Catalan combinatorics of crystallographic root systems [PhD thesis]

79 Pages·2015·0.735 MB·English

by  Marko Thiel

DISSERTATION
Titel der Dissertation
Catalan combinatorics of
crystallographic root systems
Verfasser
Marko Thiel
angestrebter akademischer Grad
Doktor der Naturwissenschaften (Dr.rer.nat)
Wien, im August 2015
Studienkennzahl lt. Studienblatt: A 791 405
Dissertationsgebiet lt. Studienblatt: Mathematik
Betreuer: Univ.-Prof. Dr. Christian Krattenthaler
2
CONTENTS
1 introduction 9
1.1 Catalancombinatorics 9
1.1.1 Catalannumbers 9
1.1.2 Coxeter-Catalannumbers 9
1.1.3 Fuß-Catalannumbers 10
1.1.4 RationalCatalannumbers 10
1.2 DiagonalHarmonics 11
1.3 Structureofthethesis 11
2 basic notions 13
2.1 Rootsystemsandreflectiongroups 13
2.1.1 Definitionofarootsystem 13
2.1.2 Positiverootsandsimpleroots 14
2.1.3 Dynkindiagramsandtheclassification 14
2.1.4 WeylgroupsasCoxetergroups 15
2.2 Hyperplanearrangements 16
2.2.1 TheCoxeterarrangement 16
2.2.2 TheaffineCoxeterarrangementandtheaffineWeylgroup 16
2.2.3 Polyhedra 18
2.2.4 Affineroots 19
2.2.5 Theheightofroots 20
3 the anderson map 21
3.1 ClassicalrationalCatalancombinatorics 21
3.1.1 RationalCatalannumbersandrationalDyckpaths 21
3.1.2 Rationalparkingfunctions 22
3.1.3 VerticallylabelledDyckpaths 23
3.2 Theaffinesymmetricgroup 24
3.3 Abaci 25
3.4 p-stableaffinepermutations 26
3.5 ThecombinatorialAndersonmap 26
3.6 TheuniformAndersonmap 27
3.6.1 p-stableaffineWeylgroupelements 27
3.6.2 FromtheSommersregiontothedilatedfundamentalalcove 28
3.6.3 Fromthedilatedfundamentalalcovetothefinitetorus 30
3.6.4 Puttingitalltogether 31
3.6.5 Thestabilizerof (w) 31
(cid:101)
A
3.7 ThecombinatorialAndersonmapandtheuniformAndersonmap 32
3.7.1 Parkingfunctionsandthefinitetorus 32
3.7.2 TheAndersonmapsareequivalent 33
3.8 RationalCoxeter-Catalannumbers 35
3.8.1 Thepartitionlattice 36
3.8.2 Thenumberoforbitsofthefinitetorus 37
3.8.3 Typesoforbits 37
3.9 Dominant p-stableaffineWeylgroupelements 38
4 the m-shi arrangement 41
4.1 TheShiarrangement 41
4.2 The m-extendedShiarrangement 41
4.3 Minimalalcovesof m-Shiregions 43
4.3.1 Theaddressofadominant m-Shialcove 43
3
4 CONTENTS
4.3.2 Floorsofdominant m-Shiregionsandalcoves 44
4.3.3 m-Shiregionsandalcovesinotherchambers 44
4.3.4 m-Shialcovesand (mh+1)-stableaffineWeylgroupelements 46
4.4 Enumerativeconsequences 47
5 the zeta map 49
5.1 Thespaceofdiagonalharmonics 49
5.1.1 TheHilbertseries 49
5.2 VerticallylabelledDyckpaths 49
5.2.1 Thestatistics 50
5.3 DiagonallylabelledDyckpaths 50
5.3.1 Thestatistics 51
5.4 Thecombinatorialzetamap 51
5.5 Theuniformzetamap 52
5.5.1 Thenonnestingparkingfunctions 52
5.5.2 m-nonnestingparkingfunctionsandthefinitetorus 54
5.5.3 Thetypeofageometricchainof m orderfilters 55
5.5.4 Therankofageometricchainof m orderfilters 55
5.6 Theuniformzetamapandthecombinatorialzetamap 56
5.6.1 NonnestingparkingfunctionsasdiagonallylabelledDyckpaths 56
5.6.2 Thezetamapsareequivalent 58
5.7 Outlook 60
6 floors and ceilings of dominant m-shi regions 61
6.1 Themainresult 61
6.2 Preliminaries 61
6.3 Proofofthemainresult 66
6.4 Corollaries 67
6.5 Outlook 67
7 chapoton triangles 69
7.1 ThreeFuss-CatalanobjectsandtheirChapotontriangles 69
7.1.1 m-nonnestingpartitions 69
7.1.2 m-noncrossingpartitions 69
7.1.3 The m-clustercomplex 70
7.2 TheH=Fcorrespondence 70
7.2.1 The h-vectorofAssoc(Φm) 71
7.3 TheBijection 71
7.4 ProofoftheH=Fcorrespondence 73
7.5 CorollariesoftheH=Fcorrespondence 74
7.6 Outlook 76
SUMMARY
ThepresentthesiscontainsfourmaincontributionstotheCatalancombinatoricsofcrystallo-
graphicrootsystems.
The first is a uniform bijection that generalises the bijection defined by Gorsky,
GMV
A A
MazinandVaziranifortheaffinesymmetricgroupintheirstudyofarationalgeneralisationof
theHilbertseriesofthespaceofdiagonalharmonics.
Thesecondisauniformbijection ζ thatgeneralisesthebijection ζ definedbyHaglundand
HL
Loehr,alsointhecontextofdiagonalharmonics.
The third is a proof of a conjecture of Armstrong relating floors and ceilings of the m-Shi
arrangement. Inthisproof,abijectionthatprovidesmorerefinedenumerativeinformationis
introduced.
Thefourth isaproof ofthe H = F correspondence, originallyconjectured by Chapotonand
thengeneralisedbyArmstrong. The H = F correspondencedescribesawayoftransforminga
refinedenumerationofthefacesofthe m-clustercomplexAssoc(Φm) toarefinedenumeration
of the set NN(Φm) of m-generalised nonnesting partitions by means of an invertible change of
variables. Theproofusesauniformbijectiontogetherwithacase-by-caseverification.
5
6 CONTENTS
ZUSAMMENFASSUNG
DievorliegendeDissertationentha¨ltvierhauptsa¨chlicheBeitra¨gezurCatalan-Kombinatorik
derkristallographischenWurzelsysteme.
Der erste ist eine einheitliche Anderson-Abbildung die die kombinatorische Anderson-
A
Abbildung verallgemeinert. Die kombinatorische Anderson-Abbildung wurde von
GMV
A
Gorsky,MazinundVaziranifu¨rdieaffinesymmetrischeGruppedefiniertumeinerationale
VerallgemeinerungderHilbertreihedesRaumesderdiagonalenharmonischenPolynomezu
erhalten.
Der zweite ist eine einheitliche Zeta-Abbildung ζ die die kombinatorische Zeta-Abbildung
ζ verallgemeinert. Die kombinatorische Zeta-Abbildung wurde von Haglund und Loehr
HL
definiert,auchimKontextderdiagonalenharmonischenPolynome.
Der dritte ist der Beweis einer Vermutung von Armstrong die die Bo¨den und die Decken
vondominantenRegionendes m-Shi-Gefu¨gesinVerbindungsetzt. IndemBeweiswirdeine
Bijektioneingefu¨hrtdienochfeinereAbza¨hlungenermo¨glicht.
Der vierte ist der Beweis der H = F Korrespondenz. Diese wurde zuerst von Chapoton
vermutetundspa¨terdurchArmstrongverallgemeinert. SiebeschreibteineinvertierbareVari-
ablensubstitutiondieeineverfeinerteAbza¨hlungderSeitendesCluster-KomplexesAssoc(Φm) in
eineverfeinerteAbza¨hlungderMengeNN(Φm) derm-nichtschachtelndenPartitionenumwandelt.
DerBeweisverwendeteineeinheitlicheBijektionsowieeineFallunterscheidung.
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8 CONTENTS
1
INTRODUCTION
1.1 catalan combinatorics
1.1.1 Catalannumbers
OneofthemostfamousnumbersequencesincombinatoricsisthesequenceofCatalannumbers
1,1,2,5,14,42,132,... givenbytheformula
(cid:18) (cid:19)
1 2n+1
Cat := .
n
2n+1 n
AvastvarietyofcombinatorialobjectsarecountedbytheCatalannumbersandmaythusjustly
be called Catalan objects. Many of them have been collected by Stanley [Sta]. Some Catalan
objectsare
(Assoc)triangulationsofaconvex (n+2)-gon,
(NC)noncrossingpartitionsof [n] := 1,2,...,n ,
{ }
(NN)nonnestingpartitionsof [n],and
(Qˇ)increasingparkingfunctionsoflength [n].
1.1.2 Coxeter-Catalannumbers
Inalgebraiccombinatorics,acommonthemeistotakecombinatorialobjectsandviewthem
as emerging from or sitting inside some algebraic structure. Doing this may reveal further
structureorsuggestpossiblegeneralisations.
Inourcase,thefourCatalanobjectsmentionedinSection1.1.1maybeseenasobjectsassociated
withthesymmetricgroup S anditsrootsystem,whichisofDynkintype A . Thismakes
n n 1
it possible to generalise each of them to all irreducible crystallographic root s−ystems Φ. For
backgroundoncrystallographicrootsystemsrefertoChapter2. Thesegeneralisationsare
(Assoc)maximalsetsofpairwisecompatiblealmostpositiveroots,
(NC)minimalfactorisationsofaCoxeterelement c intotwoWeylgroupelements,
(NN)orderfiltersintherootposetof Φ,and
(Qˇ)orbitsoftheactionoftheWeylgroupW onthefinitetorus Qˇ/(h+1)Qˇ.
Surprisingly,theCatalannumberssurvivethesegeneralisations: eachoftheseobjectsiscounted
bythesamenumberCatΦ,theCoxeter-Catalannumberof Φ. Itisdefinedas
1 ∏r
CatΦ := (h+1+ei).
W
| | i=1
Here r is the rank of Φ, W is its Weyl group, h is its Coxeter number and e ,e ,...,e are its
1 2 r
exponents. When Φ is of Dynkin type An 1, the Coxeter-Catalan number CatΦ equals the
classicalCatalannumberCat . −
n
9
10 introduction
1.1.3 Fuß-Catalannumbers
FurthergeneralisationsofCoxeter-Catalanobjectsarefoundbyintroducingapositiveinteger
m asaFußparameter. ThisgivesrisetothefollowingFuß-Catalanobjects:
(Assoc)maximalsetsofpairwisecompatible m-colouredalmostpositiveroots,
(NC)minimalfactorisationsofaCoxeterelement c into (m+1) Weylgroupelements,
(NN)geometricchainsof m orderfiltersintherootposetof Φ,and
(Qˇ)W-orbitsofthefinitetorus Qˇ/(mh+1)Qˇ.
AllofthesearecountedbytheFuß-Catalannumber
Cat(Φm) := 1 ∏r (mh+1+ei).
W
| | i=1
TheyspecialisetothecorrespondingCoxeter-Catalanobjectsinthecasewhere m =1.
If Φ is of type An 1, the Fuß-Catalan number Cat(Φm) equals the classical Fuß-Catalan num-
−
ber
(cid:18) (cid:19)
Cat(m) := 1 (m+1)n+1 .
n (m+1)n+1 n
OneunderlyingphilosophyofthefieldofFuß-Catalancombinatoricsisthatuniformtheorems
askforuniformproofs. Thatis,whenastatementholdsforallirreduciblecrystallographicroot
systems,oneshouldtrytoproveitwithoutappealingtotheirclassification.
Giventhisphilosophy,itishelpfultodividetheFuß-Catalanobjectsintotwodifferentworlds:
thenoncrossingworldcontaining(Assoc)and(NC)andthenonnestingworldcontaining(NN)and
(Qˇ). There are uniform bijections known between the objects within each world, but none
betweenobjectsofdifferentworlds.
Each world has its own unique flavour: in the noncrossing world, there are Cambrian re-
currencesandgeneralisationstononcrystallographicrootsystems[STW15]. Ontheotherhand,
in the nonnesting world there is a uniform proof of the fact that the Fuß-Catalan objects are
indeed counted by the Fuß-Catalan number Cat(Φm). Our focus in this thesis will be on the
nonnestingworld.
1.1.4 RationalCatalannumbers
A further generalisation of Fuß-Catalan numbers are rational Catalan numbers. For any irre-
duciblecrystallographicrootsystem Φ andapositiveinteger p relativelyprimetotheCoxeter
number h of Φ definetherationalCatalannumber
1 ∏r
Catp/Φ := W (p+ei).
| | i=1
ItreducestotheFuß-CatalannumberCat(Φm) when p = mh+1. TherationalCatalannumbers
count
(Qˇ)W-orbitsofthefinitetorus Qˇ/pQˇ.
For the (Assoc), (NC) and (NN) Fuß-Catalan objects there is no satisfactory generalisation to
the rational Catalan level yet. However, (Assoc) and (NC) rational Catalan objects have been
proposedfortype A [ARW13].
n 1
−
If Φ is of type An 1, the rational Catalan number Catp/Φ equals the rational (p/n)-Catalan
−
number
(cid:18) (cid:19)
1 n+p
Cat := .
p/n n+p n