This is an elementary introduction to the representation theory of real and complex matrix groups. The text is written for students in mathematics and physics who have a good knowledge of differential/integral calculus and are familiar with the basic facts from algebra, number theory and complex analysis. The goal is to present the fundamental concepts of representation theory, to describe the connection between them, and to explain some of their background. The focus is on groups which are of particular interest for applications in physics and number theory (e.g. Gell-Mann’s eightfold way and theta functions, automorphic forms). The reader finds a large variety of examples which are presented in detail and from different points of view. The examples motivate the general theory well covered already by the existing literature. Hence for complete proofs of most of the essential statements and theorems the reader is often referred to the standard sources. Many exercises are included in the text. Some of these exercises and/or the omitted proofs may give the starting point for a bachelor thesis and further studies in a master program.
In Chapter 0, “Prologue: Some groups and their actions”, the author fixes some notation concerning the groups and their actions that are later used as the first examples, namely, the general and special linear groups over the real and complex numbers and the orthogonal and unitary groups. Moreover, the author presents the symmetric group \(\mathfrak S_n\) of the permutations of \(n\) elements and some facts about its structure. The author stays on the level of very elementary algebra and sticks to the principle to introduce more general notions and details from group theory only when needed in the development of the representation theory.
The author follows this principle in Chapter 1, “Basic algebraic concepts”, where the concept of linear representations is introduced using only tools from linear algebra. The author defines and discusses the fundamental notions of equivalence, irreducibility, unitarity, direct sums, tensor product, characters, and gives some examples.
The theory developed thus far is applied in Chapter 2, “Representations of finite groups”. The author finds out that all irreducible representations may be unitarized and are contained in the regular representation. In the next step the author moves on to compact groups. To do this, he leaves the purely algebraic ground and takes in topological considerations.
Hence, in Chapter 3, “Continuous representations”, the notion of a topological and of a (real or complex) linear group is defined, the central notion for the text. Following this, the author refines the definition of a group representation by adding the usual continuity condition. The author tries to take over as much as possible from finite groups to compact groups. This requires the introduction of invariant measures on spaces with a (from now on) continuous group action, and a concept of integration with respect to these measures.
In Chapter 4, “Representations of compact groups”, it is proved that the irreducible representations are again unitarizable, finite-dimensional, fixed by their characters and contained in the regular representation. But their number is in general not finite, in contrast to the situation for finite groups. The author states, but does not prove, the Peter-Weyl theorem. To get a convincing picture, he illustrates it by reproducing Wigner’s discussion of the representations of SU(2) and SO(3). It is proved that SU(2) is a double cover of SO(3). Angular momentum, magnetic and spin quantum numbers make an appearance, but for further application to the theory of atomic spectra the author refers to the physics literature.
In the very short Chapter 5, “Representations of abelian groups”, the author assembles some material about the representations of locally compact abelian groups. The author easily gets the result that every unitary irreducible representation is one-dimensional. But as can be seen from the example \(G=\mathbb R\), their number need not be denumerable. More functional analysis than the author offers at this stage is needed to decompose a given reducible representation into a direct integral of irreducibles, a notion which is not considered here.
Before starting the discussion of representations of other noncompact groups, an important tool for the classification of representations, “The infinitesimal method”, is presented in Chapter 6. At first, the author explains what a Lie algebra is and how to associate one to a given linear group. The main ingredient is the matrix exponential function and its properties. The author also reflects briefly the notion of representations of Lie algebras. Here again the author is on purely algebraic and, at least in the examples, easily accessible ground. The author starts giving examples by defining the derived representation \(d\pi\) of a given group representation \(\pi\). The author does this for the Schrödinger representation of the Heisenberg group and the standard representation \(\pi_1\) of SU(2). Then he concentrates on the classification of all unitary irreducible representations of \(\text{SL}(2,\mathbb R)\) via a description of all (integrable) representations of its Lie algebra. Having done this, the author considers again the examples \({\mathfrak {su}}(2)\) and \(\text{heis}(\mathbb R)\) (relating them to the theory of the harmonic oscillator) and gives some hints concerning the general structure theory of semisimple Lie algebras. The way a general classification theory works is explained to some extent by considering Lie SU(3); how quarks show up is seen.
In Chapter 7, “Induced representations”, the author introduces the concept of induced representations, which allows for the construction of (sometimes infinite-dimensional) representations of a given group \(G\) starting from a (possibly one-dimensional) representation of a subgroup \(H\) of \(G\). To do this work, the author again needs a bit more Hilbert space theory and introduces quasi-invariant measures on spaces with group action. The author illustrates this by considering the examples of the Heisenberg group and \(G= \text{SU}(2)\), where the known representations are rediscovered. Then he uses the induction process to construct models for the unitary representations of \(\text{SL}(2,\mathbb R)\) and \(\text{SL}(2,\mathbb C)\). In particular, it is shown how holomorphic induction arises in the discussion of the discrete series of \(\text{SL}(2,\mathbb C)\) (here the author touches complex function theory). A brief discussion of the Lorentz group \(G^L =\text{SO}(3,1)^0\) is inserted, and it is proved that \(\text{SL}(2,\mathbb C)\) is a double cover of \(G^L\). To get a framework for the discussion of the representations of the Poincaré group \(G^P\), which is a semidirect product of the Lorentz group with \(\mathbb R^4\), the author defines semidirect products and treats Mackey’s theory in a rudimentary form. A recipe is outlined to classify and construct the irreducible representations of semidirect products if one factor is abelian. The author does not prove the general validity of this procedure, as Mackey’s imprimitivity theorem is beyond the scope of the book, but he applies it to determine the unitary irreducible representations of the Euclidean and the Poincaré groups, which are fundamental for the classification of elementary particles.
In Chapter 8, “Geometric quantization and the orbit method”, the author takes an alternative approach to some material from Chapter 7 by constructing representations via the orbit method. Here he recalls (or introduces) more concepts from higher analysis: manifolds and bundles, vector fields, differential forms, and, in particular, the notion of a symplectic form. The author again uses the information and knowledge of the examples \(G =\text{SL}(2,\mathbb R)\), SU(2) and the Heisenberg group to get a feeling what should be done here. Certain spheres and hyperboloids are identified as coadjoint orbits of the respective groups, and line bundles on these orbits and representation spaces consisting of polarized sections of the bundles are constructed.
In Chapter 9, “Outlook to number theory”, a brief presentation of some examples where representations show up in number theory is given. The author presents the notion of automorphic representation (in a rudimentary form) and explains its relation with theta functions and automorphic forms. Theta functions and the Heisenberg group, theta functions and the Jacobi group, modular forms and SL(2,\(\mathbb R\)), elements of algebraic number theory, Hecke’s and Artin \(L\)-functions are considered and the Artin conjecture is mentioned.