This is my handout for the summer school of Chinese Academy of Sciences on 20 19. Attention is little knowledge, not cold knowledge. We only mention the most important content (in our opinion). In addition, it is a handout, not a textbook, so it is all words. There are so many good books in the textbook that I have no ability to write better than others. The style of the handout is not very stable, ranging from deep to shallow. Sometimes it's funny and sometimes it's serious. It is understandable that the little knowledge of representation flies in the sky like a wild goose, sometimes S-shaped and sometimes B-shaped.

The handout is roughly divided into two parts, the first part is the representation of finite groups, and the second part is mainly the representation of compact groups. In teaching practice, we use seven lectures on finite groups, four lectures on compact groups, and the last lecture gives the basic results expressed by SL2(R). We will record the detailed course progress later.

The finite group part mainly discusses the decomposition of semi-simpleness, regular representation (that is, Peter-Weyl theorem of finite groups) and the construction of induced representation. These contents are all standard. As an application, we study the representation of second-order matrix groups over finite fields. Finally, Tadashi Tadashi's reconstruction theorem is discussed, that is, groups are constructed in turn by representation categories. Although the proof of the theorem is simple, the way of thinking here is unfamiliar to students, so in teaching, we focus on what the theorem says, not how to prove it. The whole first part can be used as a short course or reading course for undergraduates after learning abstract algebra. Here we try to be concise and to the point, and tell the basic content of finite group representation with the shortest length and the most direct logical route.

In the compact group part, we take Peter-Weyl theorem as the center. Because students are unfamiliar with topological groups, the narrative style of this part is slightly different from that of the first part. We still try to be concise, but we also try to add some discussion (fèi) to help you understand. We also emphasize examples. In some places, examples are even given directly, rather than general theories. First, we review the related theories of Fourier series and explain why it is the representation theory of S 1. Then there are two general theories, mainly to prove Peter-Weyl theorem and discuss Fourier series theory on abstract compact groups. Then we go back to the example. Understand and deepen the previous theorem with concrete group Su (2). Finally, we discuss the reconstruction theorem of compact groups, which echoes the homonym theorem of finite groups, but there is no proof of this theorem in teaching practice. At the end of the course, we stated the classification represented by SL2(R). Most of the conclusions have not been proved (and it is impossible to prove them in such a short class). We hope you can take this as the beginning of further study.

In the last section, we collected some relatively complicated materials and packaged them into exercises. These exercises can be understood by what you have learned in class, and most of them can also be done by what you have learned in class. Of course, it doesn't matter if you can't do it. It's already very difficult. It is also good to keep thinking slowly.

Our students come from the north and south of the motherland. Everyone came to Beijing for the dream of * * *, enjoying the cool breeze blowing from the representative circles in the long summer. Later, Beijing became hotter and hotter. In order to be cooler, everyone goes to class by plane or even by rocket every day, just to make the wind stronger. Most students are sophomores or juniors. Generally speaking, they learned a little about abstract algebra, measure theory and complex variable function. Some students have heard a little about group representation theory and functional analysis. The results we use are basically linear algebra of group theory, and some simple functional analysis is used in the part of compact group (that is, it is convenient to talk). As long as you have studied group theory and linear algebra, we all welcome to learn representation theory. There's always one for you.

Finally, I want to mention a phenomenon worthy of vigilance. Students are often not interested in my theoretical work of representation and are unwilling to do calculations by themselves. Instead, they are very keen on abstract nonsense without content. Here are some typical examples. First, I think GL2(Fq) is too concrete and boring, and then I see how enthusiastic Category Theory is when it comes to the reconstruction theorem.

Furthermore, when it comes to Fourier analysis of a group, Planck's formula thinks it is pure calculus. As a result, when a group algebra is C*- algebra, it shines at the moment. I'm not saying that it's not good to like abstract things, but that abstract things should be supported by concrete examples. Abstract tools should serve specific problems. The abstract mathematics we are seeing now is good at mathematics, not because its complex abstract structure looks good and sounds tall, but because it can help us solve problems that we can't solve or become complicated before, or let us have an insight into the essence of problems at a higher level. Everyone worships Grothendieck, who rewrote the whole framework of algebraic geometry. However, we must realize that Grothendieck reconstructed the abstract framework in order to solve some specific problems in algebraic geometry, especially conjecturing that this problem is positive with Weil, which is of great significance in theory and practice. So we should remember that abstraction must serve specific problems, not abstraction.

We have classes for one and a half hours at a time, three times a week, and are scheduled for Tuesday, Thursday and Friday morning. In the afternoon, there will be teaching assistants' counseling and teachers' questions and answers. Class hours will be * * * eighteen hours. The excellent quality and amazing diligence of the students make the course progress very smoothly. When I was teaching at Purdue University, some students told me that you taught like a rocket. I think the progress I'm talking about here may not be just rockets but super starships. I think it is feasible to slow down the speed by half or more in normal teaching practice. It is better to supplement students with a large number of specific examples. Some preparatory knowledge or background knowledge (which we all assume students know) should also be included in the normal teaching content (algebraic module, category theory, simple functional analysis, etc.). ).

The content of this course is reasonable as an elective course of 36 hours or 48 hours in the last semester for junior or senior students majoring in mathematics. To sum up, the progress of the lecture is roughly as follows:

1. Definition of finite group representation, semi-simplicity, statement of regular representation theorem.

2. Definition of features, orthogonality, decomposition of right regular representation and irreducible representation dimension.

3. Matrix coefficient, decomposition of regular representation; Two theorems: the irreducible representation of the order of a divisible group of dimensions, and burnside's solvability theorem.

4. Inductive representation, Frobenius's law of reciprocity, McKee's subgroup theorem.

5. The structure of 5.GL2 (FQ) group is expressed by parabolic induction.

6. Point representation, using Weil to represent structural point representation (we omit the case of splitting and skip the calculation of trace after Theorem 4.4.2).

7. Tadao theorem of finite groups.

8. Hilbert space, circle representation and Fourier series.

9. The basic facts of compact group representation and irreducible representation are limited (we skipped the proof), and Peter-Weyl theorem (we emphasized the statement of the theorem and told students that this proof can be skipped if they have not learned compact operators). It took about half an hour to prove Peter Weil's theorem.

10. Harmonic analysis on compact groups. We mainly discuss how to construct the standard orthogonal basis of L2(G) by using Peter-Weyl theorem.

Representation and harmonic analysis of 1 1 Sue 2. We skipped the proof of Weyl integral formula and did not discuss the properties of orbital integral too much. In other words, we proved Planck formula on the basis of facts, emphasized its importance, and ended it.

12. The construction of discrete sequence representation and unitary principal sequence representation of SL2 (r), and the statement of Planck formula.

Let's talk about some views on learning. First of all, it is unrealistic for students who are interested in learning mathematics to understand everything for the first time after learning new things to a certain extent (depending on everyone's talent, but generally bounded by abstract algebra of real variable functions). Secondly, it is at this stage that the requirements for learning have been greatly improved. Generally speaking, it requires us to understand the core content of a subject in a short time.

Here comes the problem. What do you mean, learned? I want to emphasize that, at least, I think it is very important to prove the theorem and even remember that it is not the same as understanding in the real sense. In my opinion, the most important thing in learning is to establish correct intuition, which is a sign of learning. What is the right intuition? I think it means that when you see the definition of a theorem, you can instinctively establish the following things in your mind: the most representative example that best reflects the essence of the problem, the rationality of the definition or theorem (why it is defined this way instead of that way), the necessity of the definition and theorem (why it is defined this way or why it is made this way), why the theorem is right (what philosophical reasons are there to explain the theorem) and so on.

Let's give a small example. What do we think of when we see the theorem that Riemannian function can be integrated if and only if the discontinuous point set of the function is a zero measurement set? The following are several important aspects: (1) Zero measure set means that there are not many points in the set, but this condition is slightly weaker than countability; (2) Riemannian integrability shows that the continuity of the function cannot be too bad. This theorem gives the exact meaning of "not too bad", so it should be correct. (3) Riemann function is discontinuous and unreasonable at all rational points, so it can be integrated. You may think of many other things, but what I want to say is that the proof of the theorem itself is not so important compared with these things. It is undoubtedly difficult to establish correct intuition. Overcoming these difficulties and finally completing the establishment of correct intuition requires a lot of accumulation. Accumulate not only knowledge but also examples. In a sense, examples are equivalent to intuition, intuition is the internal essence of examples, and examples are the external representation of intuition. Intuition is difficult to establish when there are not enough examples in our minds. So we have to study repeatedly and study examples by ourselves. Everyone should do examples by himself, and watching others do examples will always be others' examples, not their own. It's like watching others swim, and you'll never learn. Only by getting into the water or drinking a few mouthfuls of water can you learn to swim. It doesn't matter if you don't understand it the first time. It doesn't matter if you don't understand it a few times before. Do not doubt yourself. What you lack is only the accumulation of examples and experience. Examples are of course relative. What is abstract today will become an example of more abstract content tomorrow. Just as a matrix can be used as an abstraction of linear equations, the matrix itself is a concrete example of a more general operator. There is a very metaphysical thing called mathematical maturity. I think the richness of examples in your mind can also be regarded as an important aspect of mathematical maturity.

In the process of learning, we often rely on examples to move forward. When we come across a definition or theorem, we should first give ourselves an example to see if we can give a less common example. Both positive and negative examples are needed. There are too many examples, and we will naturally feel it. For example, learning groups, we first ask ourselves, how many examples of specific groups do we have in mind? What do we know about these examples? For example, we will encounter a classic result: the smallest noncommutative simple group is A5. Our first reaction, why? Try it first. Let's try several groups with orders less than 60. Then we find that if you try any example, there are always some reasons why it is not a noncommutative simple group. In the process of trial and error, you will encounter Sylow theorem, results about p- groups, and other small conclusions. When you have tried enough examples, you feel that you can psychologically accept that "the smallest simple group is A5". Although you didn't read the proof, in fact, you have basically "understood" this conclusion, because you already know roughly why other groups can't. For example, after studying Galois theory, we first saw the main theorem of Galois theory. At this time, let's try it ourselves. Let's use an equation to calculate the Galois group (for example, the Galois group of x3+x+ 1). Can it be worked out? ), or just write a domain extension and ask if it is a Galois extension? Can you write down all the intermediate fields?

The ancients said: Reading a book a hundred times is self-evident. But this is an ancient book. I don't know if you can understand it after watching it a hundred times.

It may be useless to read math for a hundred times, but it is definitely beneficial to calculate examples for a hundred times.

On the other hand, we say that we should master the core content of a subject quickly. How can we quickly master the core content of a subject? The simplest answer: let the knowledgeable people tell you. Learning from others, especially the backbone content, is the most efficient way. For this course, I try my best to be a knowledgeable person and describe the most important content of representation theory (in my opinion) in a concise and direct way. A slightly more complicated answer: thin reading. Hua told us that reading should be thick before reading. The process of reading thickness is an intuitive example process, and the process of reading thickness is a process of grasping the main content. What we said earlier is the process of reading thick books. How to read thin books? A simple exercise is to give you an hour to talk about the main content of a topic. What is the most important research object? What is the most important and essential theorem? Why is this theorem right (note that this is not proof but philosophy)? At this time, we will go to the rough and extract the essence and go straight to Huanglong. All problems only serve the core. I can't help you read this book thick. As mentioned above, this requires your own crazy efforts. I hope this course can make a little contribution to the process of thin reading.

I am far from being an excellent mathematician or a good math teacher (I want to invite my teachers and other disciples here, and I am often ashamed of my heritage because of my limited level), which makes my understanding of many problems perhaps not optimal. My literary accomplishment is also very limited, and what I write may not be meaningful, nor may I express my most essential thoughts.

In short, I didn't learn Chinese and math well. But it doesn't matter. People in this world are far more knowledgeable and cultured than I am. They wrote many things worth reading. The following are the prefaces of these two books:

Problems and theorems in 1. This is the classic of the classics. Exercise lasts forever, and a book lasts forever. The preface of the book makes people want to die.

2. Wu Hongxi, Introduction to Riemann Geometry. I'm not a geometer, so I can't evaluate the content of this book, but the preface has made me read it more than ten times. In my extremely limited reading range, it is the best preface of China's mathematics book.

Finally, I invited two textbooks about performance.

1. Searle. Finite group representation.

2. Fulton, Harris. Representation theory.

These two books are really classic teaching plans for reading thick books and thin books. Searle's book is very thin and concise, and goes straight to the point in a few words. I admire Serre for writing a book, and I almost want to give his knee directly to his old man. Serre writes documentary theorems such as chopping melons and vegetables, and the logical structure is beautifully packaged. All kinds of difficulties seem to vanish in an instant. Of course, this book is too dry. If you read it quickly, you will easily suffocate. Fulton and Harris' book is very thick, but there is no content. Rory talked a lot of nonsense. This is a classic of reading a thick book: it is full of examples! All examples! All examples! Example takes you off!

Many students are asking what to learn after this course. This of course depends on what you want to do next. But in a word, some things are still standard materials, which are worth mastering by every mathematician. There are probably two main aspects in this respect: the highest weight representation theory of compact Lie groups and the classification and representation of complex semisimple Lie algebras. The reading materials you can refer to are:

1. Introduction to Lie Algebra and Representation Theory. Concise.

2.knapp, introduction of Li Qunchao. The content is simple (Luū) and solid (suū), and it is also written clearly.

3. Plug Panski, tight Lie Qun. The content is very elementary, and it is very good as a teaching material.

The introduction of noncompact Lie group representation can also be found in Knapp's book Representation Theory of Semisimple Groups:

An overview based on examples. This book is close to 800 pages and covers everything. Of course, it is just a door. This aspect has now become a hot research direction. I suggest you read the paper directly and find a topic to do after you have a basic feeling.

Both horizontal and vertical are dry goods!

For number theorists, the representation of p-adic groups is also very important. Most of the basic contents are learned from GL2. The standard textbook is either Jacquet-Langlands or Bump. Of course, a more general and profound textbook is1Bernstein's lecture notes at Harvard in 1992.

Automorphism forms on 1.Jacquet, Langlands, GL (2). This book or its author is so famous that everyone calls it Jacquet-Langlands.

2. Bump, automatic table and demonstration. This book is a standard textbook. Generally called concave and convex. This book is full of mistakes and can barely bear it.

3. Bernstein, the lecture notes about the speech of P-adic Group can be found on Bernstein's homepage. We said we should learn from the master, and this is the master. Everything about Bernstein is worth learning. Jacquet-Langlands and Bump are teaching materials for self-control appearance. This is of course one of the most important parts in number theory. study

If you have strong self-control, just read one. In addition, you can refer to

1. Godement, Notes on Jacques-Langlands Theory. This book was not published before, but it was printed in IAS and finally published by Higher Education Society recently. It has less content and is easier to read than the original book Jacquet-Langlands.

2. Automorphism forms on 2.Gelbart, Adele groups. This book has been a standard textbook for a long time and has been widely circulated. But there are many small mistakes, and the quality of typesetting and printing is also very worrying. It's sour to read (at least I feel this way when I learn it).

Representation theory is a big knowledge, and I can only mention some parts that I am familiar with. There are many examples, such as geometric representation theory, quantum group, categorization and so on. Representation theory is also closely related to geometric topology, mathematical physics and so on. I don't understand these things, and I can't explain them to you in depth. For an overview of the theory of representation, you can take a closer look at what Academician Xi Nanhua, Dean of the School of Mathematics of China Academy of Sciences, did in the summer school.

A little nonsense, above.