On the Kernel of the Symbol Map for Multiple Polylogarithms

186 Pages·2012·1.021 MB·English

by  John Richard Rhodes

Durham E-Theses
On the Kernel of the Symbol Map for Multiple
Polylogarithms
RHODES, JOHN,RICHARD
How to cite:
RHODES, JOHN,RICHARD (2012) On the Kernel of the Symbol Map for Multiple Polylogarithms.
Doctoralthesis, DurhamUniversity. AvailableatDurhamE-ThesesOnline: http://etheses.dur.ac.uk/3905/
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On the Kernel of the Symbol Map
for Multiple Polylogarithms
John Richard Rhodes
A Thesis presented for the degree of
Doctor of Philosophy
Pure Mathematics
Department of Mathematical Sciences
Durham University
2012
Abstract
The symbol map (of Goncharov) takes multiple polylogarithms, I (x ,...,x ),
r1,...,rs 1 s
to a tensor product space where calculations are easier, but where important dif-
ferential and combinatorial properties of the multiple polylogarithm are retained.
Finding linear combinations of multiple polylogarithms in the kernel of the symbol
map is an effective way to attempt finding functional equations. We present and
utilise methods for finding new linear combinations of multiple polylogarithms (and
specifically harmonic polylogarithms) that lie in the kernel of the symbol map.
During this process we introduce a new pictorial construction for calculating the
symbol, namely the hook-arrow tree, which can be used to easier encode symbol
calculations onto a computer.
We also show how the hook-arrow tree can simplify symbol calculations where the
depth of a multiple polylogarithm is lower than its weight and give explicit expres-
sions for the symbol of depth 2 and 3 multiple polylogarithms of any weight. Using
this we give the full symbol for I (x,y,z). Through similar methods we also give
2,2,2
the full symbol of coloured multiple zeta values.
We provide introductory material including the binary tree (of Goncharov) and the
polygon dissection (of Gangl, Goncharov and Levin) methods of finding the symbol
of a multiple polylogarithm, and give bijections between (adapted forms of) these
methods and the hook-arrow tree.
ii
Declaration
The work in this thesis is based on research carried out in the Pure Mathematics
Group at the Department of Mathematical Sciences, Durham University. No part
of this thesis has been submitted elsewhere for any other degree or qualification and
it is all my own work unless referenced to the contrary in the text.
Copyright (cid:13)c 2012 by Author.
“The copyright of this thesis rests with the author. No quotations from it should be
published without the author’s prior written consent and information derived from
it should be acknowledged”.
iii
Acknowledgements
Although many would say it about themselves, I truly believe I am a very particular
kind of mathematician; one that might not suit some supervisors. Thank you,
Herbert, for communicating with me in a way that best suits my style. I have
always very much enjoyed our (often very long) conversations; the mathematical
parts were stimulating and the technology parts good fun. I truly am very grateful
for your patience and support.
Communication in the mathematics department is something it excels in. I have
very much enjoyed the small pure maths postgraduate community that has been
kept going by the GandAlF seminar. I also would like to thank Claude for some
very interesting talks concerning links between my work and the physics community.
Non-mathematical communication within the department is important to keeping
sanity levels reasonable. As many PhD students of Durham will know, coffee club is
essential to success. The ‘10.30 club’ is something I have frequently fought for and
its most loyal members are garnered with a special medal.
Outside of the mathematics department I have been lucky to have a great set of
friends providing me with a tremendous support network. I have always felt like I
had people to turn to. The times spent drinking coffee, making dinners and in the
pub have been incredibly important to my studies. It is hard to precisely define it
in words, but the way in which we discuss trivial matters is something that I never
want to be without.
iv
v
So, for the above reasons and many more, my thanks go to Alex, Ben, Ben, Ben,
Caroline, Chris, Dave, Elise, Harriet, Helen, Ian, Jack, James, James, Joey, John,
Josh, Kirsty, Luke, Mel, Nathan, Pamela, Rachel, Sarah, Scott, Simon, Steven, and
Ric.
Finally, special words of thanks to Mum, Dad, Anne-Marie, and Nan for talking to
me on the phone, and of course, everything.
Contents
Title Page i
Abstract ii
Declaration iii
Acknowledgements iv
List of Contents vi
0 Introduction 1
0.1 Opening remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.2 Definition of multiple polylogarithms . . . . . . . . . . . . . . . . . . 2
0.2.1 Other definitions of multiple polylogarithms . . . . . . . . . . 4
0.2.2 Linear combinations of multiple polylogarithms . . . . . . . . 7
0.3 Summary of Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1 The symbol of a multiple polylogarithms via binary trees and poly-
gon dissection 10
vi
Contents vii
1.1 Tensor algebra and notation conventions . . . . . . . . . . . . . . . . 11
1.1.1 Shuffle product . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.2 2-torsion of tensor products and notation conventions . . . . . 14
1.2 Outline of the symbol from binary trees . . . . . . . . . . . . . . . . . 15
1.2.1 Attaching a symbol to the binary tree . . . . . . . . . . . . . . 16
1.3 Defining the symbol from polygon dissection . . . . . . . . . . . . . . 20
1.3.1 Associating a polygon to a multiple polylogarithm . . . . . . . 20
1.3.2 Adding dissecting arrows to a polygon . . . . . . . . . . . . . 23
1.3.3 Maximal dissections . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.4 Definition of the symbol . . . . . . . . . . . . . . . . . . . . . 29
1.3.5 The symbol of products of multiple polylogarithms . . . . . . 31
1.4 Why choose the symbol to represent multiple polylogarithms? . . . . 31
1.4.1 Bar construction of the polygon algebra . . . . . . . . . . . . 32
1.4.2 Differential structure of multiple polylogarithms and the symbol 36
1.5 A simple element in the kernel of the symbol map . . . . . . . . . . . 37
1.5.1 Ho¨lder convolution . . . . . . . . . . . . . . . . . . . . . . . . 37
2 Hook-arrow trees 39
2.1 Motivation for hook-arrow trees . . . . . . . . . . . . . . . . . . . . . 39
2.2 Moving from polygons to trees . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Obtaining terms in the symbol from a hook-arrow tree . . . . . . . . 44
Contents viii
2.3.1 Step 1: Selection of first distinguished edge . . . . . . . . . . . 46
2.3.2 Step 2: Splitting the tree . . . . . . . . . . . . . . . . . . . . . 47
2.3.3 Step 3: Iterative step . . . . . . . . . . . . . . . . . . . . . . . 54
2.3.4 Recording the results of the algorithm and definition of the
symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.5 The sign of a hook-arrow tree . . . . . . . . . . . . . . . . . . 57
2.3.6 The definition of the symbol via hook-arrow trees . . . . . . . 57
2.3.7 A worked example . . . . . . . . . . . . . . . . . . . . . . . . 58
2.4 Viewing the algorithm as a ternary/4-valent tree . . . . . . . . . . . . 61
2.4.1 The definition of a ternary tree . . . . . . . . . . . . . . . . . 62
2.4.2 An isomorphism on planted plane trees . . . . . . . . . . . . . 62
2.4.3 Forming a ternary tree from the algorithm on a hook-arrow tree 65
2.4.4 Enumeration of hook-arrow trees . . . . . . . . . . . . . . . . 69
2.4.5 Schematic picture of a hook-arrow tree . . . . . . . . . . . . . 70
2.5 Simple examples of finding the symbol using hook-arrow trees . . . . 71
2.5.1 Symbol for I (x,y) . . . . . . . . . . . . . . . . . . . . . . . 71
1,1
2.5.2 Symbol for I (x,y,z) . . . . . . . . . . . . . . . . . . . . . 72
1,1,1
2.5.3 Symbol for I (x) . . . . . . . . . . . . . . . . . . . . . . . . . 73
m
3 Relating different pictorial representations of the symbol 75
3.1 Isolating single terms in the symbol . . . . . . . . . . . . . . . . . . . 75
3.1.1 Isolating a single term on a binary tree . . . . . . . . . . . . . 79
Contents ix
3.1.2 Isolating a single term on a polygon dissection . . . . . . . . . 82
3.1.3 Isolating a single term on a hook-arrow tree . . . . . . . . . . 83
3.1.4 Isolating a single term on a ternary tree . . . . . . . . . . . . 83
3.2 Bijections between pictorial representations of the symbol . . . . . . . 84
3.2.1 Hook-arrow trees to polygon dissection . . . . . . . . . . . . . 85
3.2.2 Polygon dissections to ternary trees . . . . . . . . . . . . . . . 87
3.2.3 Ternary trees to binary trees . . . . . . . . . . . . . . . . . . . 90
3.2.4 Binary trees to a hook-arrow trees . . . . . . . . . . . . . . . . 93
3.2.5 Hook-arrow trees to ternary trees . . . . . . . . . . . . . . . . 94
3.2.6 A specific example of moving between all pictorial represent-
ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4 Symbols of multiple polylogarithms of a given depth 101
4.1 The symbol of I (x ,x ) . . . . . . . . . . . . . . . . . . . . . . . . 106
r1,r2 1 2
4.2 The symbol of I (x ,x ,x ) . . . . . . . . . . . . . . . . . . . . . 109
r1,r2,r3 1 2 3
4.3 Higher depths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3.1 Discussion on the symbol of a general depth 4 multiple poly-
logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5 The symbol of coloured multiple zeta values 120
5.1 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Correspondence between Propositions 5.4 and 5.7 and Theorem 5.2 . 128