These are the notes of a “Topics in representation theory” class I not be complete without a mention of Fulton and Harris's book , that. Representation Theory: A First CourseAuthor: William Fulton, Joe Harris Published by Springer New York ISBN: DOI. William Fulton Joe Harris. Representation Theory. A First Course. With Illustrations. Springer-Verlag. New York Berlin Heidelberg London Paris. Tokyo Hong.
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Representation theory: a first eourse / William Fulton and Joe Harris. p. em. - ( Graduate texts in mathematies). IncJudes bibliographieal referenees and index. 1. Representation theory: a first course / William Fulton and Joe Harris. Representation theory is simple to define: it is the study of the ways in which a given. Representation Theory. A First Course. Authors: Fulton, William, Harris, Joe DRM-free; Included format: PDF; ebooks can be used on all reading devices Representation theory is simple to define: it is the study of the ways in which a given.
V can be identified with the right ideal of r-tuples which are zero except in one factor. Likewise, what sort of vector space is G acting on: Definitions A representation of a finite group G on a finite-dimensional complex vector space V is a homomorphism p: Xv g2 we calculate the character of Nv: By Proposition 2. For the proofs of these theorems we refer to [Se2.
Fulton, J. Harris "Representation Theory: Curtis, I. Grades policy: Your course grade will be computed as follows: Anton Khoroshkin's page. Search this site. Homology seminar. My papers with description. Operads in Algebra and Topology. Representation Theory MIM Quantum groups. D'Hoker, D. Jaffe, R. Milson, K. Rumelhart, M. Reeder, and J. Willenbring for pointing out errors in earlier printings, and to many others for teUing us about misprints.
A few words are in order about the practical use of this book. To begin with, prerequisites are minimal: A good undergraduate background should be more than enough for most of the text; some examples and exercises, and some of the discussion in Part IV may refer to more advanced topics, but these can readily be skipped.
Probably the main practical requirement is a good working knowledge of multilinear algebra, including tensor, exterior, and symmetric products of finite dimensional vector spaces, for which Appendix B may help. We have indicated, in introductory remarks to each lecture, when any back- ground beyond this is assumed and how essential it is.
For a course, this book could be used in two ways. First, there are a number of topics that are not logically essential to the rest of the book and that can be skimmed or skipped entirely. Most of the material in the Appendices is relevant only to such a long course. Again, we have tried to indicate, in the introductory remarks in each lecture, which topics are inessential and may be omitted. Another aspect of the book that readers may want to approach in different ways is the profusion of examples.
These are put in largely for didactic reasons: For the most part, however, they do not actually develop new ideas; the reader whose tastes run more to the abstract and general than the concrete and special may skip many. Of course, such a reader will probably wind up burning this book anyway. We include hundreds of exercises, of wildly different purposes and difficulties. Some are the usual sorts of variations of the examples in the text or are straightforward verifications of facts needed; a student will probably want to attempt most of these.
Sometimes an exercise is inserted whose solution is a special case of something we do in the text later, if we think working on it will be useful motivation again, there is no attempt at "efficiency," and readers are encouraged to go back to old exercises from time to time. Many exercises are included that indicate some further directions or new topics or standard topics we have omitted ; a beginner may best be advised to skim these for general information, perhaps working out a few simple cases.
In exercises, we tried to include topics that may be hard for nonexperts to extract from the literature, especially the older literature. In general, much of the theory is in the exercises-and most of the examples in the text. We have resisted the idea of grading the exercises by expected difficulty, although a "problem" is probably harder than an "exercise. This may be a hint, a statement of the answer, a complete solution, a reference to where more can be found, or a combination of any of these.
We hope these miscellaneous remarks, as haphazard and uneven as they are, will be of some use. Preface v Using This Book ix. Abelian Groups; 6 3 8. More Projection Formulas; More Consequences Real Representations and Representations over Subfields of C The Proofs 84 Part D: Lie Groups and Lie Algebras 89 7.
Rank 3 A Little Geometric Plethysm The Exponential Map 9. Representations of 6 d: Simple Lie Algebras Lie Algebras: Proof of Frobenius's Fonnula 54 5.
Two Constructions 8. Lie Groups: Lie Algebras in Dimensions One. Weyl's Construction for Symplectic Groups Weyt's Construction for Orthogonal Groups Representations of sI3C. The General Set-up: Representations of C. Representations of SP4 C About the Killing Form Contents xiii SOmC and SO. Part II: Part I Representations of GLnC Representations of 5[3 C.
C and Spin. Tensor Products and Restrictions to Subgroups Bruhat Decompositions The Spin Groups Spin". Other Determinantal Identities Proof of WCF. Lie Theory Applications to Classical Lie Algebras and Groups Clifford Algebras and Spin Representations of so. Algebraic Constructions of the Exceptional Lie Algebras Complex Lie Groups. Proof of Capelli's Identity Hints. Contents xv B. On Derivations D. On the Weyl Group E. Duals and Contractions C. Ado's Theorem F.
To give an analogy. Second is the companion question: In both cases. In the 19th century. In differential geometry. Questions of existence or non- existence. Representation theory is very much a 20th-century subject. There are certainly valid reasons from a logical point of view: The real reason for us.
Only in the 20th century was the notion of an abstract group given. When we analyze. There are other topics. For example. Given this point of view. In general. We will spend the first six lectures on the case of finite groups. Finite Groups or mapped to a linear group GL V? It is largely for this reason that we are starting otT with the representation theory of finite groups: Many of the techniques developed for finite groups will carryover to Lie groups.
A map cp between two representations V and W of G is a vector space map cp: Definitions A representation of a finite group G on a finite-dimensional complex vector space V is a homomorphism p: In the latter case we give an analysis that will turn out not to be useful for the study of finite groups. Abelian groups. We work out as examples the case of abelian groups. The dimension of V is sometimes called the degree of p. Complete reducibility. When there is little ambiguity about the map p and.
We will call this a G-linear map when we want to distinguish it from an arbitrary linear map between the vector spaces V and W We can then define Ker cp.
W as a linear map cp from V to W. If Vand Ware representations. A representation V is called irreducible if there is no proper nonzero invariant subspace W of V. C of V is also a repre- sentation. A subrepresentation of a representation V is a vector subspace W of V which is invariant under G. W is also a representation.
In other words. This in tum forces us to define the dual representation by for all 9 E G. For a representation V. Representations of Finite Groups commutes for every 9 E G. Having defined the dual of a representation and the tensor product of two representations.
Exercise 1. Note that the dual representation is. Schur's Lemma As in any study. R is the space of complex-valued functions on G. Complete Reducibility. More generally. W G of elements of Hom V. Verify that in general the vector space of G-linear maps between two representations V and W of G is just the subspace Hom V. Show that this is an isomorphic representation. If X is any finite set and G acts on the left on X. We have. Let p: W fixed under the action of G.
This subspace is often denoted HomG V. The failure. Any representation is a direct sum of irreducible representations. Representations of Finite Groups built up out of other representations by linear algebraic operations. There are two ways of doing this.
This property is called complete reducibility. We should focus. If W is a subrepresentation of a representation V of a finite group G.
The key to all this is Proposition 1. We will see that. One can introduce a positive definite Hermitian inner product H on V which is preserved by each 9 E G i. Alterna- tively but similarly. L is complementary to W in V.
The additive group IR does not have this property: Occasionally the decomposition is written 1. It follows from Schur's lemma that if W is another representation of G. One more fact that will be established in the following lecture is that a finite group G admits only finitely many irreducible representations J'i up to iso- morphism in fact.
The decomposition of V into a direct sum of the k factors is unique. For any representation V of a finite group G. Schur's Lemma 7 of complete reducibility is one of the things that makes the subject of modular representations. The extent to which the decomposition of an arbitrary representation into a direct sum of irreducible ones is unique is one of the consequences of the following: Schur's Lemma 1.
For the second. AI has a nonzero kernel. It does not take long. In particular if G is abelian. Our first goal. The irreducible representations of an abelian group G are thus simply elements of the dual group..
This is known generally as iii Plethysm: Describe the decompositions. Abelian Groups.. Every subspace of V is thus invariant. Representations of Finite Groups the irreducible representations of G.
Note that if V decomposes into a sum of two represen- tations. Once we have done this. V will be G-linear for every p if and only if 9 is in the center Z G of G.
Symk V. We would like. We want to know where J sends an eigenvector v for the action of t. Let us now turn to the problem of describing an arbitrary representation of 6 3. The real reason we are doing it is that it will serve to introduce an idea that. This has some virtues as a didactic technique in the present context admittedly dubious ones. To see how this goes. Abelian Groups. We will see in the next lecture a wonderful tool for doing this. I 2 ' Zg-I 3 ' This representation.
This two-dimensional representation V is easily seen to be irreducible. To begin with. The idea is a very simple one: U' and V. The conclusion. If it is not. If the eigenvalue of v is Wi: In fact.
Representations of Finite Groups this. V' is isomorphic to the standard repre- sentation. Suppose now that we start with such an eigenvector v for t. Use this approach to find the decomposition of the represen- tations Sym 2 Vand Sym 3 v. This idea will turn out. Is Symm Symnv isomorphic to Symn Symmv?
As we have indicated. Let V be an irreducible representation of the finite group G. Show that. We can thus use this redundancy. More projection formulas.
Characters As we indicated in the preceding section. Of course. There wiII be more examples and more constructions in the following lectures. We compute the values of these characters on a fixed element g E G. Proposition 2. Exercise 2. If V is a representation of G. For the action of g. Let V and W be representations of G. Compute the characters of Sym k V and N v. Characters 13 to simplify the data we have to specify. D Exercise 2. In particular. The key observation here is it is enough to give.
For Sym 2 V. This then suggests the following: Show that if we know the character Xv of a representation V. The original fixed-point formula. Its character is XV 2. To see the character of the standard representation. This is a table with the conjugacy classes [g] of G listed across the top. As we have said. We compute the character table of 6 3. Characters know the coefficients of the characteristic polynomial of g: In sum. If V is the permutation repre- sentation associated to the action of a group G on a finite set X.
This is easy: This suggests expressing the basic information about the irreducible representations of a group G in the form of a character table. Xu' and Xv are independent. Carry this out explicitly for elements g E G of orders 2. Characters will be similarly useful for larger groups. The map cp is a projection of V onto VG. The idea behind our solution to this is already implicit in the previous lecture. We observed there that for any representation V of G and any g E G. If we just want to know the number m of copies of the trivial representation appearing in the decomposition of V.
In this section we will start by giving an explicit formula for the projection of a representation onto the direct sum of the trivial factors in this decomposition. We ask for a way of finding V G explicitly. On the other hand. To start. If V is irreducible then by Schur's lemma dim Hom V. W G is the multiplicity of V in W. We can do much more with this idea.
The number of irreducible representations of G is less than or equal to the number of conjugacy classes. W of the representation Hom V. The numbers over each conjugacy class tell how many times to count entries in that column. W G is the multiplicity of W in V. Corollary For example.. The key is to use Exercise 1. In terms of this inner product. Corollary 2.
The multiplicity ai of V.
Example 3. Xv' ' We obtain some further corollaries by applying all this to the regular representation R of G.
Any representation is determined by its character. The First Projection Formula and Its Consequences 17 We will soon show that there are no nonzero class functions orthogonal to the characters. Any irreducible representation V of G appears in the regular representation dim V times. E9 ai. A representation V is irreducible if and only if Xv.
The orthogonality of the rows of the character table is equiv- alent to an orthogonality for the columns assuming the fact that there are as.
Written out. Show that the number of elements in each of these conjugacy classes is. IGI L. As for the irreducible representations of 6 4. The character of the trivial representation on the five conjugacy classes is of course 1. The character table so far looks like. As with any symmetric group 6 d. To find the character of the standard representation. Characters many rows as columns. We obtain then the complete character table for 6 4: Verify the last row of this table from 2. Since there are by Corollary 2.
W is really a representation of the quotient group! The key is the 2 in the last column for Xw: We now get a dividend: The latter of these is easy to locate: We can see that this is irreducible either from its character since IXv. As for the remaining representation of degree two. Three representations U.
From the table. It follows. Rotation by about a line joining the midpoints of two opposite edges is a transposition in 6 4 and fixes no faces.
We start. Decompose the permutation representation of 6 4 on i the vertices and ii the edges of the cube. We may thus ask how these representations decompose. As we said above. Example 2. Compute the character table. Now X. The differences of opposite faces therefore span V'.
Which pairs of nonisomorphic representations of 6 4 become isomorphic when restricted? More Consequences In this section. More Projection Formulas. More Consequences 21 Exercise 2. The answer is given by Proposition We simply write out the condition that CPa.
Then CPa. By Schur's lemma.
Characters Proposition 2. The number of irreducible representations of G is equal to the number of conjugacy classes of G. V EEl W. We can express most of what we have learned so far about representations of a finite group G in these terms. To begin. Suppose a: The ring structure is then given simply by tensor product. The representation ring R G of a group G is easy to define. The statement that a representation in determined by its character then says that X is injective.
We will see in more examples below how we can use this information to build up the character table of a given group. For now. More Consequences 23 the images of X are called virtual characters and correspond thereby to virtual representations.
By Proposition 2. The argument for Proposition 2. In terms of representation rings. X is the character of an irreducible representation. The following problem. L g-I.. In these lectures we will often be given a subgroup G of a general linear group GL V. Let V and W be irreducible representations of G. If GI and G2 are groups.
We conclude this lecture with some exercises that use characters to work out some standard facts about representations. Define L: Another challenge: Problem 2. Show that if V is a faithful representation of G.
More Consequences 25 Problem 2. Show that the dimension of an irreducible representation of G divides the order of G. Show that the character of any irreducible representation of dimension greater than 1 assumes the value 0 on some conjugacy class of the group.
Real Representations This lecture is something of a grabbag. We will see quite a bit more about the representations of the symmetric groups in general later.
Induced Representations. Group Algebras. Before turning to a more systematic study of symmetric and alter- nating groups. Section 3. As for the irreducible representations. We can now find the remaining two representations in either of two ways. Where should we look for these? On the basis of our previous experience and Problem 2. To start with. Xv g2 we calculate the character of Nv: Find the characters of the representations V and V'.