Representation Theory Jock McOrist Lecture notes, perpetually in progress MATM035 Department of Mathematics University of Surrey Guildford GU2 7XH, United Kingdom Copyright c 2018 by Jock McOrist. All rights reserved. E-mail address: [email protected] Abstract Symmetries are a powerful method for easily understanding properties of otherwise complicated mathematical and physical objects. Group theory is a branch of mathematics developed to un- derstand symmetries, however it often leads to complicated abstract quantities. Representation theory turns such abstract algebraic concepts into linear transformations of vector spaces; the linearity making the system much easier to solve. In doing, representation theory can unveil deep symmetry properties of physical systems as well as leading to powerful and compact solutions to otherwise difficult and intractable problems. There will be on average 3 lectures per week. There are two unassessed courseworks and one class test. These will be marked and returned to you. The coursework and tests are important feedback for you, telling you how you are doing in the course, what your strengths and weaknesses are, and what type of questions to expect in the exam. All important information will be disseminated in the usual channels via SurreyLearn. There is a syllabus on SurreyLearn, and I encourage to look at it. It describes the course content, what you are expected to know, and all of the assessments. The lecture notes are structured to closely follow what we do in class, but I cannot emphasise enough the value of attending lectures. It is where all important information is disseminated. I will give explanations complementary to that written in the notes, and where appropriate fill in omitted steps in deriving formulae. There are also many worked examples throughout the notes. There are exercises at the end of the chapter which you should work through as a minimum. Some of these exercises may be used as unassessed coursework, marked and returned to you for feedback. Key concepts are boldfaced when they are introduced. Many ideas and examples in representation theory are developed in the exercises. They are also an important platform to practice proving results. There will be solutions to some (but not necessarily all) questions, and I will release them throughout the semester once you have had a serious attempt at the questions. There are also many wonderful textbooks out there which I encourage you to read through. I first learnt representation theory from Georgi's book `Lie Algebras in Particle Physics' and some of the material here are based on his explanations. Although the course is largely self-contained, we will draw heavily on your knowledge of group theory and linear algebra. In particular you need to be familiar eigensystems, diagonalisation of matrices, properties of inner products, traces and vector spaces. Finally, my email [email protected] is always on and door (05AA04) is always open, so if you have any questions or comments then I will try to respond as quickly as possible. i Contents 1. Introduction . .1 1.1. Why representation theory? . .1 2. Symmetries and finite groups . .4 2.1. Some preliminary definitions . .4 2.2. Some examples of groups . .5 2.2.1. Abelian groups . .5 2.2.2. Non-abelian groups . .6 2.3. Subgroups and cosets . .8 2.4. Conjugacy classes . 10 2.5. Automorphisms . 10 2.6. Exercises . 12 3. Representation theory basics . 14 3.1. What is representation theory? . 15 3.2. Some examples . 16 3.3. Reducible and irreducible representations . 19 3.4. Useful theorems . 20 3.5. Characters . 23 3.6. Tensor Products . 24 3.7. Exercises . 26 4. Lie groups and Lie algebras . 29 4.1. Lie groups . 29 4.2. Examples of Lie groups . 31 4.3. Multiplication and Lie algebras . 34 4.4. Adjoint representation . 38 4.5. An inner product and compact algebras . 38 4.6. Subalgebras, simple and semi-simple algebras . 39 4.7. Exercises . 41 5. The key example: su(2)................................... 44 5.1. From the group SU(2) to the algebra su(2) . 44 5.2. Highest weight construction of representations of su(2) . 45 5.3. Examples . 47 5.4. Exercises . 49 6. Roots and weights . 51 6.1. Weights . 51 6.2. The adjoint representation . 52 6.3. Roots . 54 6.4. Raising and lowering operators . 55 6.5. Heaps and heaps of su(2)s . 56 6.6. Example: su(3).................................... 57 6.7. Exercises . 61 7. Simple roots . 63 ii 7.1. Positive weights . 63 7.2. Simple roots . 64 7.3. Constructing the algebra . 69 7.4. Exercises . 72 8. Dynkin Diagrams . 74 8.1. Dynkin Diagrams . 74 8.2. Example: g2 ...................................... 75 8.3. Cartan Matrix . 76 8.4. Constructing the g2 algebra . 83 8.5. Example: c3 ...................................... 85 8.6. Fundamental weights . 85 8.7. Example: more representations su(3) . 87 8.8. Complex conjugation . 89 8.9. Exercises . 92 9. A tour of compact simple Lie algebras and their physical applications . 93 ∼ 9.1. su(N) = aN−1 ..................................... 93 ∼ 9.2. The so(2N) = dN algebra . 95 ∼ 9.3. so(2N + 1) = bN ................................... 97 ∼ 9.4. sp(2N) = cN ...................................... 98 9.5. Exotic Lie algebras: e6, e7, e8, f4, and g2 ...................... 99 9.6. Exercises . 101 iii 1. Introduction 1 1. Introduction 1.1. Why representation theory? To understand the importance and relevance of representation theory in mathematics and phys- ics, we first must ask the question: why group theory? The answer to this comes from trying to understanding real world phenomena through math- ematics, also known as physics. It often amounts to solving difficult, if not intractable equations. For example, most equations of motion that arise in classical dynamics, as you may have seen in Lagrangian and Hamiltonian Dynamics (MAT3008), are complicated non-linear partial dif- ferential equations. The properties of these equations are often elusive, even numerically. There is no known formalism for systematically solving the equations, and analytic solutions are rare. Similar conclusions arise in modern mathematical physics: studies of quantum dynamics (small scale phenomena) and large scale phenomena (general relativity) involve equations that are ridiculously hard to solve. So if life is so tough, how can we make progress? A remarkably successful approach comes from the study of symmetries. Symmetries can be an incredible labour saving device { they are often shortcuts to understanding physical systems even before understand exactly what the systems are! For example, even before knowing how to define a model of the hydrogen atom, using spherical symmetry (and some other symmetries), we are able to determine the orbits of electrons and their spectra. Group theory is the study of such symmetries. In studying the real world, the information we often extract from symmetries is not group theory itself, but rather a representation of the group. For example, a hydrogen atom is spherically symmetric, meaning that it looks the same as you change the angles θ; φ that ap- pear in spherical polar coordinates (r; θ; φ). Experimentally, we do not observe this symmetry directly, despite cartoons you may have seen representing the hydrogen atom as spherical ball. Instead this symmetry is seen experimentally in studying the atom in various excited states, as shown in 1.1. What scientists have noticed is that the excited states are observed in different representations of the group SO(3). Representations are the physically important thing to study. Mathematically speaking, a representation of a group is a way of expressing a group in terms of linear transformation of a vector space. That is, we turn abstract group elements, whose properties are often rather opaque, into matrices, something far more intuitive. We will see what this means in later chapters, but suffice it to say for now, the problem is now vastly simpler. So we study representations of groups because they are simpler than groups themselves and because this is what turns up in the real world. The representation theory of groups divides up according to the type of group being studied: • Finite groups: these are groups with a finite number of elements. The representation 1. Introduction 2 Figure 1.1: A numerical picture of the hydrogen atom in various excited states. The plots are really three-dimensional, and what is illustrated are two-dimensional slices. The excited states are labelled by the three integers at the top, and each uniquely corresponds to a representation of a Lie group, including that of SO(3). (Technically speaking, these are numerical solutions of Schrodinger's wave equation. The plots are probability densities arising from the resulting wavefunction.) theory is important to crystallography, some areas of quantum mechanics and geometry. • Compact groups: groups that are continuous but of bounded domain. Representation theory of compact groups is important as many of the results of finite group representation theory generalise in a nice way. • Lie groups: these are the most prominent category of continuous groups. Many results of compact groups apply to Lie groups. The representation theory of Lie groups is important in many areas of physics and chemisty, as well as mathematics. • Linear algebraic groups: these are Lie groups but not over the fields R or C but over 1. Introduction 3 a finite field. The representation theory of these groups is far less well-understood, and quite difficult involving sophisticated tools in algebraic geometry. • Non-compact groups: non-compact groups are too broad and opaque in structure to construct any systematic form of representation theory. Several special cases have been studied on an ad hoc basis. Clearly, group representation theory is a huge subject, and an incredibly active area of research within mathematics. We cannot possibly hope to cover the entire of the subject.