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Representations of SO(3) in C [X, Y, Z]

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Representations of SO(3) in C [X, Y, Z] DEGREE PROJECT IN TECHNOLOGY, FIRST CYCLE, 15 CREDITS STOCKHOLM, SWEDEN 2019 Representations of SO(3) in C [x, y, z] OTTO SELLERSTAM KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2019 Representationer av SO(3) i C [x,y,z] OTTO SELLERSTAM KTH SKOLAN FÖR TEKNIKVETENSKAP Abstract This paper will present a concrete study of representations of the special orthogonal group, SO(3), a group of great importance in physics. Specifically, we will study a natural representation of SO(3) in the space of polynomials in three variables with complex coefficients, C[x; y; z]. We will find that this special case provides all irreducible representations of SO(3), and also present some corollaries about spherical harmonics. Some preparatory theory regarding abstract algebra, linear algebra, topology and measure theory will also be presented. Sammanfattning Denna rapport kommer att redog¨ora en konkret unders¨okning av representationer av den speciella ortogonala gruppen, SO(3), en grupp med viktiga till¨ampningar inom fysik. Mer specifikt kommer vi att studera en naturlig representation av SO(3) i rummet av polynom i tre variabler med komplexa koefficienter, C[x; y; z]. Vi kommer att se att alla irreducibla representationer av SO(3) dyker upp ur detta specialfall, samt presentera n˚agraf¨oljdsatser ang˚aendeklotytefunktioner. F¨orberedande teori ang˚aende abstrakt algebra, linj¨ar algebra, topologi och m˚atteorikommer ¨aven att redovisas. 1 Contents 1 Introduction 4 2 Some Algebraic Structures 5 3 Linear Algebra 8 3.1 Basic concepts . .8 3.2 Multilinear algebra . 11 4 Topology 15 4.1 Point-set topology . 15 4.1.1 Topological spaces and continuity . 15 4.1.2 Compactness . 16 4.2 Topological groups . 17 5 Measure Theory 19 5.1 Construction of the Lebesgue integral . 19 5.1.1 Lp-spaces . 20 5.2 Borel algebras and the Haar measure . 21 6 Representation Theory 23 6.1 Basic definitions . 23 6.2 Some important theorems . 24 6.3 Character theory . 26 6.4 Canonical decomposition of a representation . 30 6.5 From finite to compact groups . 30 7 The 3D Rotation Group SO(3) 32 7.1 Parameterization . 32 7.2 Conjugacy classes of SO(3) ..................................... 33 7.3 The Haar measure on SO(3) .................................... 34 8 Representations of SO(3) 35 8.1 Representations in C[x; y; z]..................................... 35 8.1.1 Introduction . 35 8.1.2 The natural representation . 36 8.1.3 Characters of SO(3)..................................... 38 8.1.4 An inner product on C[x; y; z]................................ 39 8.2 Tensor products of irreducible representations . 44 2 3 1 Introduction The goal of representation theory is to reduce the study of complicated algebraic structures to the study of vector spaces; elements are represented as bijective linear operators on a vector space, which often helps to concretize the underlying algebraic structure. Some of the most common structures to study using representation theory include groups, associative algebras and Lie algebras. In this paper we will discuss an analysis of the 3D rotation group from a representation theoretic per- spective. We will go through some fundamental concepts of abstract and linear algebra, which will be used to define representations and to present some basic ideas from representation theory of finite groups. SO(3) is, however, not a finite group, so we will also need to discuss a framework of how to translate the ideas from the finite case to the infinite case. To discuss this framework, we also need to develop some concepts from topology and measure theory, since notions such as compactness and integration with respect to a measure are required. 4 2 Some Algebraic Structures We start of by defining some basic algebraic structures, together with some examples. Definition 1 (Group). A group is a set G equipped with a binary operation ? : G × G ! G which satisfies the following axioms for all a; b; c 2 G. 1. Associativity: a ? (b ? c) = (a ? b) ? c 2. Identity: there exists an element e 2 G such that a ? e = e ? a = a 3. Inverse: there exists an element a−1 2 G such that a−1 ? a = a ? a−1 = e When we want to emphasize both the underlying set G and the group operation ?, we will write (G; ?). A group where the elements commute, i.e. a ? b = b ? a for all a; b 2 G, is called an abelian group, named after Niels Henrik Abel. Example 1. We provide some examples of groups. Notice that the first two examples are abelian, while the last one is not. 1. The integers Z under addition, (Z; +) form a group. Note however that the integers under multi- plication, (Z; ·), does not, since the multiplicative inverse of for example 2 is 1=2, which is not an integer. 2. The rational numbers without the number 0 form a group under multiplication, denoted (Q − f0g; ·). 3. The general linear group GL(n) consisting of all invertible n × n-matrices with real entries is a group 2 under matrix multiplication. We will later on consider this group to be a subset of Rn . 4. The orthogonal group O(n) = fA 2 GL(n): AT A = AAT = Ig ⊂ GL(n) and the special orthogonal group SO(n) = fA 2 O(n) : det(A) = 1g ⊂ GL(n) form subgroups of the general linear group, i.e. they are subsets of GL(n) that also form groups. Next we define a field, a structure which is a bit more complex than a group, since its definition involves two binary operations instead of one, together with more axioms. Definition 2 (Field). A field is a set F together with two binary operations, addition + : F × F ! F and multiplication · : F × F ! F, that satisfies the following axioms for all a; b; c 2 F. 1. Commutativity of addition: a + b = b + a 2. Associativity of addition:(a + b) + c = a + (b + c) 3. Additive identity: there exists an element 0 2 F such that a + 0 = a 4. Additive inverse: there exists an element −a 2 F such that a + (−a) = 0 5. Commutativity of multiplication: a · b = b · a 6. Associativity of multiplication:(a · b) · c = a · (b · c) 7. Multiplicative identity: there exists an element 1 2 F such that 1 · a = a 8. Multiplicative inverse: for a 6= 0, there exists an element denoted a−1 such that a−1 · a = 1 9. Distributivity: a · (b + c) = (a · b) + (a · c) Notice that this definition can be summarized by saying that a field is an abelian group under addition with additive identity denoted 0, and that the nonzero elements form an abelian group under multiplication, with multiplicative identity denoted 1. The multiplication also needs to be distributive over the addition. Example 2. Next, we provide some examples of fields. 5 1. The rational numbers Q, the real numbers R and the complex numbers C together with regular addition and multiplication all form a field. The reader should be comfortable with the notion of a vector space, but we include the definition for the sake of completeness. Definition 3 (Vector space). A vector space over a field F is a set V together with two binary operations, addition + : V × V ! V and scalar multiplication · : F × V ! V that satisfies the following axioms for all x; y; z 2 V and a; b 2 F.1 1. Commutativity of addition: x + y = y + x 2. Associativity of addition:(x + y) + z = x + (y + z) 3. Additive identity: there exists a vector 0 such that 0 + x = x 4. Additive inverse: there exists a vector −x such that −x + x = 0 5. First distributive law:(a + b) · x = a · x + b · x 6. Second distributive law: a · (x + y) = a · x + a · y 7. Multiplicative identity: 1 · x = x, where 1 denotes the multiplicative identity in F 8. Relation to ordinary multiplication:(ab) · x = a · (b · x) Example 3. Again, we include some examples, also for the sake of completeness. 1. One of the most simple examples of a vector space is R3, the space in which we live, consisting of 3-tuples with vector addition and scalar multiplication defined coordinate wise. 2. More generally, by letting F be an arbitrary field we can construct the set Fn, where n is some positive whole number, of n-tuples of elements of F. Next we define a special kind of vector space with additional structure. Definition 4. Let A be a vector space over a field F equipped with an additional binary operation × : A × A ! A. We call A an algebra if it satisfies the following for all x; y; z 2 A and a; b 2 F. 1. Right distributivity:(x + y) × z = x × z + y × z 2. Left distributivity: x × (y + z) = x × y + x × z 3. Compatibility with scalars:(ax) × (by) = (ab)(x × y) A substructure is a subset of some algebraic structure, that itself is the same type of structure. Examples of substructures are 1. Subgroups, a subset of a group that is a group 2. Subfield, a subset of a field that is a field 3. Subspace, a subset of a vector space that is a vector space 4. Subalgebra, a subset of an algebra that is an algebra For algebras, we also have a more strict class of substructures: a left ideal of an algebra A is a subalgebra I ⊂ A such that aI = fax : x 2 Ig ⊂ I for all a 2 A. Analogously, a right ideal of A is a subalgebra I ⊂ A such that Ia = fxa : x 2 Ig ⊂ I for all a 2 A.A two-sided ideal is an ideal that is both a left and a right ideal.
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