Progress in Mathematics

Series EditorsHyman Bass Joseph Oesterlé

Volume 29

Alan WeinsteinYuri Tschinkel

9

T

Jean Marcel Pallo

Associahedra, Tamari Lattices and Related Structures

amari Memorial Festschrift

EditorsJim Stasheff

Folkert M ller-Hoissen ü

© Springer Basel 2012

DOI 10.1007/978-3- 0 -Springer Basel Heidelberg New York Dordrecht London

ISBN 978-3-0348-0404-2 ISBN 978-3-0348-0405-9 (eBook)

Editors

0348- 405 9

Folkert Müller-Hoissen

Dynamics and Self-OrganizationMax-Planck-Institute for

Germany

Jean Marcel PalloDépartement d’Informatique, LE2IUniversité de Bourgogne

Department of MathematicsJim Stasheff

Chapel Hill, NC

Göttingen Dijon France

University of North Carolina

USA

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Preface

On the occasion of the centennial birthday of the mathematician Dov Tamari (1911–2006), born as the German Bernhard Teitler, this book commemorates his groundbreaking work resulting in an associativity theory, with important contributions tothe “word (decision) problem”, as well as lattice theory and geometric combinatorics.The editors of this book invited designated researchers to present modern areas ofmathematics that are related to Tamari’s work.

To a monomial (word) formed from a set (of letters), one can assign differentmeanings by properly distributing brackets. If the bracketing expresses a binaryoperation on the set, associativity becomes an issue. Interpreting associativity asa (left- or rightward) substitution rule leads to what is known as a Tamari lattice.This partial order on a Catalan set (i.e., the number of its elements is a Catalannumber) first appeared in 1951 in Dov Tamari’s thesis at the Sorbonne in Paris. Itturned out that these Tamari lattices possess realizations on special polytopes, calledassociahedra, which appeared in a different context in Jim Stasheff’s thesis in 1961.In fact, associahedra already appeared in Tamari’s thesis, but not in the part that waspublished. Since then these beautiful structures, and quite a number of importantgeneralizations, have made their appearance in many publications in different areasof pure and applied mathematics, such as Algebra, Combinatorics, Computer Science,Category Theory, Geometry, Topology, and more recently also in Physics. It is thisinterdisciplinary nature of these structures that provides much of their fascinationand value.

In the first chapter of this book, Folkert Muller-Hoissen and Hans-Otto Waltherdescribe Tamari’s extremely troubled life. When the Nazis came to power in Germanyof 1933, he saw himself forced to leave Germany, losing the possibility of a smoothacademic career he could have had in a less cruel political and social environment.All the obstacles along his further way luckily could not break his dedication andpassion for mathematics. His uncompromising demand for honesty and fairness on alllevels, including politics, surely did not make his life easier. The chapter about DovTamari also offers an elementary introduction to some aspects of his mathematicalwork. It is supplemented by Carl Maxson’s reminiscences as a student of Tamari.

v

vi Preface

Jim Stasheff, whose name is firmly connected with associahedra, traces the latterback to Tamari’s 1951 thesis, reviews their history, and leads the reader to moderndevelopments. Jean-Louis Loday develops an arithmetic of (planar rooted binary)trees, a framework in which Tamari lattices find a natural place. He also reviewsrealizations of the latter as polytopes, the associahedra.

The further chapters in this book are of a somewhat more advanced nature. SusanGensemer reviews the problem of extending a partially defined binary operationon a set (partial groupoid) to a completely defined associative binary operation(semigroup). This problem has been at the very roots of Tamari’s mathematicalresearch.

We grouped together articles that deal primarily with associahedra and relatedfamilies of polytopes, and then those that center more around Tamari lattices andrelated families of posets.

Satyan Devadoss, Benjamin Fehrman, Timothy Heath and Aditi Vashist treat geo-metric and combinatorial aspects of the moduli space of “particles” on the Poincaredisk. In this framework, a generalization of associahedra shows up, the cyclohe-dra. Cesar Ceballos and Gunter Ziegler summarize some mysteries and questionsconcerning realizations of the associahedra. A well-known class of polytopes, thepermutahedra (also called permutohedra), and moreover polytopes obtained fromCambrian lattices, which generalize Tamari lattices, are the subject of ChristopheHohlweg’s article, highlighting the role of finite reflection groups in their realiza-tions. Further classes of polytopes, like flag nestohedra, graph-associahedra andgraph-cubeahedra, arise from truncations of cubes, and their properties are describedin the article by Victor Buchstaber and Vadim Volodin. Stefan Forcey reports onan extension of the Tamari order to families of polytopes called multiplihedra andcomposihedra, and explores the interplay between lattice structures and Hopf algebrastructures.

Patrick Dehornoy presents an exhaustive study of the connection between Tamarilattices and the Thompson group F , which consists of a special class of piecewiselinear homeomorphisms of the unit interval onto itself. Ross Street looks at the Tamarilattice as an example of an operad and dives into monoidal categories. FredericChapoton considers the category of modules over the incidence algebra of a Tamarilattice, and also the derived category of the former. Hugh Thomas explains how theTamari lattice arises in the context of the representation theory of quivers. NathanReading traces the way from Tamari lattices to Cambrian lattices, in the contextof finite Coxeter groups, reviews the construction of Cambrian fans, and moreovermakes contact with the important concept of cluster algebras. Filippo Disanto, LucaFerrari, Renzo Pinzani and Simone Rinaldi present a unified setting for Dyck andTamari lattices. Dyck lattices are a refinement of Tamari lattices. The restriction ofthe weak (strong) Bruhat order on permutations of a fixed length to “312-avoidingpermutations” leads to the Tamari (Dyck) order. A generalization of the Tamariorder to a partial order on the set of “tubings” of a simple graph is described inMarıa Ronco’s work. Jorg Rambau and Victor Reiner present a survey of higherStasheff-Tamari orders (which first appeared in the work of Mikhail Kapranov and

Preface vii

Vladimir Voevodsky). These are posets defined on triangulations of cyclic polytopesand there is a relation with higher Bruhat orders (first introduced by Yuri Manin andVadim Schechtman).

A physical realization of maximal chains of Tamari lattices in terms of tree-shaped soliton solutions of the Kadomtsev-Petviashvili (KP) equation, describing,e.g., shallow water waves, is the subject of the article by Aristophanes Dimakis andFolkert Muller-Hoissen. The analysis of KP solitons naturally leads to a reduction ofhigher Bruhat orders to higher Tamari orders, which is different from the relationdescribed by Rambau and Reiner.

We hope that this book will convey to the reader a bit of the fascination that wasexperienced by those who contributed to the foundations and modern developmentsdescribed in it.

Finally, we would like to thank Dr. Thomas Hempfling for his efficient help duringthe publishing process.

Gottingen, Dijon, Chapel Hill Folkert Muller-HoissenOctober 2011 Jean Pallo

Jim Stasheff

Contents

Dov Tamari (formerly Bernhard Teitler) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Folkert Muller-Hoissen and Hans-Otto Walther

1 Germany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Palestine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 France . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Tamari’s thesis and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Israel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Via Princeton to Brazil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 The Netherlands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 USA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Tamari-Technion Affair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

9.1 Freudenthal’s initiative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.2 AMS and CAFTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

10 After retirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.1 Mathematical activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.2 Back to the roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

11 Further recollections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

On Being a Student of Dov Tamari . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Carl Maxson

How I ‘met’ Dov Tamari . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Jim Stasheff

1 Introduction/History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.1 The so-called Stasheff polytope . . . . . . . . . . . . . . . . . . . . . . 461.2 Giving due credit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.3 Outline of this paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2 Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 Other polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1 Multiplihedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

ix

x Contents

3.2 Cyclohedra and generalized associahedra . . . . . . . . . . . . . . 543.3 Compactified configuration spaces . . . . . . . . . . . . . . . . . . . 55

4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Coderivation interpretation and curvature . . . . . . . . . . . . . . 564.2 Pentagonal algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3 The pentagonal algebra of open string field theory . . . . . . 584.4 An-structures for 4 < n≤ ∞ . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Opportunities – missed and not . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Dichotomy of the Addition of Natural Numbers . . . . . . . . . . . . . . . . . . . . . . 65Jean-Louis Loday

1 About the formula 1+1 = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 Splitting the integers into pieces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 Trees and addition on trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 Where we show that 1+1 = 2 and 2+1 = 3 . . . . . . . . . . . . . . . . . . 695 The integers as molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 Multiplication of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 Trees and polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 Realizing the associahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8.1 Tamari poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738.2 Associahedron and regular pentagons . . . . . . . . . . . . . . . . . 748.3 Realizations of the associahedron . . . . . . . . . . . . . . . . . . . . 75

9 Associahedron and the trefoil knot . . . . . . . . . . . . . . . . . . . . . . . . . . . 77References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Partial Groupoid Embeddings in Semigroups . . . . . . . . . . . . . . . . . . . . . . . . 81Susan H. Gensemer

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812 Definitions and preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.1 Embedding into a partially ordered semigroup . . . . . . . . . 894.2 Embedding into a commutative semigroup . . . . . . . . . . . . 95

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Moduli Spaces of Punctured Poincare Disks . . . . . . . . . . . . . . . . . . . . . . . . . 99Satyan L. Devadoss, Benjamin Fehrman, Timothy Heath, and Aditi Vashist

1 Motivation from physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992 Particles on the Poincare disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013 The Fulton-McPherson compactification . . . . . . . . . . . . . . . . . . . . . . 1064 Group actions on screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105 Combinatorial results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Contents xi

Realizing the Associahedron: Mysteries and Questions . . . . . . . . . . . . . . . . 119Cesar Ceballos and Gunter M. Ziegler

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192 Realization space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 Vertices on a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234 Realizing the multiassociahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Permutahedra and Associahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Christophe Hohlweg

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1292 Permutahedra and finite reflection groups . . . . . . . . . . . . . . . . . . . . . 131

2.1 Finite reflection groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312.2 Permutahedra as V -polytopes . . . . . . . . . . . . . . . . . . . . . . . 1322.3 Root systems and permutahedra as H -polytopes . . . . . . . 1362.4 Faces of permutahedra and the weak order . . . . . . . . . . . . . 141

3 Coxeter generalized associahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.1 Coxeter elements and Coxeter singletons . . . . . . . . . . . . . . 1433.2 Coxeter generalized associahedra as H -polytopes . . . . . . 1453.3 Faces and almost positive roots . . . . . . . . . . . . . . . . . . . . . . 1473.4 Edges and Cambrian lattices . . . . . . . . . . . . . . . . . . . . . . . . 1503.5 Isometry classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.6 Integer coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4 Further developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Combinatorial 2-truncated Cubes and Applications . . . . . . . . . . . . . . . . . . . 161Victor M. Buchstaber and Vadim D. Volodin

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1612 Simple polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

2.1 Enumerative polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633 Class of 2-truncated cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644 Smooth toric varieties over 2-truncated cubes . . . . . . . . . . . . . . . . . . 1685 Small covers of 2-truncated cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686 Nestohedra and graph-associahedra . . . . . . . . . . . . . . . . . . . . . . . . . . 1697 Building set as a structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718 Flag nestohedra as 2-truncated cubes . . . . . . . . . . . . . . . . . . . . . . . . . 1749 Recursion formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

9.1 Associahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1789.2 Cyclohedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1799.3 Permutohedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1809.4 Stellohedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

10 Bounds of face polynomials for flag nestohedra andgraph-associahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

11 Nested polytopes and graph-cubeahedra as 2-truncated cubes . . . . 183References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

xii Contents

Extending the Tamari Lattice to Some Compositions of Species . . . . . . . . . 187Stefan Forcey

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1871.1 Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

2 Several flavors of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1902.1 Ordered, bi-leveled and painted trees . . . . . . . . . . . . . . . . . 1902.2 Trees with corrollas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

3 Interval retracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1954 Lattices and polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2005 Algebraic implications of interval retracts . . . . . . . . . . . . . . . . . . . . . 205

5.1 Coalgebras of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2065.2 Cofree composition of coalgebras . . . . . . . . . . . . . . . . . . . . 2075.3 The coalgebra of painted trees . . . . . . . . . . . . . . . . . . . . . . . 208

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Tamari Lattices and the Symmetric Thompson Monoid . . . . . . . . . . . . . . . 211Patrick Dehornoy

1 The framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2121.1 Parenthesized expressions and associativity . . . . . . . . . . . . 2121.2 Trees and rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2131.3 Richard Thompson’s group F . . . . . . . . . . . . . . . . . . . . . . . 2151.4 The action of F on trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2161.5 The generators aα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2191.6 Presentation of F in terms of the elements aα . . . . . . . . . . 220

2 A lattice structure on the Thompson group F . . . . . . . . . . . . . . . . . . 2232.1 The symmetric Thompson monoid F+

sym . . . . . . . . . . . . . . . . 2232.2 Subword reversing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2252.3 The lattice structure of F+

sym . . . . . . . . . . . . . . . . . . . . . . . . . . 2302.4 Computing the operations . . . . . . . . . . . . . . . . . . . . . . . . . . 232

3 The Polish normal form on F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2353.1 The Polish encoding of trees . . . . . . . . . . . . . . . . . . . . . . . . 2363.2 The Polish algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2373.3 The covering relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2393.4 The Polish normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

4 Distance in Tamari lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2444.1 The diameter of Tn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2444.2 Syntactic invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2454.3 The embedding of F+

sym into F . . . . . . . . . . . . . . . . . . . . . . . . 247References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Parenthetic Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Ross Street

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2512 Parengebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2523 Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2554 Well-formed words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Contents xiii

5 Free monoidal categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

On the Categories of Modules Over the Tamari Posets . . . . . . . . . . . . . . . . 269Frederic Chapoton

1 Representations of posets in general . . . . . . . . . . . . . . . . . . . . . . . . . . 2702 Dimension of the incidence algebra of Yn . . . . . . . . . . . . . . . . . . . . . 2713 Main conjecture on the category DbmodYn . . . . . . . . . . . . . . . . . . . 2724 At the level of Grothendieck groups . . . . . . . . . . . . . . . . . . . . . . . . . . 2735 Derived equivalences with other posets . . . . . . . . . . . . . . . . . . . . . . . 2756 Quasi-homogeneous isolated singularities . . . . . . . . . . . . . . . . . . . . . 277References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

The Tamari Lattice as it Arises in Quiver Representations . . . . . . . . . . . . . 281Hugh Thomas

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2812 Quiver representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2823 Subrepresentations, quotient representations, and extensions . . . . . 2824 Pullbacks of extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2835 Indecomposable representations of the quiver An . . . . . . . . . . . . . . . 2836 Morphisms and extensions between indecomposable

representations of An . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2857 Subcategories of repQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2868 Quotient-closed subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2869 Subcategories ordered by inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 28710 Torsion classes in repAn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28811 Related posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

From the Tamari Lattice to Cambrian Lattices and Beyond . . . . . . . . . . . . 293Nathan Reading

1 A map from permutations to triangulations . . . . . . . . . . . . . . . . . . . . 2932 Bringing lattice congruences into the picture . . . . . . . . . . . . . . . . . . . 2973 Cambrian lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3014 Cambrian fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3065 Sortable elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3106 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

6.1 Coxeter-Catalan combinatorics . . . . . . . . . . . . . . . . . . . . . . 3156.2 Sortable elements in quiver theory . . . . . . . . . . . . . . . . . . . 3156.3 Combinatorial Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . 3166.4 Cluster algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

xiv Contents

Catalan Lattices on Series Parallel Interval Orders . . . . . . . . . . . . . . . . . . . 323Filippo Disanto, Luca Ferrari, Renzo Pinzani, Simone Rinaldi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3242 Series-parallel interval orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

2.1 Preorder linear extensions of series parallelinterval orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

3 Catalan lattices on series parallel interval orders . . . . . . . . . . . . . . . . 3293.1 The Dyck lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3303.2 The Tamari lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

4 Series parallel interval orders and patternavoiding permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3334.1 The Tamari lattice and the weak Bruhat

order on Av(312) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3354.2 The Dyck lattice and the strong Bruhat

order on Av(312) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3355 Further works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Generalized Tamari Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339Marıa Ronco

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3392 Graph associahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3403 Binary operations on tubings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3424 Generalization of Tamari order to tubings . . . . . . . . . . . . . . . . . . . . . 346References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

A Survey of the Higher Stasheff-Tamari Orders . . . . . . . . . . . . . . . . . . . . . . 351Jorg Rambau and Victor Reiner

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512 Cyclic polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3523 The two orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3534 Encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

4.1 Submersion sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3604.2 Snug rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.3 Non-interlacing separated d

2 -faces . . . . . . . . . . . . . . . . . . . . 3635 Lattice property, homotopy types and Mobius function . . . . . . . . . . 3646 Connection to flip graph connectivity . . . . . . . . . . . . . . . . . . . . . . . . . 367

6.1 Bistellar flips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3676.2 The flip connectivity question . . . . . . . . . . . . . . . . . . . . . . . 3696.3 The flip graph of a cyclic polytope . . . . . . . . . . . . . . . . . . . 3696.4 Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3716.5 The case d = 2: the rotation graph of binary trees . . . . . . . 372

7 Subdivisions and the Baues problem . . . . . . . . . . . . . . . . . . . . . . . . . 3737.1 Subdvisions and secondary polytopes . . . . . . . . . . . . . . . . . 3737.2 Baues’s original problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3747.3 Cellular strings and the generalized Baues problem . . . . . 375

Contents xv

8 Connection to the higher Bruhat orders . . . . . . . . . . . . . . . . . . . . . . . 3788.1 Definition of higher Bruhat orders . . . . . . . . . . . . . . . . . . . . 3788.2 Some geometry of higher Bruhat orders . . . . . . . . . . . . . . . 3808.3 The map from higher Bruhat to higher Stasheff-Tamari

orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3859 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

KP Solitons, Higher Bruhat and Tamari Orders . . . . . . . . . . . . . . . . . . . . . . 391Aristophanes Dimakis and Folkert Muller-Hoissen

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912 KP solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3943 Tropical approximation of a subclass of KP line solitons . . . . . . . . . 3954 Higher Bruhat and higher Tamari orders . . . . . . . . . . . . . . . . . . . . . . 402

4.1 Higher Bruhat orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4034.2 Higher Tamari orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

5 KP line soliton evolutions in the case M = 5 . . . . . . . . . . . . . . . . . . . 4126 Some insights into the case M > 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4157 An algebraic construction of higher Bruhat and Tamari orders . . . . 417

7.1 Higher Bruhat orders and simplex equations . . . . . . . . . . . 4177.2 Equations associated with higher Tamari orders . . . . . . . . 420

8 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

Appendix: Dov Tamari’s Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

List of Contributors

Buchstaber, VictorGeometry and Topology Department, Steklov Institute of Mathematics, Gubkina str.8, 119991 Moscow, Russiae-mail: buchstab@mi.ras.ru

Ceballos, CesarInstitut fur Mathematik, Freie Universitat Berlin, Arnimallee 2, 14195 Berlin,Germanye-mail: ceballos@math.fu-berlin.de

Chapoton, FredericInstitut Camille Jordan, Universite Claude Bernard Lyon 1, Batiment Braconnier, 21Avenue Claude Bernard, 69622 Villeurbanne Cedex, Francee-mail: chapoton@math.univ-lyon1.fr

Dehornoy, PatrickLaboratoire de Mathematiques Nicolas Oresme, Universite de Caen, 14032 Caen,Francee-mail: patrick.dehornoy@gmail.com

Devadoss, SatyanDepartment of Mathematics and Statistics, Williams College, Williamstown, MA01267, USAe-mail: satyan.devadoss@williams.edu

Dimakis, AristophanesDepartment of Financial and Management Engineering, University of the Aegean,41 Kountourioti Str., GR-82100 Chios, Greecee-mail: dimakis@aegean.gr

Disanto, FilippoInstitut fur Genetik, Universitat Koln, Zulpicher Str. 47a, 50674 Koln, Germanye-mail: disafili@yahoo.it

xvii

xviii List of Contributors

Fehrman, BenjaminDepartment of Mathematics, University of Chicago, Chicago, IL 60637, USAe-mail: bfehrman@math.uchicago.edu

Ferrari, LucaDipartimento di Sistemi e Informatica, Universita degli Studi di Firenze, Viale G.B.Morgagni 65, 50134 Firenze, Italye-mail: ferrari@dsi.unifi.it

Forcey, StefanDepartment of Mathematics, Buchtel College of Arts and Sciences, The Universityof Akron, Akron, OH 44325-4002, USAe-mail: sforcey@gmail.com

Gensemer, Susan HeleneDepartment of Economics, Maxwell School of Citizenship and Public Affairs,Syracuse University, Syracuse, NY 13244-1020, USAe-mail: gensemer@maxwell.syr.edu

Heath, TimothyDepartment of Mathematics, Columbia University, New York, NY 10027, USAe-mail: timheath@math.columbia.edu

Hohlweg, ChristopheDepartement de Mathematiques – LaCIM, Universite du Quebec a Montreal, CP8888 Succ. Centre-Ville, Montreal, Quebec, H3C 3P8 Canadae-mail: hohlweg.christophe@uqam.ca

Loday, Jean-LouisInstitut de Recherche Mathematique Avancee, CNRS et Universite de Strasbourg, 7rue Rene-Descartes, 67084 Strasbourg, Francee-mail: loday@math.unistra.fr

Maxson, CarlMathematics Department, Texas A&M University, College Station, TX 77843-3368,USAe-mail: cjmaxson@math.tamu.edu

Muller-Hoissen, FolkertMax-Planck-Institute for Dynamics and Self-Organization, Bunsenstrasse 10,D-37073 Gottingen, Germanye-mail: folkert.mueller-hoissen@ds.mpg.de

Pallo, Jean MarcelDepartement d’Informatique, LE2I, Universite de Bourgogne, B.P. 47870, 21078Dijon Cedex, Francee-mail: pallo@u-bourgogne.fr

List of Contributors xix

Pinzani, RenzoDipartimento di Sistemi e Informatica, Universita degli Studi di Firenze, Viale G.B.Morgagni 65, 50134 Firenze, Italye-mail: pinzani@dsi.unifi.it

Rambau, JorgLehrstuhl fur Wirtschaftsmathematik, Universitat Bayreuth, D-95440 Bayreuth,Germanye-mail: joerg.rambau@uni-bayreuth.de

Reading, NathanDepartment of Mathematics, North Carolina State University, SAS Hall 4118, Box8205, Raleigh, NC 27695, USAe-mail: nathan reading@ncsu.edu

Reiner, VictorSchool of Mathematics, University of Minnesota, Minneapolis, MN 55455, USAe-mail: reiner@math.umn.edu

Rinaldi, SimoneDipartimento di Scienze Matematiche e Informatiche, Pian dei Mantellini, 44, 53100,Siena, Italye-mail: rinaldi@unisi.it

Ronco, MariaInstituto de Matematica y Fısica, Universidad de Talca, 2 norte 685 Talca, Chilee-mail: mariaronco@inst-mat.utalca.cl

Stasheff, JimMathematics Department, University of North Carolina at Chapel Hill, CB #3250,Phillips Hall Chapel Hill, NC 27599, USAe-mail: jds@math.upenn.edu

Street, RossMathematics Department, Macquarie University, New South Wales 2109, Australiae-mail: ross.street@mq.edu.au

Thomas, HughDepartment of Mathematics and Statistics, University of New Brunswick,Fredericton, NB, E3B 5A3, Canadae-mail: hugh@math.unb.ca

Vashist, AditiDepartment of Mathematics, University of Michigan, Ann Arbor, MI 48109, USAe-mail: avashist@umich.edu

Volodin, VadimSteklov Institute of Mathematics, Gubkina str. 8, 119991 Moscow, Russiae-mail: volodinvadim@gmail.com

xx List of Contributors

Walther, Hans-OttoMathematisches Institut, University of Gießen, Arndtstraße 2, D-35392 Gießen,Germanye-mail: hans-otto.walther@math.uni-giessen.de

Ziegler, Gunter M.Institut fur Mathematik, Freie Universitat Berlin, Arnimallee 2, 14195 Berlin,Germanye-mail: ziegler@math.fu-berlin.de