GENERALIZED NONCROSSING PARTITIONS AND COMBINATORICS OF COXETER GROUPS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Drew Douglas Armstrong August 2006 c 2006 Drew Douglas Armstrong ALL RIGHTS RESERVED GENERALIZED NONCROSSING PARTITIONS AND COMBINATORICS OF COXETER GROUPS Drew Douglas Armstrong, Ph.D. Cornell University 2006 This thesis serves two purposes: it is a comprehensive introduction to the “Cata- lan combinatorics” of finite Coxeter groups, suitable for nonexperts, and it also introduces and studies a new generalization of the poset of noncrossing partitions. This poset is part of a “Fuss-Catalan combinatorics” of finite Coxeter groups, generalizing the Catalan combinatorics. Our central contribution is the definition of a generalization NC(k)(W ) of the poset of noncrossing partitions corresponding to each finite Coxeter group W and positive integer k. This poset has elements counted by a generalized Fuss-Catalan number Cat(k)(W ), defined in terms of the invariant degrees of W . We develop the theory of this poset in detail. In particular, we show that it is a graded semilattice with beautiful structural and enumerative properties. We count multichains and maximal chains in NC(k)(W ). We show that the order complex of NC(k)(W ) is shellable and hence Cohen-Macaulay, and we compute the reduced Euler character- istic of this complex. We show that the rank numbers of NC(k)(W ) are polynomial in k; this defines a new family of polynomials (called Fuss-Narayana) associated to the pair (W, k). We observe some fascinating properties of these polynomials. We study the structure NC(k)(W ) more specifically when W is a classical type A or type B Coxeter group. In these cases, we show that NC(k)(W ) is isomorphic to a poset of “noncrossing” set partitions in which each block has size divisible by k. Hence, we refer to NC(k)(W ) in general as the poset of “k-divisible noncrossing partitions”. In this case, we prove rank-selected and type-selected enumeration formulas for multichains in NC(k)(W ). We also describe new bijections between multichains of classical noncrossing partitions and classical k-divisible noncrossing partitions. It turns out that our poset NC(k)(W ) shares many enumerative features in common with the generalized nonnesting partitions of Athanasiadis and the gen- eralized cluster complex of Fomin and Reading. We give a basic introduction to these topics and describe several new conjectures relating these three families of “Fuss-Catalan objects”. We mention connections with the theories of cyclic sieving and diagonal harmonics. BIOGRAPHICAL SKETCH Drew Armstrong was born February 26, 1979, during a total eclipse of the sun, and grew up near the edge of civilization, on a Canadian island between lake Huron and lake Superior. From 1998 to 2002, he attended Queen’s University in Kingston, Ontario, where he received a Bachelor of Science in Mathematics. In 2002, he began graduate studies in Mathematics at Cornell University. He received his MS in Mathematics in 2005, and completed his PhD in August 2006, under the supervision of Professor Louis Billera. While at Cornell, Drew met his wife Heather. They are expecting their first child in November 2006. iii For my first child. iv ACKNOWLEDGEMENTS This thesis has been in production for several years, and I owe my gratitude to many people. First, I thank my wife Heather for her support, both moral and mathematical. She suggested to me the bijection from nonnesting partitions to noncrossing partitions at the end of Section 5.1.2. Second, I thank Lou Billera, my advisor at Cornell University, for his patience and sage advice. He introduced me to the world of algebraic combinatorics, and I have benefitted from his perspective and deep knowledge. Third, I thank my collaborator Hugh Thomas for his advice on how to shell things. His suggestion of Theorem 3.7.2 put the finishing touch on Chapter 3. I also thank him for many helpful discussions, and his hospitality at the University of New Brunswick in November 2005. Fourth, there are many people who have contributed to my understanding of this subject through helpful discussions, and their mark can be found in these pages. I want to thank: (in alphabetical order) Christos Athanasiadis, David Bessis, Tom Brady, Ken Brown, Fr´ed´eric Chapoton, Sergey Fomin, Christian Krat- tenthaler, Jon McCammond, Nathan Reading, Vic Reiner and Eleni Tzanaki. In addition, I thank the American Institute of Mathematics at Palo Alto for their hos- pitality at the January 2005 workshop, where my relationships with several of the above named researchers began. I also want to thank John Stembridge for making available his posets and coxeter packages for Maple. These have provided me with valuable experimental data, including the Fuss-Narayana polynomials for the exceptional finite irreducible Coxeter groups (Figure 3.4). Fifth, I want to thank the people who got me started in this subject. I thank Roland Speicher, my undergraduate advisor at Queen’s University, for introducing v me to noncrossing partitions; and I thank Paul Edelman, my first professional contact, for his comments on my undergraduate thesis, and his encouragement and advice. Finally, although I was too late to speak with her, Rodica Simion’s work and passion for noncrossing partitions inspired me in the beginning, and I still follow her example. vi TABLE OF CONTENTS 1 Introduction 1 1.1 Coxeter-CatalanCombinatorics . 1 1.2 OutlineoftheThesis .......................... 14 2 Coxeter Groups and Noncrossing Partitions 24 2.1 CoxeterSystems ............................ 24 2.2 RootSystems .............................. 29 2.3 ReducedWordsandWeakOrder . 33 2.4 AbsoluteOrder ............................. 38 2.5 ShiftingandLocalSelf-Duality . 46 2.6 Coxeter Elements and Noncrossing Partitions . ... 52 2.7 InvariantTheoryandCatalanNumbers. 62 3 k-Divisible Noncrossing Partitions 71 3.1 MinimalFactorizations . 71 3.2 Multichains and Delta Sequences . 73 3.3 Definition of k-Divisible Noncrossing Partitions . 79 3.4 Basic Properties of k-Divisible Noncrossing Partitions . 83 3.4.1 Small values of k ........................ 83 3.4.2 GradedSemilattice . 85 3.4.3 IntervalsandOrderIdeals . 88 3.4.4 Meta-Structure ......................... 94 3.4.5 CoverRelations......................... 95 3.4.6 Automorphisms ......................... 99 3.5 Fuss-Catalan and Fuss-Narayana Numbers . 104 3.6 The Iterated Construction and Chain Enumeration . 115 3.7 Shellability and Euler Characteristics . 126 4 The Classical Types 139 4.1 Classical Noncrossing Partitions . 140 4.2 The Classical Kreweras Complement . 147 4.3 Classical k-Divisible Noncrossing Partitions . 157 4.3.1 ShufflePartitions . 158 4.3.2 TheMainIsomorphism. 162 4.3.3 CombinatorialProperties. 165 4.3.4 Automorphisms . 170 4.4 Type A .................................173 4.5 Type B .................................177 4.6 Type D .................................188 vii 5 Fuss-Catalan Combinatorics 193 5.1 NonnestingPartitions. 194 5.1.1 Classical Nonnesting Partitions . 194 5.1.2 AntichainsintheRootPoset . 195 5.1.3 Partitions and the Lattice of Parabolic Subgroups . 200 5.1.4 Geometric Multichains of Filters . 205 5.1.5 k-Noncrossing and k-Nonnesting Parabolic Subgroups . 216 5.2 ClusterComplexes . .. .. 218 5.2.1 Polygon Triangulations and the Associahedron . 218 5.2.2 TheClusterComplex. 222 5.2.3 The Generalized Cluster Complex . 231 5.3 ChapotonTriangles. 243 5.4 FutureDirections .. .. .. 250 5.4.1 NoncrystallographicRootPoset . 251 5.4.2 CyclicSieving. .. .. 254 5.4.3 DiagonalHarmonics . 256 Bibliography 261 viii LIST OF FIGURES 1.1 The finite irreducible Coxeter groups . 2 1.2 The cycle diagram of permutation (124)(376)(58) . .... 11 1.3 The interval between 1 and (1234) in the Cayley graph of A3 with respect to transpositions is isomorphic to the lattice of noncrossing partitionsoftheset[4]. 12 1.4 2-divisible noncrossing partitions of the set [6] . ....... 13 2.1 Coxeter diagrams of the finite irreducible Coxeter groups...... 26 2.2 The root system Φ(I2(5)) ....................... 31 2.3 The Hasse diagram of Weak(A3); 1-skeleton of the type A3 per- mutohedron; and Cayley graph of A3 with respect to the adjacent transpositions S = (12), (23), (34) ................. 36 { } 2.4 Weak order versus absolute order on A2 ............... 41 2.5 Kν is an anti-automorphism of the interval [µ, ν] Abs(W ) .... 51 µ ⊆ 2.6 NC(A3) as an interval in Abs(A3) .................. 56 2.7 Degrees of the finite irreducible Coxeter groups . .... 63 2.8 The numbers Cat(W ) for the finite irreducible Coxeter groups . 69 3.1 A multichain / delta sequence pair in NC(W )............ 76 3.2 The Hasse diagram of NC(2)(A2)................... 82 (2) 3.3 The Hasse diagram of NC (A2)................... 82 3.4 Fuss-Narayana polynomials for the finite irreducible Coxeter groups 109 3.5 NC(k)(W ) as an order filter in NC(W k)...............116 4.1 A noncrossing and a crossing partition of the set [6]. 141 4.2 NC(4) is isomorphic to NC(A3) ...................144 4.3 Illustration of the Kreweras complement . 149 4.4 Illustration of the relative Kreweras complement. ...... 155 4.5 The poset NC(2)(3) of 2-divisible noncrossing partitions of the set [6].....................................158 4.6 2-shuffle noncrossing partitions of the set [6] . 161 4.7 Anexampleoftheisomorphism(4.15) . 164 4.8 The snake generating set for A7 ....................170 4.9 A type B noncrossingpartition