On tree hook length formulae, Feynman rules and B-series [Master thesis]

110 Pages·2014·1.414 MB·English

by  Bradley Robert Jones

On tree hook length formulae, Feynman rules and B-series
by
Bradley Robert Jones
B.Sc., Simon Fraser University, 2012
Thesis Submitted in Partial Fulfillment
of the Requirements for the Degree of
Master of Science
in the
Department of Mathematics
Faculty of Science
© Bradley Robert Jones 2014
SIMON FRASER UNIVERSITY
Fall 2014
All rights reserved.
However, in accordance with the Copyright Act of Canada, this work may be
reproduced without authorization under the conditions for “Fair Dealing.”
Therefore, limited reproduction of this work for the purposes of private study,
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with the law, particularly if cited appropriately.
APPROVAL
Name: Bradley Robert Jones
Degree: Master of Science (Mathematics)
Title of Thesis: On tree hook length formulae, Feynman rules and B-series
Examining Committee: Dr. Paul Tupper, Associate Professor
Chair
Dr. Karen Yeats, Associate Professor
Senior Supervisor
Dr. Cedric Chauve, Professor
Supervisor
Dr. Tamon Stephen, Associate Professor
Examiner
Date Approved: September 25, 2014
ii
Partial Copyright Licence

iii
Abstract
This thesis relates similar ideas from enumerative combinatorics, Hopf algebraic quantum field theory
anddifferentialanalysis. Hooklengthformulae, fromenumerativecombinatorics, areequationsthatcan
lead to bijections between tree classes and other combinatorial classes. Feynman rules are maps used in
quantumfieldtheorytogenerateintegralsfromparticleinteractiondiagrams. HereweconsiderFeynman
rules from the Hopf algebra perspective. B-series are powers series that sum over trees and are used in
differential analysis to analyze Runge-Kutta methods. The aim of this thesis is to bring together the
ideas of the three communities. We show how to use differential equations to obtain new hook length
formulae. Some of these new hook length formulae result in new combinatorial bijections. We use hook
length formulae to express differential equations combinatorially. We also provide a generalization to
hook length. Finally we include a catalogue of known hook length formulae.
iv
To my parents.
v
“Vix prece finita torpor gravis occupat artus,
mollia cinguntur tenui praecordia libro,
in frondem crines, in ramos bracchia crescunt,
pes modo tam velox pigris radicibus haeret,
ora cacumen habet: remanet nitor unus in illa.”
Metamorphoses, Ovid
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Acknowledgements
I would foremost like to thank my senior supervisor, Karen Yeats. Without you I would not have found
this topic. I am very appreciative of all the input regarding this thesis, without which this thesis would
merely be a mass of theorems and equations. Thank you for motivating me to write paragraphs and
actually finish this work.
Iwouldalso liketo thankmy othersupervisor, CedricChauve. Thankyou forremindingmethatI have
to justify my work. My thanks to Marni Mishna who kindled my interest in combinatorics. Thanks
to Philippe for reassurance and advice and to Justin for using every chance he could get to tell people
that I study trees. Finally, I wish to thank my parents for their love, support and learning how to say
com·bi·na·or·ics.
IwouldliketothanktheOnlineEncyclopediaofIntegerSequences(OEIS)[50],whichIusedmanytimes
to check for sequences. Sequences I looked up include: A003319 (connected permutations), A000311
(Schröder’s 4th problem), A000111 (Euler numbers), A006963 (planar embedded trees), A000312 (nn),
A113583 (permutations with no local minima at even pos), A038037 (mobiles), A048802 (labelled dec-
orated trees), A007317 (decorated plane tree), A007106 (number of labelled odd degree trees with 2n
nodes) and A151374 (a class of walks n the quarter plane). I also created a new entry to the OEIS,
A227917, forlabelledbinary treeswhere eachlabelis greaterthan thelabelsof itsancestors ofdegree2.
I used Maple’s [41] dsolve function to solve most of the differential equations appearing in this paper.
I also developed a small script to calculate hook length formulae coefficients using Maple.
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Contents
Approval ii
Abstract iv
Dedication v
Quotation vi
Acknowledgements vii
Contents viii
List of Tables xi
List of Figures xii
I Introduction 1
II Hook length formulae 6
2.1 Combinatorial classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Simple tree classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Hook length series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Hook length formulae and isomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Some new hook length formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Decorated trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 General hook length operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
IIIThe Connes-Kreimer Hopf algebra and hook length 40
3.1 Hopf algebra of rooted trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 The L∗ operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
B
3.2.1 Examples of L∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
B
3.2.2 The universal property and L∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
B
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IVDifferential equations of hook length series 50
4.1 B-series and Runge-Kutta methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Mazza’s differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Hypergeometric differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4 New hook length formulae using Mazza’s differential equation . . . . . . . . . . . . . . . 57
4.5 The differential equation for decorated trees and general
hook length operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.6 New methods of finding hook length formulae . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6.1 Leafless method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6.2 System method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6.3 Scaled method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.7 Representing differential equations combinatorially . . . . . . . . . . . . . . . . . . . . . . 70
4.7.1 More on the theta operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7.2 Combinatorial solutions to differential equations. . . . . . . . . . . . . . . . . . . 73
V Conclusion 77
VICatalogue of hook length formulae 81
6.1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3 Complete binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.4 Semicomplete binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.5 Fibonacci trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.6 Motzkin trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.7 r-ary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.8 Complete r-ary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.9 Plane trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.10 Fat plane trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.11 Labelled unordered trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.12 Labelled unordered binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.13 Labelled unordered complete binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.14 Labelled unordered even trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.15 Labelled unordered odd trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.16 Cyclic trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.17 Schröder trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.18 Plane forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.19 Labelled unordered forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.20 Cyclic forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.21 Single forest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Bibliography 93
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Definitions 97
x