
References
Combinatorial 2-truncated Cubes and Applications
1 Introduction
2 Simple polytopes
2.1 Enumerative polynomials
3 Class of 2-truncated cubes
4 Smooth toric varieties over 2-truncated cubes
5 Small covers of 2-truncated cubes
6 Nestohedra and graph-associahedra
7 Building set as a structure
8 Flag nestohedra as 2-truncated cubes
9 Recursion formulas
9.1 Associahedra
9.2 Cyclohedra
9.3 Permutohedra
9.4 Stellohedra
10 Bounds of face polynomials for flag nestohedra and graph-associahedra
11 Nested polytopes and graph-cubeahedra as 2-truncated cubes
References
Extending the Tamari Lattice to Some Compositions of Species
1 Introduction
1.1 Species
2 Several flavors of trees
2.1 Ordered, bi-leveled and painted trees
2.2 Trees with corrollas
3 Interval retracts
4 Lattices and polytopes
5 Algebraic implications of interval retracts
5.1 Coalgebras of trees
5.2 Cofree composition of coalgebras
5.3 The coalgebra of painted trees
5.3.1 Primitives in the coalgebras of trees and painted trees
References
Tamari Lattices and the Symmetric Thompson Monoid
Introduction
1 The framework
1.1 Parenthesized expressions and associativity
1.2 Trees and rotations
1.3 Richard Thompson's group F
1.4 The action of F on trees
1.5 The generators aa
1.6 Presentation of F in terms of the elements aa
2 A lattice structure on the Thompson group F
2.1 The symmetric Thompson monoid F+sym
2.2 Subword reversing
2.3 The lattice structure of F+sym
2.4 Computing the operations
3 The Polish normal form on F
3.1 The Polish encoding of trees
3.2 The Polish algorithm
3.3 The covering relation
3.4 The Polish normal form
4 Distance in Tamari lattices
4.1 The diameter of Tn
4.2 Syntactic invariants
4.3 The embedding of F+sym into F
References
Parenthetic Remarks
1 Introduction
2 Parengebras
3 Operads
4 Well-formed words
5 Free monoidal categories
References
On the Categories of Modules Over the Tamari Posets
Introduction
1 Representations of posets in general
2 Dimension of the incidence algebra of Yn
3 Main conjecture on the category DbmodYn
4 At the level of Grothendieck groups
5 Derived equivalences with other posets
6 Quasi-homogeneous isolated singularities
Appendix
References
The Tamari Lattice as it Arises in Quiver Representations
1 Introduction
2 Quiver representations
3 Subrepresentations, quotient representations, and extensions
4 Pullbacks of extensions
5 Indecomposable representations of the quiver An
6 Morphisms and extensions between indecomposable representations of An
7 Subcategories of repQ
8 Quotient-closed subcategories
9 Subcategories ordered by inclusion
10 Torsion classes in repAn
11 Related posets
References
From the Tamari Lattice to Cambrian Lattices and Beyond
1 A map from permutations to triangulations
2 Bringing lattice congruences into the picture
3 Cambrian lattices
4 Cambrian fans
5 Sortable elements
6 Applications
6.1 Coxeter-Catalan combinatorics
6.2 Sortable elements in quiver theory
6.3 Combinatorial Hopf algebras
6.4 Cluster algebras
References
Catalan Lattices on Series Parallel Interval Orders
1 Introduction
2 Series-parallel interval orders
2.1 Preorder linear extensions of series parallel interval orders
3 Catalan lattices on series parallel interval orders
3.1 The Dyck lattice
3.2 The Tamari lattice
4 Series parallel interval orders and pattern avoiding permutations
4.1 The Tamari lattice and the weak Bruhat order on Av(312)
4.2 The Dyck lattice and the strong Bruhat order on Av(312)
5 Further works
References
Generalized Tamari Order
1 Introduction
2 Graph associahedra
3 Binary operations on tubings
4 Generalization of Tamari order to tubings
References
A Survey of the Higher Stasheff-Tamari Orders
1 Introduction
2 Cyclic polytopes
3 The two orders
4 Encodings
4.1 Submersion sets
4.2 Snug rectangles
4.3 Non-interlacing separated d/2 -faces
5 Lattice property, homotopy types and M¨obius function
6 Connection to flip graph connectivity
6.1 Bistellar flips
6.2 The flip connectivity question
6.3 The flip graph of a cyclic polytope
6.4 Diameter
6.5 The case d = 2: the rotation graph of binary trees
7 Subdivisions and the Baues problem
7.1 Subdvisions and secondary polytopes
7.2 Baues's original problem
7.3 Cellular strings and the generalized Baues problem
8 Connection to the higher Bruhat orders
8.1 Definition of higher Bruhat orders
8.2 Some geometry of higher Bruhat orders
8.3 The map from higher Bruhat to higher Stasheff-Tamari orders
9 Enumeration
References
KP Solitons, Higher Bruhat and Tamari Orders
1 Introduction
2 KP solitons
3 Tropical approximation of a subclass of KP line solitons
4 Higher Bruhat and higher Tamari orders
4.1 Higher Bruhat orders
4.2 Higher Tamari orders
5 KP line soliton evolutions in the case M = 5
6 Some insights into the case M & 5
7 An algebraic construction of higher Bruhat and Tamari orders
7.1 Higher Bruhat orders and simplex equations
7.2 Equations associated with higher Tamari orders
8 Further remarks
References
Appendix Dov Tamari's Publications
Index

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