Product Description
This paperback edition of the standard algebra text reviews the development of the representation theory of finite groups and associative algebras and discusses applications and connections with other parts of mathematics. The authors incorporate a self-contained account of the three main branches of representation theory - ordinary, modular, and integral - and demonstrate the connections between them. There is a wealth of illustrative examples and auxiliary results, and almost every section contains exercises. ... Read more

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Applying Cognitive-Behavioral Therapy, one of psychology's most popula r theories, to group work. Provides practical techniques for conducti ng CBT therapy in groups and shows how to troubleshoot common problems . One of the first books to address the use of CBT therapy in group s ettings -- a unique book that will appeal to a wide range of clinician s. ... Read more

Customer Reviews (1)

Cognitive-Behavioral Group Therapy for Specific Problems and Populations
This was a great book. I am a CBT therapist and it was helpful to see a group CBT book out there. It is a great tool. I recommend it highly. ... Read more

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This book is aimed at graduate students in physics who are studying group theory and its application to physics. It contains a short explanation of the fundamental knowledge and method, and the fundamental exercises for the method, as well as some important conclusions in group theory.

The book can be used by graduate students and young researchers in physics, especially theoretical physics. It is also suitable for some graduate students in theoretical chemistry. ... Read more

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This book develops the combinatorics of Young tableaux and shows them in action in the algebra of symmetric functions, representations of the symmetric and general linear groups, and the geometry of flag varieties.The first part of the book is a self-contained presentation of thebasic combinatorics of Young tableaux, including the remarkable constructions of "bumping" and "sliding", and several interesting correspondences.In Part II the author uses these results to study representations with geometry on Grassmannians and flag manifolds, including their Schubert subvarieties, and the related Schubert polynomials.Much of this material has never before appeared in book form.There are numerous exercises throughout, with hints and answers provided. Researchersin representation theory and algebraic geometry as well as in combinatorics will find this book interesting and useful, while students will find the intuitive presentation easy to follow. ... Read more

Customer Reviews (1)

A Fine Synthesis of Combinatorics, Geometry, and Algebra
With his usual lucidity, Fulton brings together the surprisingly wide area of mathematics concerned with Young tableaux.These are combinatorial patterns which index basis vectors of group representations (either of the symmetric group or the general linear group).These vectors can be seen as Plucker coordinate functions on non-linear representations, namely homogeneous spaces (Grassmannians and flag varieties).Thus, Young tableaux form an invaluable tool to examine these representations and varieties in concrete detail.Fulton also gives a good exposition of the combinatorial operations on tableaux which reflect the crystal basis structure from quantum GL(n), though Fulton does not explicitly discuss quantum groups.Other good expositions of these topics, from a more algebraic and combinatorial point of view, are Sagan's newly revised "The Symmetric Group", and Stanley's "Enumerative Combinatorics", Vol 2. ... Read more

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This is a well-balanced introduction to topology that stresses geometric aspects.Focusing on historical background and visual interpretation of results, it emphasizes spaces with few dimensions, where visualization is possible, and interaction with combinatorial group theory via the fundamental group.It also present algorithms for topological problems. Most of the results and proofs are known, but some have been simplified or placed in a new perspective. Over 300 illustrations, many interesting exercises, and challenging open problems are included.New in this edition is a chapter on unsolvable problems, which includes the first textbook proof that the main problem of topology, the homeomorphism problem, is unsolvable. ... Read more

Customer Reviews (3)

Does not deformation retract onto Munkres et al.
This is a wonderfully intellectual, semi-historical approach to classical topology.

Chapter 0 gets some fundamentals out of the way. Chapter 1 is very intriguing and contains lots of ideas. First we are given a taste of the Riemann surfaces of complex analysis. These are complemented by the nonorientable surfaces, and it all leads to the classification of surfaces, which is achieved through the fundamental group and the realisations of surfaces as polygons with identifications, and this in turn leads picturesquely to covering surfaces. These simply and concisely presented ideas provide the seeds for much of the later chapters. The short chapter 2 sets up the two-way connection between topology and combinatorial group theory, which proves fruitful when the fundamental group grows into two chapters of its own (3 and 4). Then follows a sort of supplementary chapter 5 which touches on homology theory (otherwise largely neglected, but with good reason, Stillwell argues) to motivate abelianisation, which is the method we use to formally tell the fundamental groups of all surfaces apart. Chapters 2-5 were a bit slowed down by foundational issues, but now in chapters 6-8 it's all topology all the time. There are nice accounts of the classical theories of curves on surfaces (chapter 6) and knots (chapter 7). In chapter 8 we see how some of our previous ideas carry over to 3-manifolds. But ultimately 3-manifolds are deep water, with the homeomorphism problem being unsolved and all. Neither would it help to move up to 4-manifolds or higher, but at least that's not our fault so to speak because there the homeomorphism problem is in fact unsolvable. The homeomorphism problem and other fundamental problems are essentially algorithmic (i.e., given two spaces, decide whether they are different or the same) so unsolvability (non-existence of algorithms) is indeed a force to be reckoned with, so it is given its own chapter 9, naturally culminating with the unsolvability of the homeomorphism problem.

There are many ways to destroy the soul of topology. Stillwell says in the preface: "In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams." Stillwell protects us from such dangers by his emphasis on low dimensions, his insistence on the fundamental group as the best unifying tool, visualisation and illustrations, and his great respect for primary sources. The latter is reflected in frequent references and in the commented, chronological bibliography, which is very useful.

An introduction well worth reading
Ideas from topology are now manifested everywhere in physics, engineering, computer graphics, and many, many other applications. Consequently, a thorough understanding of topology has become necessary for those who are involved in these applications. This book gives an introduction to "classical" topology that emphasizes the geometric intuition behind the subject, and is thus very suitable for those who need such an understanding. That is not to say that aspiring mathematicians will not gain from the reading of the book. It still maintains a standard of rigor that graduate students in mathematics need to advance in more in-depth courses in topology. The author does not hesitate to use diagrams in the book, which makes it an even better one for those interested in applications. Most of the results he discusses were known in the late 1800's and early 1900's, but they are still important today, especially in physics.

In chapter 0, the author introduces the fundamental concepts in topology with a discussion of the homoemorphism problem. Confining attention to three dimensions and less, the author does mention the impossibility of solving the problem in dimensions greater than or equal to four. He then gives an overview of open and closed sets, continuous functions, identification spaces, and elementary homotopy theory. The building blocks of the main objects he considers in the book, namely simplicial complexes, are discussed in detail. The Haupvermutung is briefly discussed, and a full proof, due to E.E. Moise, of the Jordan curve theorem is given. The proof is the first example of the general approach that the author takes in the book, namely of reducing general topology to combinatorial topology. A brief introduction to algorithms is given, and the author introduces the group theory needed for the rest of the book.

Since he is taking an historical approach in this book, the author begins the study of surfaces with the study of Riemann surfaces. He motivates the ideas of Riemann surfaces, such as branched coverings of the 2-sphere, very nicely, and gives a very understandable theorem for surfaces is proved in detail. In addition, the concept of a universal covering space is described beautifully, and the author shows how to obtain it for orientable surfaces of genus greater than one. The author also gives a brief taste of Fuchsian groups.

Chapter 2 is devoted completely to the group theory of graphs, as a warm up to the study of the fundamental group in the next chapter. The fundamental group is defined to be an equivalence class of maps, and with the exception of the circle, it is calculated using deformation retraction and the Seifert-Van Kampen theorem. The fundamental group of complexes are then calculated in chapter 4, using first a method due to Poincare, and then directly. Some knot theory is introduced, and the Wirtinger presentation of knot groups is discussed.

Homology theory is presented in chapter 5 as an abelianization of the fundamental group, and the abelianization is shown to be independent of the presentation of the fundamental group. The author does not spend much time in homology, arguing (correctly) that in dimensions less than or equal to three the fundemental group contains all the information obtained from homology.

The study of curves on surfaces is the subject of chapter 6, with the contractibility problem studied first using Dehn's algorithm. Some methods for "simplifying" simple curves on closed orientable surfaces by homeomorphisms are also discussed. These techniques could be considered an elementary warm-up to the handle calculus procedures done in higher-dimensional topology. The physicist-string-theorist reader will appreciate the discussion of the mapping class group of the torus.

All of chapter 7 is devoted to knot theory, and this subject, now of enormous importance in physics and computational biology, is treated in great detail here. The braid group is defined, and Artin's solution of the word problem is given.

A very short overview of 3-dimensional manifolds is given in chapter 8, wherein the important concept of a Heegaard splitting is discussed, along with other methods for constructing 3-manifolds. The recognition problem for the 3-sphere, and the famous Poincare conjecture, are mentioned, and the author outlines one method, called shelling a simplicial decomposition, for recoginizing a 3-sphere. He shows however the existence of an unshellable triangulation of the 3-ball (Bing's cube). The author is incorrect though when he states that an algorithm for disproving the conjecture, i.e. an algorithm for enumerating all the 3-manifolds not homeomorphic to the 3-sphere, does not exist. Since the date of publication of this book, such an algorithm has been constructed by Rourke and Sanderson.

The book ends with a discussion of unsolvable problems in combinatorial topology. This is an unusual topic in a book on topology, but given the importance currently of computer algorithms in the growing field of "computational topology" such an inclusion is appropriate and useful. The author discusses Turing machines, Church's thesis, and the Halting problem. The author discusses the unsolvable problems in group theory, and shows how the halting problem can be reduced to the word problem in groups. He leaves as an exercise, using the formalism developed in the chapter on presentations of groups, the problem of showing the homeomorphism problem is unsolvable for closed 4-manifolds.

An accessible introduction to low-dimensional topology.
This book treats (geometric) topology of dimensions two and three together with necessary machinery of group theory. The account is accessible if you have the basic knowledge of general topology and algebra. It well illustrates the inherent difficulties in low-dimensional topology. A mild surprise with this book is that Novikov's proof of the unsolvability of the word problem is fully described. ... Read more

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This thoroughly revised and updated version of the popular textbook on abstract algebra introduces students to easily understood problems and concepts. John Humphreys and Mike Prest include many examples and exercises throughout the book to make it more appealing to students and instructors. The second edition features new sections on mathematical reasoning and polynomials. In addition, three chapters have been completely rewritten and all others have been updated. First Edition Pb (1990): 0-521-35938-4 ... Read more

Customer Reviews (2)

Excellent book
This book is ideal for a first brush with abstract algebra. Ideally it should be read before one tackles linear algebra - however, this is rarely the case. The book is extremely well written, contains many exercises and some solutions, consistently points the reader towards more advanced texts in the appropriate areas, and gives many brief and interesting historical anecdotes.
The approach is fairly elementary in many places - for instance, one proves many results for the integers alone, which are in fact true for rings in general. However, this serves the important pedagogical purpose of introducing topics in a context with which we are all familiar - later, when reading more advanced texts, the ideas are easily generalized.
The book is a great buy for the price. I highly recommend it.

The most fascinating math book I've ever read.
This a university-level math textbook, but it can be read by anyone with a high school-level knowledge of algebra.It covers subjects such as set theory, error-correcting codes, properties of prime numbers, and booleanalgebra, and it's FUN!!I especially liked the section on congruenceclasses.To give an example: the authors explain why a number of the form4k + 3 (k is an integer) can never be the sum of squares, and they give youthe tools to discover your own odd facts such as that.A verywell-written math book. ... Read more

Product Description

This book consists of three parts, rather different in level and purpose. The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and characters. The second part is a course given in 1966 to second-year students of l’Ecole Normale. It completes in a certain sense the first part. The third part is an introduction to Brauer Theory.

Customer Reviews (2)

Extremely concise -- no illustrative examples.
I'm using this book as an undergraduate, so my rating is clearly skewed, as evidenced by the huge "Graduate Texts in Mathematics" on the cover.We've only covered the first five chapters so far, and while the overarching ideas are quite clear, I find the notation confusing.No (even small) reviews of the linear algebra you studied years ago; it just dives in.Perhaps graduate students can follow it all quickly with no concrete examples, but it takes me a few readings through each section to begin to understand what is being said.By concrete I mean a real-world see-it put-your-hands-on-it example, or at least an example involving a few numbers as elements.

Here is an excerpt (so you can judge for yourself how helpful the first chapter will be for you) of a representation example from section 1.2 entitled 'Basic Examples:' "Leg g be the order of G, and let V be a vector space of dimension g, with a basis (e-sub-t)sub-t-in-g indexed by the elements t of G.For s-in-G, let rho-sub-s be the linear map of V into V which sends e-sub-t to e-sub-st; this defines a linear representation, which is called the regular representation of G.Its degree is equal to the order of G.Note that e-sub-s = rho-sub-s(e-sub1); hence note that the images of e-sub1 form a basis of V.Conversely, let W be a representation of G containing a vector w such that the rho-sub-s(w), s-in-G, form a basis of W; then W is isomorphic to the regular representation (an isomorphism tau: V --> W is defined by putting tau(e-sub-s) = rho-sub-s(w))."

The language is very concise and usually quite clear, and I suppose for someone with a sophisticated math background it could be a preferred book.For someone like me who has had only one semester of introductory linear algebra two years ago, I would prefer a more "bridging" text -- that is, one which often and quickly reviewed basic concepts from linear algebra and was less concise in its explanations of definitions and examples.

Typical Serre, concise, clean, clear.
This is a an excellent introduction to the subject.The book really breaks into 3 distinct parts.The first 5 chapters are a rapid introduction to the basics, similar to what one would get from any indroductory text.They are most notable for actually going through the details on D_n, S_n cyclic groups... The second section (chapters 6-13) gives a more graduate level presentation of the material.Starting with a discussion of group algebras, moving onto inducted representations Artin's theorem (the existence of virtual characters) The third section is Brauer Theory. The book is by Serre so it goes without saying it one of the best if not the best book on the market.His failure to deal with the additional complexities of the infinite group case (which he indicates in the title) is a small problem.He could have spent at least 1 chapter addressing how the results of the book could be extended.The index of notation is a fantastic asset for a subject where notation plays such a large role. ... Read more

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The aim of this book is to provide a concise treatment of some topics from group theory and representation theory for a one term course. It focuses on the non-commutative side of the field emphasizing the general linear group as the most important group and example. The book will enable graduate students from every mathematical field, as well as strong undergraduates with an interest in algebra, to solidify their knowledge of group theory. The reader should have a familiarity with groups, rings, and fields, along with a solid knowledge of linear algebra. Close to 200 exercises of varying difficulty serve both to reinforce the main concept of the text and to expose the reader to additional topics. ... Read more

Customer Reviews (1)

An excellent supplement or a wonderful continuation
I used this book as a 2nd year grad student with a little more than a year of graduate algebra. It worked very well as tool to solidifying my knowledge of group theory. i think it would also work as a great book to read while taking a first year algebra course. The idea of the book is to teach theory grounded in examples. In particular, Alperin uses the matrix groups as the main example for the entire book. The writing is very user friendly and the proofs are adequate. It covers basic group theory concepts, local structure (p-groups, Sylow, solvability), Parabolic Subgroups of Gl(n,F) and at the end goes into modules and representation theory. ... Read more

Product Description
This book is a useful and accessible introduction to symmetry principles in particle physics.Concepts of group theory are clearly explained and their applications to subnuclear physics brought up to date.Many worked examples are included.There is a growing interest in the quark structure of hadrons and in theories of particle interactions based on the principle of gauge symmetries. Students and researchers on theoretical physics will make great strides in their work with the ideas and applications found here. ... Read more

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The number of areas of chemistry where the application of simple group theoretical ideas is important for undergraduate and post-graduate students has greatly increased over recent years. This book, written as part of the MacMillan Physical Science Series, covers the essential group theory encountered in under-graduate and postgraduate chemistry courses, with the emphasis throughout on the application of theory. The aim has been to include a wide range of examples without unnecessarily increasing the level of difficulty for the reader. Only a very modest mathematical background is needed to be able to use this book, although an elementary knowledge of quantum mechanics and bonding theory would be helpful. ... Read more

Customer Reviews (1)

Quaint and curious volume of forgotten lore
This book by George Davidson is - or perhaps it's "was" now, not having looked at another book on group theory since buying this one - an ideal way to learn how to apply group theory to spectral interpretation. The book is a model of clarity. The presentation is crystal clear: Symmetry elements and operations, groups, matrices (primarily useful for computer simulations), spectroscopy, etc. The only proviso being the final chapter on the Woodward-Hoffman rules is neither clear nor helpful. Each chapter ends with a series of questions (answers in back) of what you've just been presented. Character tables and correlation tables are included. The comparison of Raman and IR spectral data is well covered, as is the construction of molecular orbital energy level diagrams. Numerous examples include trans-butadiene, octahedral metal complexes, nitrite and carbonate ions, etc.

Cotton's book on this topic, though considered a classic, really is more of a virtuosic acrobatic performance directed to academic colleagues rather than toward assisting tyros or working chemists hoping or needing to master this recondite field of chemistry. While Group Theory for Chemists may be pejoratively referred to as "a bit rule of thumb" by old alchemists, I would recommend it not only to beginners but to all desiring a clear presentation considerate of the reader. ... Read more

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Customer Reviews (2)

Somewhat disappointed
I bought this book because I needed a modern treatise on Group Theory and because I would like to plug in some holes in my knowledge of it. Originally I gave it 4 stars rating but after reading it for a month I am changing it to 3 stars.

PROS:
(1) Very nice uniform and modern approach to group theory based on projection operators.
(3) Well structured presentation

CONS:
(1) Book is short counting only 273 pages. Applications are covered rather lightly. I wish the book was twice the size with second half devoted to gory details with applications in physics, geometry, etc.
(2) Book is terse and intense. It is assumed that a reader knows Linear Algebra very well including spectral theory. Paul Halmos "Final Dimensional Vector Spaces" should help there. I guess it is also implied that a reader has been exposed to group theory already.
(3) The book is not a text book, it is a monograph on the subject.
(4) Diagrammatic tools used, aka "birdtracks", are not very useful at least to me. I find myself constantly rewriting the birdtracks diagrams back into tensor notations. This is in contrast with Feynman diagrams that I use and love.

Overall I gave it 3 (three) stars. It definitely gave me food for thought but I learned little new from it.

A beautiful work you must have
Any practitioner of the famous Bourbaki group would have a brain implosion if they tried to read this book. Their aversion to "practical" and "visual" tools in mathematics as opposed to the rigorous formalisms for which they are justifiably famous is well documented by Benoit Mandelbrot, amongst others. They would not like this work.

This book is, first and foremost, a book about beauty with beautiful ideas and striking visuals. This is deliberately so. As the author states, "if diagrammatic notation is to succeed, it need not only be precise, but also beautiful". Of course one would expect that a book whose subject matter is primarily the many elegant symmetries of nature would end up dealing with beauty, not only at the mathematical but also at the visual level. I can testify to the success of Professor Cvitanovic's endeavors on this front. When the book arrived I showed it to my wife, who has some artistic training and a plus-critical eye for visuals, (but who certainly can't tell the difference between a casimir and a casserole). Her response said it all.

To fully appreciate this book you need postgraduate qualifications in a slew of highly technical subjects and I won't pretend that my qualifications are anywhere nearly sufficient for this. But I am mathematically literate enough and have enough formal training, (and enough interest in many of the topics covered), to recognize that this is a beautiful and brilliant work which will give me and other readers years of additional pleasure. I say "additional" because I have been tracking Professor Cvitanovic's web book, from which this print originates, for a long time now and the contents of this work are not new to me.

Take note that this is not a textbook or even a standard reference of any sort and it makes no concessions whatsoever to lack of prior knowledge. It is primarily a highly technical research monograph about problem solving. Each chapter launches headfirst into its subject matter without offering any but the most cursory introduction. The reader is expected to understand the nomenclature, the context of the material and be familiar with the mathematics and physics dealt with. But even with this, the amazing clarity of Professor Cvitanovic's writing shines through. In contrast to the dense and murky scribblings of many (I might even suggest, most) texts at this level, the writing in this book as well as the flow of logic, the layout and everything else which matters is breathtakingly simple and, may I say, beautiful. The work deserves 10 stars out of 5. (I'm still working on the mathematics of that!)

I only wish that the good professor would see his way clear to publishing in hard cover his other web book, ChaosBook.pdf, which is an equally elegant and valuable work and which aligns much better with material that I am very familiar with. Perhaps one day...

I strongly urge all interested readers to purchase this book in hard cover, even if they have the web book version. This is not only for the considerable value of the book itself, but also as a way of showing appreciation for the remarkable generosity and altruism of the author in making the fruits of years of hard labor available for free.
... Read more

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Customer Reviews (8)

Lie groups, not just for particle physics
Having read, and loved, Lie Groups for Pedestrians, I picked up this book to further my knowledge of this wonderful subject.
I am not a particle physicist nor am I mathematician, I am a spectroscopist and had read some about Lie groups and their applications to spectroscopy. However to read and digest the material that was contained in the books and articles I was coming across, it was clear that I needed to know more about Lie groups and algebras. This book was exactly what I needed. It gave very clear and concise definitions (if you have had an introduction to group theory) of what Lie groups and algebras are and the tools that are needed to use them.
The exercises at the end of the sections were a real joy for me. Working problems is the best way to learn a subject like this, and they helped to clarify what the preceding chapter had talked about. The writing is anything but dry and an easy read.
To start this book I would recommend that if you are a scientist you have taken, and understood, a good introductory course to QM and group theory; if you are a mathematician that you have taken and understood a good abstract algebra course. Do not do yourself a disservice by trying to digest this book without the proper background. You will most likely turn yourself off from a very beautiful and exciting area.
This book is not for someone who has taken an intro to physics course and wants to know about all the riddles of the universe. They will be lost, frustrated and otherwise flummoxed by this book.

Not fair on non-physicist mathematicians
Couldn't get into this, I gave up in the first chapter after failing to understand how he was applying his Kronecker product to his vectors. He just failed to explain his notation adequately. *And* there were mistakes in that first bit I read up to then.

I appreciate that physicists and mathematicians use different language, and I also appreciate that this was an advanced work, i.e. postgrad plus, but it would have been nice to have seen a glossary of terms and a little more background.

This may be a competent and erudite work, but unfortunately impenetrable without unspecified previous knowledge, and that's not the way these books ought to be.

Lie groups, examples and exercises
An excellent overview of Lie Groups and Algebras. Gilmore, as he notes himself, has concentrated on producing a self contained course for physicists. The mathematical treatment is generally detailed and shows most steps. He notes the omission of various topics in physics and mathematics, but refers the reader to specialized texts in his comprehensive bibliography. My course of Lie Groups at university was focused on mathematical applications and differential equations and this text by Gilmore provides a satisfying broader appreciation of Lie Groups and Algebras in their own right and their applications to fields and problems I wasn't previously aware of. I'm especially pleased with the many exercises which I find a great help in developing greater understanding and testing my grasp of the text.

This book becomes my reference on group theory in physics
I've waited many years to find a book like this.
It may take me many years to master everything in it,
but at least with this book I have a chance to try.
I contrast this text to books and papers by Gell-Mann, Richard Feynman,
and Steven Weinberg and these great men come off second best
when it comes to exposition of the relationships between groups.
I have found what appear to be factor of two difference
between the examples and the tables for A(n)
but those once corrected seem to leave this the complete
reference on group theory for physics that I've been looking for for a long time.
I congratulate Robert Gilmore for his well written book.

Rave Review
I haven't read this whole book cover to cover, because of time constraints.However, I can say that it is extremely clear in it's exposition.The material is very well chosen for use by physicists.I have read pure math books on this topic, and while they can be more sophisticated and thorough, they are rarely as straight forward, nor do they cover the breadth of material in this book.

In sum I would have to agree with what I was told: "this is the book on Lie Algebra for a physicist". ... Read more

Product Description
This is the second edition of the popular textbook on representation theory of finite groups. The authors have revised the text greatly and included new chapters on Characters of GL(2,q) and Permutations and Characters. The theory is developed in terms of modules, since this is appropriate for more advanced work, but considerable emphasis is placed upon constructing characters. The character tables of many groups are given, including all groups of order less than 32, and all but one of the simple groups of order less than 1000. Each chapter is accompanied by a variety of exercises, and full solutions to all the exercises are provided at the end of the book. ... Read more

Customer Reviews (3)

Excellent introductory text on representations and characters
James and Liebeck have done a wonderful job presenting the material in a concise, straightforward, easily handled fashion.The book is well organized, the exercises vary from basic to difficult, and the solutions are provided in the back of the book so that you don't have to bang your head on the wall for too long because of the tough problems!My only complaint is that their writing style is very dry and that this makes the material seem somehow "lifeless".This is true of most math books anyway, but it is possible for math texts to be exciting literature even if you aren't head-over-heels in love with the topic you want to read up on (Simmons' Introduction to Modern Analysis is a good example).

A Nice Introduction to Representation Theory
Not having a formal background in
pure mathematics, I approached the
subject of the representation theory
of finite groups with some trepidation.
Having looked at various books in the
field, I found that the book by
James and Liebeck was the clearest
and most readable exposition of
the subject.

There is little fuss or abstract
formalism that might obscure the
meaning of the fundamental concepts
and theorems.The material is clearly
written and very well organized.
The chapters are very short, thankfully,
and the best thing is that there are
complete worked solutions to all the
chapter exercises.

The book ends with a nice application
of the theory to molecular vibration.

(As always, it helps to know the basic
facts about groups, and linear algebra -
vector spaces, linear transformations,
matrices etc.)

An excellent book!

-Very- highly recommended
Representations and characters of finite groups are a classical subject, with substantial developments well underway a century ago.As a result there has been gradual refinement and polishing of the field. The beauty of this subject is shown well in this book.The style is clear and order of exposition excellent.Moreover, the authors write with great patience, both in respect to presenting calculations in detail and in providing the steps to proofs.This is not to say the book is easy, but rather that the distance from step to step is very well suited for advanced undergraduates and beginning graduate students. The book is highly mathematical in its outlook, except for the final chapter, which deals with normal modes of vibration of a molecule.There are some real gems.In a certain finite group, is it possible to find an element of order two and one of order four whose product has order three?The answer may be found in the chapter, "An Application to Group Theory."The presentation of the Frobenius Reciprocity Theorem is a exceptional delight.As for prerequisites, a good grounding in general group theory at an undergraduate level and a reasonably good acquaintance with rings and modules should suffice. ... Read more

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Techniques of physics find wide application in biology, medicine, engineering and technology generally.This series is devoted to techniques which have found and are finding application.The aim is to clarify the principles of each technique, to emphasize and illustrate the applications and to draw attention to new fields of possible employment. ... Read more

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The third edition of this text is expanded and embellished by the addition of chapters by noted group experts. It is logically organized into chapters that present the merits, rationale, dynamics, process and developmental tasks of group counseling. It discusses leader and member dynamics in depth, provides technical guidance for organizing and running groups, and gives special emphasis to the use of structured activities in groups and the relevance of family theory as a resource to group leaders.

Readers will not only learn about groups but will learn how to lead groups. The group process model presented is explained clearly using language and diagrams that are easy to follow. The activities at the end of each chapter provide an experiential extension to the content so that readers can internalize and apply concepts. The book is intended to be a hands-on tool that will give credence to groups as a helping process in which clients learn to solve personal and interpersonal problems, learn and grow in personally meaningful ways. ... Read more

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The Representation Theory of the Symmetric Group provides an account of both the ordinary and modular representation theory of the symmetric groups. The range of applications of this theory is vast, varying from theoretical physics, through combinatories to the study of polynomial identity algebras; and new uses are still being found. ... Read more

Product Description
This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this idea to its fullest extent, Chevalley incorporated a broad range of topics, such as the covering spaces of topological spaces, analytic manifolds, integration of complete systems of differential equations on a manifold, and the calculus of exterior differential forms.

The book opens with a short description of the classical groups: unitary groups, orthogonal groups, symplectic groups, etc. These special groups are then used to illustrate the general properties of Lie groups, which are considered later. The general notion of a Lie group is defined and correlated with the algebraic notion of a Lie algebra; the subgroups, factor groups, and homomorphisms of Lie groups are studied by making use of the Lie algebra. The last chapter is concerned with the theory of compact groups, culminating in Peter-Weyl's theorem on the existence of representations. Given a compact group, it is shown how one can construct algebraically the corresponding Lie group with complex parameters which appears in the form of a certain algebraic variety (associated algebraic group). This construction is intimately related to the proof of the generalization given by Tannaka of Pontrjagin's duality theorem for Abelian groups.

The continued importance of Lie groups in mathematics and theoretical physics make this an indispensable volume for researchers in both fields. ... Read more

Customer Reviews (1)

A landmark, but a bit dated.
This is really a landmark in textbooks on Lie groups, and once was "the" book to be read together with Pontryagin's "Topological Groups". Unfortunately, the writing style has become a bit "dated" (this was written in the early forties!). Of course, that doesn't mean Chevalley's book has become useless, but for the first reading, I would recommend Adams' book (Benjamin / Addison Wesley?) instead. ... Read more

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This book is a study of how a particular vision of the unity of mathematics, often called geometric function theory, was created in the 19th century. The central focus is on the convergence of three mathematical topics: the hypergeometric and related linear differential equations, group theory, and on-Euclidean geometry.The text for this second edition has been greatly expanded and revised, and the existing appendices enriched with historical accounts of the Riemann–Hilbert problem, the uniformization theorem, Picard–Vessiot theory, and the hypergeometric equation in higher dimensions. The exercises have been retained, making it possible to use the book as a companion to mathematics courses at the graduate level."If you want to know what mathematicians like Gauss, Euler and Dirichlet were doing...this book could be for you. It fills in many historical gaps, in a story which is largely unknown...This book is the result of work done by a serious historian of mathematics...If you are intrigued by such topics studied years ago but now largely forgotten...then read this book."--The Mathematical Gazette (on the second edition)"One among the most interesting books on the history of mathematics... Very stimulating reading for both historians of modern mathematics and mathematicians as well."--Mathematical Reviews (on the first edition)"The book contains an amazing wealth of material relating to the algebra, geometry, and analysis of the nineteenth century.... Written with accurate historical perspective and clear exposition, this book is truly hard to put down."--Zentralblatt fur Mathematik (review of 1st edition) ... Read more

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A deep and important heritage
The story begins with the hypergeometric series, studied by Euler and Gauss. This is a power series in x depending on three parameters. It is deeply rooted in classical analysis and it solves a linear differential equation, the hypergeometric equation. Kummer pushed the classical approach to its end by finding the 24 explicit solutions to this equation. These solutions are intricately related to each other; and one solution is defined here, another there, and so on. In short, the situation is clamouring for a Riemann to explain that all of this makes perfect sense complexly in terms of analytic continuation and monodromy relations. This is the way to go. Fuchs developed a general theory of linear differential equations along these lines. Then it's back to the hypergeometric series for more inspiration. For which parameter values is the hypergeometric series an algebraic function? Schwarz discovered that this condition on the three parameters may be expressed as that they correspond to a triangular tessellation. What is this clamouring for if not group theory? Well, that's easy for us to say. Actually, generalising Schwarz's results became a battle between the old and the new. Fuchs and Gordan went at it with invariant theory, but Klein carried the day with group theory and geometry. And the victorious march of these ideas was only just beginning. Dedekind and Klein used them to transform the theory of elliptic modular functions, which old fossils like Fuchs and Hermite had only been able to approach via elliptic functions. Indeed, the basic idea, that of periodicity with respect to a group, "was to prove to be the way historically towards the 'right' generalization of elliptic functions", namely automorphic functions. This is the culmination of the book, and here the story is told with more zeal, through correspondence highlights and so on. Poincare's interest in differential equations lead him to Fuchs's work. Despite "ignorance, even quite astounding ignorance", of much of the above literature, he still immediately discovered the connection with hyperbolic geometry (while boarding a bus, no less). This naturally caught the eye of Klein, who, being "deliberately well-read", felt that he had to inform Poincare about these works and his own perspective "that the task of modern analysis was to find all functions invariant under linear transformations". The famous competition that followed was really "more of a cooperative effort". Eventually Poincare's papers concluded this whole remarkable development, through which solid problems of classical analysis prompted a beautiful theory of complex functions deeply unified with group theory and geometry. ... Read more

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This book gives the complete theory of the irreducible representations of the crystallographic point groups and space groups. This is important in the quantum-mechanical study of a particle or quasi-particle in a molecule or crystalline solid because the eigenvalues and eigenfunctions of a system belong to the irreducible representations of the group of symmetry operations of that system. The theory is applied to give complete tables of these representations for all the 32 point groups and 230 space groups, including the double-valued representations. For the space groups, the group of the symmetry operations of the k vector and its irreducible representations are given for all the special points of symmetry, lines of symmetry and planes of symmetry in the Brillouin zone. Applications occur in the electronic band structure, phonon dispersion relations and selection rules for particle-quasiparticle interactions in solids. The theory is extended to the corepresentations of the Shubnikov (black and white) point groups and space groups. ... Read more