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An Algebraic Approach to Harmonic Polynomials on S3

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An Algebraic Approach to Harmonic Polynomials on S3 AN ALGEBRAIC APPROACH TO HARMONIC POLYNOMIALS ON S3 Kevin Mandira Limanta A thesis in fulfillment of the requirements for the degree of Master of Science (by Research) SCHOOL OF MATHEMATICS AND STATISTICS FACULTY OF SCIENCE UNIVERSITY OF NEW SOUTH WALES June 2017 ORIGINALITY STATEMENT 'I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.' Signed .... Date Show me your ways, Lord, teach me your paths. Guide me in your truth and teach me, for you are God my Savior, and my hope is in you all day long. { Psalm 25:4-5 { i This is dedicated for you, Papa. ii Acknowledgement This thesis is the the result of my two years research in the School of Mathematics and Statistics, University of New South Wales. I learned quite a number of life lessons throughout my entire study here, for which I am very grateful of. I would like to express my endless gratitude to my supervisor, Associate Professor Norman Wildberger for his constant support and guidance throughout my research study in the School of Mathematics and Statistics, University of New South Wales. His unorthodox view towards mathematics enables me to appreciate mathematics even more. I would also like to thank all of the staff in the school for their help towards my study. I am also indebted to Lembaga Pengelola Dana Pendidikan (Indonesian Endowment Fund for Education) for awarding me the scholarship during my master study in Sydney. Without the aid from them, my dream to pursue this study would have been impossible. Finally, I would like to thank both my parents and my brothers for their constant encouragement while I am studying here. Sydney, June 2017 Author, Kevin Mandira Limanta iii Abstract In this thesis we are going to study harmonic polynomials on spheres, with the particular attention to the 3-sphere 3. As a Lie group, the algebraic structure of 3 SU (2) enables us to study harmonic polynomials from an algebraic point ' of view. A purely algebraic approach has several advantages, such as the possi- bilities to extend to more general matrix groups and work over general fields such as finite fields, and importantly to rational numbers. This algebraic approach allows us to construct an explicit basis for the space of harmonic polynomials on 3 and describe them from a geometrical point of view. Finally, we give an explicit algebraic formula for zonal harmonic polynomials on 3 which can be identified as characters of Lie group 3 SU (2) ' Contents 0Introduction 1 1 The Algebra of Complex Numbers and Quaternions 5 1.1 The Complex Numbers C ...................... 5 1.2 The Quaternions H .......................... 8 1.3IdentifyingQuaternionswithMatrices............... 10 2 Spherical Rotation 16 2.1 Rotation in 1 ............................. 16 2.2 Describing Rotations by Quaternions . ............. 18 2.3SphericalParametrization...................... 21 2.3.1 StereographicProjections.................. 22 2.3.2 The Standard Theory: When F = R ............ 25 3 Theory of Harmonic Polynomials 27 3.1 Introduction . ............................. 27 3.2HarmonicPolynomialsontheSphere................ 31 3.3TheKelvinTransform........................ 39 3.3.1 InversionintheUnitSphere................. 39 3.3.2 MapsPreservingHarmonicFunctions............ 39 3.4 Spherical Harmonics via Differentiation............... 43 3.5BasesforHarmonicPolynomials:AnalyticApproach....... 49 4 Theory of Harmonic Polynomials: Algebraic Approach 55 4.1 Harmonic Polynomials on 2 ..................... 57 4.1.1 ComplexFactorialBasis................... 61 4.1.2 Zonal Harmonic Polynomials on 2 ............. 65 4.1.3 GeneratingFunction..................... 72 4.2 Harmonic Polynomials on 3 ..................... 73 4.2.1 ComplexFactorialBasis................... 74 4.3ZonalHarmonicsintheFactorialBasis............... 79 4.4ZonalHarmonicsintheComplexFactorialBasis.......... 84 i CONTENTS ii 2 1 A Spherical Harmonic Decomposition of ( − ) 90 A.1 Functions on 1 andFourierTheory................ 91 B Integrating Polynomial over a Sphere 93 B.1 Integrating monomials over 1 .................... 94 B.2 Integrating monomials over 2 .................... 96 B.3Integratingmonomialsoverageneralsphere............ 99 Chapter 0 Introduction In this thesis we are going to study harmonic analysis on spheres, with particular attention to the 3-sphere 3. This sphere is particularly interesting because it possesses a group structure, albeit a non-commutative one. In fact, 1 and 3 are the only two spheres that have a group structure. The algebraic structure of 3 enables us to talk about harmonic analysis on the sphere from a purely algebraic point of view. This thesis will be mainly focused on the study of harmonic polynomials, as we can use the fact that every function defined on a sphere can be decomposed into a series of harmonic polynomials, which serve as the building blocks of spherical functions. The main aim is to introduce the reader to an algebraic way of describing harmonic polynomials on 3. We will be particulary interested in the zonal harmonic polynomials — harmonic polynomials on 3 that are invariant under rotation. These polynomials are compelling because in Lie group theory, zonal harmonic polynomials are characters of the group 3 SU (2). ' This algebraic way of describing harmonic polynomials on 3 connects nat- urally with the theory of quaternions. There are already various studies about spherical harmonic polynomials for a general sphere, using a standard analytical point of view, such as Axler [6] and Müller [22], so one will have options to go with the standard view or the algebraic one. However, since the standard view often incorporates transcendental functions such as the square root function and trigonometricfunctions,itisnotalwayscomputationallyeffective, and does not extend easily to other fields. The algebraic point of view that will be developed here will extend to 2 and also often to orthogonal group of matrices. However, that extension will not be discussed in this thesis and could be a suggestion for 1 0. Introduction 2 future work. Moreover, one may be able to generalize this theory to work over general fields such as finite fields, and importantly to rational numbers. This thesis will be developed as follows. Chapter 0 will be reserved as the introduction to the thesis to give the reader the general idea of where the direc- tionofthethesisisheading.InChapter1,wewillintroducethebasicmaterials needed to explain the algebraic structure of 3 by discussing the algebraic struc- ture of 1 first. In Chapter 2 we see that multiplication of complex numbers or quaternions can be interpreted geometrically as spherical rotations. We will see that incorporating quaternions will be useful to explain rotations on 2,which has no group structure. In Chapter 3 we present the standard theory of har- monic functions. We will introduce first the concept of decomposition of general polynomials into homogeneous polynomials and finally into harmonic polynomi- als. The standard theory constructed later will be a decent tool to explain the decomposition of homogeneous polynomials into harmonic polynomials, as given below. Theorem 1 Any homogeneous polynomial (x)=(12) (A ) ∈ can be uniquely written in the form = + (x) 2 + + (x) 2 − ··· − where = and each (A );thatis, is harmonic homogeneous 2 ∈ polynomial of degree . ¥ ¦ At the end of Chapter 3, we will give an explicit basis for harmonic poly- nomials of degree . It is important to note here that although we present the standard theory, we want to extract as much theory as possible to work in a more general algebraic setting. If a certain theory is possible for generalization over a more general field, we will make a distinction. For example, instead of working on Euclidean spaces R,weturnourattentionto-dimensional affine space A over a general field, which might also be the field of rational numbers. There are, however, several steps in which we will resort to an analytical point of view. We will mention it whenever we do so. In the last chapter, we will turn to our purely algebraic setting and extend the ideas developed by Wildberger in the preprint [31]. We will focus on harmonic polynomials on 3 by first discussing harmonic polynomials on 2. We will attempt to approach the study of harmonic polynomials on 2 from an algebraic 0. Introduction 3 point of view and see that the lack of group structure will still be a necessary reason for us to use transcendental functions in describing zonal harmonics. However, that is not the case in 3. The highlight of this thesis will be the derivation of an algebraic version of zonal harmonic polynomials in 3, given in the Theorem below. Theorem 2 Up to a scalar, there is a unique zonal harmonic polynomial of degree namely 2 b c !(2 2)! (2 2)! (2)! 2 2 2 2 2 2 (x)= ( 1) − − − − − − (2 +1)!( )!( )!! =0 =0 =0 X X X − − With the introduction of the
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