Cayley-Catalan Combinatorics of Affine Permutation Groups [PhD thesis]

130 Pages·2017·1.11 MB·English

by  Robin Sulzgruber

DISSERTATION / DOCTORAL THESIS
Titel der Dissertation / Title of the Doctoral Thesis
“Cayley–Catalan Combinatorics of Affine Permutation
Groups”
verfasst von / submitted by
Robin Sulzgruber MSc
angestrebter akademischer Grad / in partial fulfilment of the requirements for the degree of
Doktor der Naturwissenschaften (Dr. rer. nat.)
Wien, 2017 / Vienna 2017
Studienkennzahllt.Studienblatt/ A796605405
degreeprogrammecodeasitappearsonthestudent
recordsheet:
Dissertationsgebietlt. Studienblatt/ Mathematik
fieldofstudyasitappearsonthestudentrecordsheet:
Betreutvon/Supervisor: Univ.-Prof.Dr.ChristianKrattenthaler

Contents
Table of Contents iii
Acknowledgements v
Abstract vii
Zusammenfassung ix
Chapter 0. Introduction 1
Chapter 1. Notation 5
1.1. Partitions 5
1.2. Lattice paths 6
1.3. Roots, hyperplanes and reflections 6
1.4. Affine permutations 9
1.5. Affine signed permutations 12
Chapter 2. Exploring the Catalan-cube 17
2.1. Dyck paths and Catalan numbers 17
2.2. Rational Catalan numbers 18
2.3. Cayley numbers and parking functions 20
2.4. Rational parking functions 22
2.5. The q,t-Catalan numbers 23
2.6. The q,t-Cayley numbers 27
2.7. Rational q,t-Catalan numbers 29
Chapter 3. Core partitions 31
3.1. Cores and the affine symmetric group 31
3.2. Affine inversions 37
3.3. Simultaneous core partitions 41
3.4. The skew-length statistic 43
Chapter 4. The finite torus 49
4.1. Coxeter–Catalan numbers 49
4.2. The finite torus 51
4.3. Lattice path models for the finite torus 52
4.4. p-stable elements of the affine Weyl group 57
4.5. The dinv-statistic 60
Chapter 5. The Shi arrangement 63
5.1. Non-nesting parking functions and the Shi arrangement 63
5.2. Lattice path models for non-nesting parking functions 65
iii
iv CONTENTS
5.3. The area-statistic 71
5.4. Rational Shi tableaux 72
Chapter 6. The zeta map 83
6.1. The uniform zeta map 83
6.2. From area to dinv 85
6.3. Combinatorics in type C 87
n
6.4. Combinatorics in type D 97
n
6.5. Combinatorics in type B 108
n
Bibliography 113
Index 117
Acknowledgements
FirstandforemostIwanttothankChristianKrattenthalerwhoalwaysgavemethefreedomto
pursuemyownideas,butatthesametimeprovidedadviseanddirectionwheneveritwasneeded.
Thank you also for encouraging and making it possible for me to travel and visit conferences
during my studies. I could not have hoped for a better advisor.
I gained a lot from working and discussing with numerous people generously sharing their
knowledgewithme. ThankyoutoCesarCeballos,TomDenton,IlseFischer,JimHaglund,Myrto
Kallipoliti, Christoph Neumann, Vivien Ripoll, Carsten Schneider, Christian Stump, Marko
Thiel, Nathan Williams and Meesue Yoo.
I gratefully acknowledge the financial support from the Austrian Science Fund FWF in the
framework of the special research program SFB F50. I would also like to thank the University
of Vienna and in particular the Faculty of Mathematics. Throughout my time here I have
experienced an environment that was both fascinating mathematically and very enjoyable in
general.
IcountmyselfveryluckybepartofalargegroupofawesomeyoungpeoplewhoworkedinVienna
whileIwashere: Alex,Andrei,Cesar,Chen,Christoph,Daniel,Emma,Florian,Gwendal,Hans,
Henri, Judith, Koushik, Lukas, Manjil, Marko, Meesue, Michael, Myrto, Sabine, Stephan, Ting,
Tomack, Viviane, Vivien, Yvonne, Zhicong and Z´ofia. Thank you for drinks had, games played,
riddling and travelling the world together!
Finally, thankyoutoallmyMugglefriends(non-mathematicalfriends)andmyfamilyfortheir
genuine interest in what a parking function might be!
v

Abstract
Cayley–Catalan combinatorics refers to the study of combinatorial objects counted by (gener-
alised)CatalannumbersorCayleynumbers. Examplesofclassicalcombinatorialobjectstreated
in this thesis that fall into this category are Dyck paths, parking functions and core partitions.
These objects turn out to be closely related to the group of affine permutations and their inver-
sions. ManyinvolvedideascarryovertoarbitraryaffineWeylgroups. Exploringthisconnection
wereviewthefinitetorus,theShiarrangementandnon-nestingparkingfunctions. Inparticular,
we define new combinatorial models for these objects in terms of labelled lattice paths when the
crystallographic root system is of classical type.
Several combinatorial statistics on Catalan objects have been introduced to give combinatorial
interpretations for polynomials appearing in representation theory or algebraic geometry. For
example, Haglund’s bounce-statistic, Haiman’s dinv-statistic or Armstrong’s skew-length of a
partitionhaveallbeenusedtodefineq-analoguesofCatalannumbers. Westrengthenandexpand
on previously known symmetry properties of the skew-length statistic. The dinv-statistic is
generalisedtoastatisticonthefinitetorus,allowingforanewdefinitionofq-Catalannumbersfor
arbitraryWeylgroups. Furthermore,weextendthenotionofShitableauxtogiveageneralisation
of the skew-length statistic for affine Weyl groups, thereby enabling us to give a combinatorial
definition of rational q-Catalan numbers for Weyl groups.
An important bijection in this field is the so called zeta map. The original zeta map is a
bijectiononthesetofDyckpaths,however,itcanbegeneralisedtoauniformbijectionattached
to any Weyl group. We prove that this bijection transforms the dinv-statistic on elements of the
finite torus into the area-statistic on non-nesting parking functions. Furthermore, we develop
the lattice path combinatorics of the zeta map for the infinite families of crystallographic root
systems in analogy to the connection to Dyck paths when the Weyl group is the symmetric
group. This leads to the discovery of two new bijections between ballot paths and lattice paths
in a square, both of which are known to be counted by central binomial coefficients.
vii

Zusammenfassung
Die Bezeichnung Cayley–Catalan Kombinatorik bezieht sich auf das Studium von kombina-
torischen Objekten, die von (verallgemeinerten) Catalan-Zahlen oder Cayley-Zahlen abgez¨ahlt
werden. Beispiele fu¨r klassische kombinatorische Objekte, die in dieser Dissertation behandelt
werden und in diese Kategorie fallen, sind Dyck-Pfade, Parkfunktionen oder Kernpartitionen.
Alle diese Objekte sind eng verwandt mit der Gruppe affiner Permutationen und deren Inver-
sionen. Viele der zu beobachtenden Konzepte lassen sich auf allgemeine affine Weyl-Gruppen
u¨bertragen. In diesem Zusammenhang besprechen wir den endlichen Torus, das Shi-Gefu¨ge und
nichtverschachtelte Parkfunktionen. Insbesondere definieren wir in den F¨allen, in denen das zu-
grundeliegendekristallographischeWurzelsystemvonklassischemTypist, neuekombinatorische
Modelle fu¨r diese Objekte anhand von bezeichneten Gitterpunktwegen.
Verschiedene kombinatorische Statistiken wurden auf Catalan-Objekten definiert, um Poly-
nomen, dieinderDarstellungstheorieoderderalgebraischenGeometrieauftauchen, einekombi-
natorische Bedeutung zu verleihen. Zum Beispiel wurden Haglunds bounce-Statistik, Haimans
dinv-Statistik und Armstrongs Schie߬ange einer Zahlenpartition benutzt, um q-Analoga der
Catalan-Zahlen zu definieren. Wir verfeinern und verallgemeinern bisher bekannte Symme-
trieeigenschaften der Schie߬ange. Die dinv-Statisik wird zu einer Statistik auf den Elementen
des endlichen Torus verallgemeinert. Dies erlaubt es uns q-Catalan-Zahlen fu¨r beliebige Weyl-
Gruppen zu definieren. Zus¨atzlich erweitern wir den Begriff von Shi-Tafeln, um eine Verallge-
meinerungderSchiefl¨angefu¨raffineWeyl-Gruppenzuerm¨oglichen. DadurchsindwirinderLage,
eine kombinatorische Definition von rationalen q-Catalan-Zahlen fu¨r Weyl-Gruppen anzugeben.
Eine wichtige Bijection in diesem Gebiet ist die sogenannte Zeta-Abbildung. Die urspru¨ngliche
Zeta-Abbildung ist eine Bijektion auf der Menge der Dyck-Pfade, jedoch wurde sie einheitlich
zu einer einer beliebigen Weyl-Gruppe zugeh¨origen Bijektion verallgemeinert. Wir beweisen,
dassdieseBijektiondiedinv-StatistikaufdenElementendesendlichenTorusaufdasGebietder
entsprechendennichtverschachteltenParkfunktionu¨berfu¨hrt. WeitersentwickelnwirinAnalogie
zuderVerbindungzuDyck-PfadenimFalledersymmetrischenGruppedieGitterpunktwegskom-
binatorik der Zeta-Abbildung fu¨r die unendlichen Familien von kristallographischen Wurzelsys-
temen. Dies fu¨hrt zur Entdeckung von zwei neuen Bijektionen zwischen Abstimmungspfaden
und Gitterpunktwegen, die einem Quadrat eingeschrieben sind, von welchen jeweils bekannt ist,
dass sie von Zentralbinomialkoeffizienten abgez¨ahlt werden.
ix
CHAPTER 0
Introduction
Three of the most ubiquitous sequences of numbers in combinatorics are the factorials n!, the
Catalannumbers(cid:0)2n(cid:1)/(n+1)andtheCayleynumbers(n+1)n−1. Factorialscountpermutations
n
of an n-set which can be represented as bijections between n-sets, as linear orders of an n-set or
as collections of labelled cycles. Catalan numbers count a wealth of combinatorial objects such
asbinarytrees,Dyckpaths,non-crossingpartitions,non-crossingperfectmatchings,certainpat-
tern avoiding permutations, rooted plane trees, triangulations of polygons and many more [70].
Cayleynumberscountmostobviouslymapsfroman(n−1)-settoan(n+1)-set, mostfamously
labelled trees, but also parking functions and regions of the Shi arrangement among others.
Itisoneofthebeautifulcoincidencesinalgebraiccombinatoricsthatthesenumbersarerelated
to each other elegantly by way of a group G acting on a set X such that
(cid:18) (cid:19)
1 2n
#G=n!, #X =(n+1)n−1 and #{Gx:x∈X}= .
n+1 n
As we shall see, a convenient choice is to take the symmetric group G = S acting on the set
n
X =PF of parking functions. An integer vector with non-negative entries (f ,...,f )∈Nn is
n 1 n
called parking function if there exists a permutation σ ∈ S such that f < σ(i) for all i. For
n i
example, there are sixteen parking functions of length three.
(cid:110) (cid:111)
PF = (0,0,0),(0,0,1),(0,1,0),(1,0,0),(0,0,2),(0,2,0),(2,0,0),(0,1,1),
3 (1,0,1),(1,1,0),(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1),(2,1,0)
Clearly the symmetric group acts on the set of parking functions via permutation of entries. At
this point the orbits {Gx:x∈X} are easily seen to be indexed by increasing parking functions,
which can be identified with Dyck paths. There are five increasing parking functions of length
three.
(0,0,0) (0,0,1) (0,0,2) (0,1,1) (0,1,2)
Of course there are other equivalent choices for G and X.
The study of the parking function representation has given rise to a whole new branch in
algebraic combinatorics. It connects a variety of topics in mathematics – from Macdonald poly-
nomials, and symmetric functions in general [29, 19], and the representation theory of Weyl
groups and related algebras [9, 33], to hyperplane arrangements [5], Hilbert schemes [41, 42]
andknotinvariants[34]. Itisinpartresponsiblefortherevivalofclassicalcombinatorialobjects
such as core partitions and abacus diagrams.
Over the years Catalan numbers have been generalised in many different directions, which I
prefer to view as dimensions. In this thesis the focus is on the following four dimensions.
The first is the introduction of a rational parameter, which encompasses the introduction of a
Fuß-parameter. Many of the objects counted by Catalan numbers listed above can be assigned
1