Generalized noncrossing partitions and combinatorics of Coxeter groups

176 Pages·2009·1.29 MB·English

by  Drew Armstrong

MEMOIRS
of the
American Mathematical Society
Number 949
Generalized Noncrossing Partitions
and Combinatorics of Coxeter Groups
Drew Armstrong
November 2009 • Volume 202 • Number 949 (third of 5 numbers) • ISSN 0065-9266
American Mathematical Society
Number 949
Generalized Noncrossing Partitions
and Combinatorics of Coxeter Groups
Drew Armstrong
November2009 • Volume202 • Number949(thirdof5numbers) • ISSN0065-9266
2000MathematicsSubjectClassification.
Primary05E15,05E25,05A18.
Library of Congress Cataloging-in-Publication Data
Armstrong,Drew,1979-
Generalizednoncrossingpartitionsandcombinatoricsofcoxetergroups/DrewArmstrong.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 949)
Revisionoftheauthor’sthesis(Ph.D.)–CornellUniversity,2006.
“Volume202,number949(thirdof5numbers).”
Includesbibliographicalreferences.
ISBN978-0-8218-4490-8(alk.paper)
1.Combinatorialenumerationproblems. 2.Combinatorialanalysis. 3.Groupactions(Math-
ematics). I.Title.
QA164.A76 2009
511(cid:1).62—dc22 2009029178
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Contents
Acknowledgements ix
Chapter 1. Introduction 1
1.1. Coxeter-Catalan combinatorics 1
1.2. Noncrossing motivation 6
1.3. Outline of the memoir 9
Chapter 2. Coxeter Groups and Noncrossing Partitions 13
2.1. Coxeter systems 13
2.2. Root systems 16
2.3. Reduced words and weak order 19
2.4. Absolute order 22
2.5. Shifting and local self-duality 26
2.6. Coxeter elements and noncrossing partitions 29
2.7. Invariant theory and Catalan numbers 35
Chapter 3. k-Divisible Noncrossing Partitions 41
3.1. Minimal factorizations 41
3.2. Multichains and delta sequences 42
3.3. Definition of k-divisible noncrossing partitions 46
3.4. Basic properties of k-divisible noncrossing partitions 48
3.5. Fuss-Catalan and Fuss-Narayana numbers 59
3.6. The iterated construction and chain enumeration 66
3.7. Shellability and Euler characteristics 72
Chapter 4. The Classical Types 81
4.1. Classical noncrossing partitions 81
4.2. The classical Kreweras complement 87
4.3. Classical k-divisible noncrossing partitions 92
4.4. Type A 102
4.5. Type B 104
4.6. Type D 111
Chapter 5. Fuss-Catalan Combinatorics 115
5.1. Nonnesting partitions 115
5.2. Cluster complexes 128
5.3. Chapoton triangles 143
5.4. Future directions 147
Bibliography 155
v

Abstract
This memoir is a refinement of the author’s PhD thesis — written at Cornell
University (2006). It is primarily a desription of new research but we have also
included asubstantial amount of background material. Atthe heart of thememoir
we introduce and study a poset NC(k)(W) for each finite Coxeter group W and
each positive integer k. When k =1, our definition coincides with the generalized
noncrossingpartitionsintroducedbyBradyandWattinK(π,1)’s for Artin groups
offinitetypeandBessisinThedualbraidmonoid. WhenW isthesymmetricgroup,
we obtain the poset of classical k-divisible noncrossing partitions, first studied by
Edelman in Chain enumeration and non-crossing partitions.
Ingeneral,weshowthatNC(k)(W)isagradedjoin-semilatticewhoseelements
are counted by a generalized “Fuss-Catalan number” Cat(k)(W) which has a nice
closed formula in terms of the degrees of basic invariants of W. We show that
this poset is locally self-dual and we also compute the number of multichains in
NC(k)(W), encoded by the zeta polynomial. We show that the order complex
of the poset is shellable (hence Cohen-Macaulay) and we compute its homotopy
type. Finally, we show that the rank numbers of NC(k)(W) are polynomials in k
with nonzero rational coefficients alternating in sign. This defines a new family of
polynomials (called “Fuss-Narayana”) associated to the pair (W,k). We observe
some interesting properties of these polynomials.
In the case that W is a classical Coxeter group of type A or B, we show that
NC(k)(W) is isomorphic to a poset of “noncrossing” set partitions in which each
blockhassizedivisiblebyk. Thismotivatesourgeneraluseoftheterm“k-divisible
noncrossingpartitions”fortheposetNC(k)(W). IntypesAandB weprove“rank-
selection”and“type-selection”formulas refining the enumeration ofmultichains in
NC(k)(W). Wealsodescribebijectionsrelatingmultichainsofclassicalnoncrossing
partitions to “k-divisible” and “k-equal” noncrossing partitions. Our main tool is
the family of Kreweras complement maps.
Along the way we include a comprehensive introduction to related background
material. Before defining our generalization NC(k)(W), we develop from scratch
ReceivedbytheeditorDecember8,2006;andinrevisedformOctober18,2007.
ArticleelectronicallypublishedonJuly22,2009;S0065-9266(09)00565-1.
2000 MathematicsSubjectClassification. Primary05E15,05E25,05A18.
Key words and phrases. Noncrossing partition, Coxeter group, Coxeter element, Catalan
number,Fuss-Catalannumber,nonnestingpartition,clustercomplex.
ThisworkwassupportedinpartbyNSFgrantDMS-0603567.
(cid:1)c2009 American Mathematical Society
vii
viii ABSTRACT
thetheoryofthegeneralizednoncrossingpartitionsNC(1)(W)asdefinedbyBrady
and Watt in K(π,1)’s for Artin groups of finite type and Bessis in The dual braid
monoid. This involves studying a finite Coxeter group W with respect to its gen-
erating set T of all reflections, instead of the usual Coxeter generating set S. This
is the first time that this material has appeared together.
Finally, it turns out that our poset NC(k)(W) shares many enumerative fea-
tures in common with the generalized nonnesting partitions of Athanasiadis in
Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes and
in On a refinement of the generalized Catalan numbers for Weyl groups; and the
generalized cluster complexes of Fomin and Reading in Generalized cluster com-
plexes and Coxeter combinatorics. Wegiveabasicintroductiontothesetopicsand
wemakeseveralconjecturesrelatingthesethreefamiliesof“Fuss-Catalanobjects”.
Acknowledgements
This work has taken several years and gone through several phases to arrive
in its present form. The following people have contributed along the way; they
deserve my sincerest thanks.
First, IthankLouBillera, myadvisoratCornellUniversity, underwhoseguid-
ance the original draft [2] was written. I thank my wife Heather for her thorough
readingofthe manuscript andfor helpful mathematical discussions. Thankstothe
anonymous referee for many valuable suggestions. And thanks to Hugh Thomas
who made several contributions to my understanding; in particular, he suggested
the proof of Theorem 3.7.2.
Ihavebenefittedfromvaluableconversationswithmanymathematicians; their
suggestionshaveimprovedthismemoirincountlessways. Thanksto: (inalphabet-
ical order) Christos Athanasiadis, David Bessis, Tom Brady, Ken Brown, Fr´ed´eric
Chapoton, Sergey Fomin, Christian Krattenthaler, Cathy Kriloff, Jon McCam-
mond, Nathan Reading, Vic Reiner and Eleni Tzanaki. I would especially like to
thank Christian Krattenthaler for his close attention to the manuscript and for his
subsequent mathematical contributions [84, 85, 86, 87].
Thank you to the American Institute of Mathematics, Palo Alto, and to the
organizers — Jon McCammond, Alexandru Nicaand Vic Reiner — of the January
2005conference[1]atwhichIwasabletomeetmanyoftheabove-namedresearchers
for the first time. Thanks also to John Stembridge for the use of his software
packages posets and coxeter for Maple [128].
Finally, I thank the people who introduced me to noncrossing partitions. My
first research experience was under the supervision of Roland Speicher at Queen’s
University. His papers on noncrossing partitions, as well as those of Paul Edel-
man and Rodica Simion, inspired me to look further. In particular, I thank Paul
Edelman for his encouragement and advice.
ix