1.1. From the Publisher.

First published in 1962, this classic book remains a remarkably complete introduction to various aspects of the representation theory of finite groups. One of its main advantages is that the authors went far beyond the standard elementary representation theory, including a masterly treatment of topics such as general non-commutative algebras, Frobenius algebras, representations over non-algebraically closed fields and fields of non-zero characteristic, and integral representations. These and many other subjects are treated extremely thoroughly, starting with basic definitions and results and proceeding to many important and crucial developments. Numerous examples and exercises help the reader of this unsurpassed book to master this important area of mathematics.

1.2. From the Preface.

Representation theory is the study of concrete realisations of the axiomatic systems of abstract algebra. It originated in the study of permutation groups, and algebras of matrices. The theory of group representations was developed in an astonishingly complete and useful form by Frobenius in the last two decades of the nineteenth century. Both Frobenius and Burnside realised that group representations were sure to play an important part in the theory of abstract finite groups. The first book to give a systematic account of representation theory appeared in 1911 (Burnside, The Theory of Groups of Finite Order) and contained many results on abstract groups which were proved using group characters. Perhaps the most famous of these is Burnside's theorem that a finite group whose order has at most two distinct prime divisors must be solvable. Recently, a purely group-theoretic proof of Burnside's theorem has been obtained by Thompson. The new proof is of course important for the structure theory of groups, but it is at least as complicated as the original proof by group characters.

The second stage in the development of representation theory, initiated by E Noether in 1929, resulted in the absorption of the theory into the study of modules over rings and algebras. The representation theory of rings and algebras has led to new insights in the classical theory of semi-simple rings and to new investigations of rings with minimum condition centring around Nakayama's theory of Frobenius algebras and quasi-Frobenius rings.

Another major development in representation theory is R Brauer's work on modular representations of finite groups. Like the original work of Frobenius, Brauer's theory has many significant applications to the theory of finite groups. At the same time it draws on the representation theory of algebras and suggests new problems on modules and rings with minimum condition. It also emphasises the fundamental importance of number-theoretical questions in group theory and representation theory.

During the past decade there has been increased emphasis on integral representations of groups and rings, motivated to some extent by questions arising from homological algebra. This theory of integral representations has been a fruitful source of problems and conjectures both in homological algebra and in the arithmetic of non-commutative rings.

The purpose of this book is to give, in as self-contained a manner as possible, an up-to-date account of the representation theory of finite groups and associative rings and algebras. This book is not intended to be encyclopaedic in nature, nor is it a historical listing of the entire theory. We have instead concentrated on what seem to us to be the most important and fruitful results and have included as much preliminary material as necessary for their proofs.

In addition to the classical work given in Burnside's book, we have paid particular attention to the theory of induced characters and induced representations, quasi-Frobenius rings and Frobenius algebras, integral representations, and the theory of modular representations. Much of this material has heretofore been available only in research articles. We have concentrated here on general methods and have built the theory solidly on the study of modules over rings with minimum condition. Enough examples and problems have been included, however, to help the research worker who needs to compute explicit representations for particular groups. We have included some applications of group representations to the structure theory of finite groups, but a definitive account of these applications lies outside the scope of this book. In Section 92 we have given a survey of the present literature dealing with these applications and have included in this book all the representation-theoretic prerequisites needed for reading this literature, though not all the purely group-theoretic background which might be necessary.

No attempt has been made to orient the reader toward physical applications. For these we may refer the reader to recent books and articles dealing with that part of group theory relevant to physics, and in particular to Wigner, Gelfand-Sapiro, Lomont, and Boerner.

It has also been necessary to omit the vast literature on representations of the symmetric group. Fortunately the reader is now able to consult the excellent book on this topic by Robinson.

Many of the results on group representations have been generalised to infinite groups and also to infinite-dimensional representations of topological groups. We have felt that these generalisations do not properly fall within the scope of this book and, in fact, would require a lengthy separate presentation.

This book has been written in the form of a textbook; a preliminary version has been used in several courses. We have assumed that the reader is familiar with the following topics, which are usually treated in a "standard" first-year graduate course in algebra: elementary group theory, commutative rings, elementary number theory, rudiments of Galois theory, vector spaces, and linear transformations. We are confident that the expert as well as the student will find something of interest in this book. We offer no apology, however, for writing to be understood by a reader unfamiliar with the subject. In keeping with this objective, we have not always presented results in their greatest generality, and we have included details which will sometimes seem tedious to the experienced reader. After serious deliberation, we decided not to introduce the full machinery of homological algebra. Although it would have simplified several sections of the book, we felt that many readers were not likely to be well-grounded in homological algebra, and this book was not intended to be a first course in the subject.

1.3. Contents.

I. Background from Group Theory,
II. Representations and Modules,
III. Algebraic Number Theory,
IV. Semi-simple Rings and Group Algebras,
V. Group Characters,
VI. Induced Characters,
VII. Induced Representations,
VIII. Non-semi-simple Rings,
IX. Frobenius Algebras,
X. Splitting Fields and Separable Algebras,
XI. Integral Representations,
XII. Modular Representations.

1.4. Review by: W E Jenner.
Mathematical Reviews MR0144979 (26 #2519).

The appearance of this masterful book is very timely, especially in view of the recent renewal of activity in the representation theory of finite groups. It provides a remarkably complete introduction to the subject and contains for the first time a large amount of material that has been available hitherto only in the original memoirs. It undoubtedly will be the canonical reference work on its subject for a long time, and should do much to revive interest in non-commutative algebras. Contents: (I) Background from group theory; (II) Representations and modules; (III) Algebraic number theory; (IV) Semisimple rings and group algebras; (V) Group characters; (VI) Induced characters; (VII) Induced representations; (VIII) Non-semisimple rings; (IX) Frobenius algebras; (X) Splitting fields and separable algebras; (XI) Integral representations; (XII) Modular representations. The authors have made a conscientious and successful effort to be clear even to beginners. (For instance, some of the hazards of tensor products are carefully set forth for the benefit of neophytes.) There are many exercises which it would be a mistake to ignore even on first reading: they contain important things. Many examples are given in detail except in cases where, of necessity, they would be of unwieldy length. As for matters of taste, the approach is highly conceptual - via module theory - although the authors exhibit no doctrinaire revulsion to the use of matrices when it is appropriate. In conformity with the Bourbaki canon, operators are written on the left. The Wedderburn theorems are cleanly done. {The reviewer was amused by the proof that the full matric ring over a division ring is simple; the elementary computational proof is avoided.} The Wedderburn principal theorem is given, along with Malcev's refinement on the equivalence of the semi-simple components under automorphisms induced by the radical. The question of existence of unit elements is put into proper perspective; this is not really a serious problem for representative theory until topological groups are encountered. The coherent account of induced representations and Frobenius algebras is a good job of necessary organisation which should prove valuable. The chapter on integral representations is especially appealing; it contains Maranda's fundamental results, as well as others more recent. (The requisite theory of modules over Dedekind rings of Chevalley and Kaplansky which is necessary here, is given in Chapter III.) This, hopefully, should do much to stimulate further interest in this subject which is here expounded with the highest authority. There is an extensive bibliography which is valuable on its own merits. In brief, the authors have written a book worthy of the subject and have rendered an important service to the mathematical community.

1.5. Review by: G de B Robinson.
Canadian Mathematical Bulletin 7 (1) (1964), 161-164.

An examination of the above Table of Contents makes it clear that to write an adequate review of this comprehensive volume is a formidable task. Though the authors do not claim to be encyclopaedic, they have succeeded in correlating two large and important fields of modern mathematics, showing how many ideas which arise in the study of the representation theory of finite groups have a more general interpretation for associative algebras. One thing which strikes one immediately and with increasing force as the development proceeds is the correctness of the initial approach of Frobenius, Burnside, Schur and Dickson, since few of the ideas which have been introduced in the last fifty years are fundamentally new. Though the notion of a finite group is almost trivial it has proved to be so important in many different contexts that an enormous amount of time and energy has been devoted to exploring its representation theory and other properties. Foremost in these explorations of recent years has been Richard Brauer, a pupil of Schur, to whom many of the deep results recorded here are due.

How should a review of a book like this be written? In order to break down the problem let us divide the chapters as follows I-II, III, IV-V, VI-VII, VIII-IX, X-XII. Clearly chapter III is in a separate category, it could almost have been an Appendix but where to place it is unimportant; what is important is that the authors have devoted much pains to make it adequate for their needs and modern in treatment. It is not easy reading without some background in the classical theory of algebraic numbers, but its study is rewarding and many of the results obtained are fundamental in subsequent applications.

Chapters I, II, IV and V cover the representation theory of Frobenius, Burnside and Schur set in the algebraic mould of Emmy Noether. The following sentence from the introduction to chapter IV describes what the authors are trying to do. "Some of the main results in this theory depend not so much on special properties of group algebras as on properties which group algebras have because they belong to the large class of rings with minimum condition. " Put in this way, the development cannot become a race to reach the famous formulae of character theory before the reader loses interest, - it must rather be an elaboration, and deepening of Schur's 'Neue Begründung' of 1905. For this the reader should be grateful since here are the rolling hills and wide flowing streams which lead us to the crags and precipices to be encountered later.

In § 28 the authors give a brief account of Young's approach to the group algebra and the representation theory of the symmetric group Sn. It remains a tantalising fact that in this case the group algebra yields the actual matrices of the irreducible representations with all their interest and significance, while character theory plays a secondary role. Clearly, there is still much to be learned and the fact that these ideas have such wide application in physics and chemistry acts as a spur to the mathematicians to continue their study, since the rainbow may lead to the pot of gold 'the other side of the mountain'.

Chapters VI and VII describe the upward trail, first marked by Frobenius' Reciprocity Theorem, and recently followed by Brauer, Mackey, Clifford and Shoda. The notion of an induced representation ties together two fundamental aspects of group theory, namely that of a linear representation of a subgroup $H$ of $G$ and the abstract relationship between $H$ and $G$ described by the permutation representation of $G$ induced by $H$. That every finite group is isomorphic to a subgroup of $S_{n}$ has been known for a long time, but it has not contributed greatly to our knowledge of group theory up to the present. Is it a rash conjecture that this relationship, properly expressed, is the rainbow which could lead to the pot of gold?

Chapters VIII-IX develop the properties of semi-simple rings and algebras. Much of this work is due to Nakayama and the levels into which it naturally falls can best be indicated by the chain of inclusion relations (p. 440). over a field $K$. If the characteristic $p$ of $K$ divides the order of the group $G$ then the group algebra $KG$ is no longer semi-simple and we are involved in the modular representation theory of $G$ treated in chapter XII.

{group algebras} ⊆ {symmetric algebras} ⊆ {Frobenius algebras} ⊆ {quasi Frobenius algebras}

In the last three chapters of the book we are in the midst of the mountains, trying to follow the streams to their sources whose directions were so clear on the plains. In chapter X the authors give a lucid account of the effect on reducibility and decomposability of extending the field K. This aspect of representation theory goes back to Schur but the significance of the 'Schur index' (\70) has only been appreciated lately, largely as a result of Brauer's work. Though the ideas are difficult they hold promise for the future.

Chapter XI on integral (Z-) representations is the longest in the book (75 pages). Perhaps this is not surprising since every effort has been made to explain the difficulties inherent in a study of the group ring. To clarify ideas the authors give some simple examples which show how $\mathbb{Z}$ equivalence differs from $\mathbb{Q}$- equivalence and in § 74 discuss the $\mathbb{Z}$ representation theory of a cyclic group of order $p$. The extension of these results to more general groups is one of the outstanding problems of the theory. This work goes back to Zassenhaus and Maranda, while many recent results are due to Reiner and Swan.

The last chapter on modular representations tells the story of the work of Brauer and his students carried out in Toronto, Michigan and Harvard over the past 30 years. It is an exciting story for those who have been connected with it. The road has become increasingly difficult, but analogy has played a strong role and character theory, appropriately modified and extended to include the Cartan invariants, leads to a very beautiful system of formulae. As in the case of the ordinary theory, it is possible to use the modular theory to deduce deep properties of abstract groups as Feit and Thompson have recently shown.

And what of the 'pot of gold'? Will we ever really understand the implications of the simple axioms which define a group? Are the difficulties inherently similar to those which have made the theory of numbers so fascinating and so obscure? Whatever the future holds in store, we are deeply indebted to Professor Curtis and Professor Reiner for completing a tremendous task and producing a book which fills a long felt need. It is a trivial comment to say that few books are free of typographical errors, and few mathematical books are free of mathematical errors. Though some of both kinds have been detected, one may be sure that they will be corrected in future editions of this important work.

1.6. Review by: F E J Linton.
The American Mathematical Monthly 72 (2) (1965), 223.

The most succinct way to describe this exhaustive text is to gather its chapters into four clusters. The first cluster of four chapters, about which more is written below, consists of preliminary material prerequisite to the remainder. There follow three chapters dealing with characters and representations of finite groups in the semisimple situation, where the order and characteristic are relatively prime. The next three chapters develop the analogue of the Wedderburn decomposition theory for rings having minimum condition but not semisimple, and its applications to group representations in the non-semisimple case. The final pair of chapters deals with integral and modular representations.

The space available here is inadequate to do justice to the tremendous scope of this book. The interested reader is encouraged to consult the finely detailed table of contents of the book itself, and perhaps also page vii of its preface. It should be said, however, that so called homological techniques, other than use of the notions of projective and injective modules, play a very minor role. It should also be remarked that, with the addition of some material on Galois theory, the first four chapters could well serve as the basis for a good upper-undergraduate first course in algebra.

To be more explicit, these initial chapters present brief, well-written, reasonably self-contained expositions of groups (including solvable and nilpotent groups, and some extension theory), their representations as groups of linear transformations (with heavy emphasis on the interpretation of a representation as a module over the group ring), algebraic number theory (primes and valuations in algebraic number fields, etc.), and the semisimple decomposition theory of Wedderburn. The manner in which the latter material is presented is, of course, somewhat influenced by the needs of the representation theory, so that the definition of the radical, for example, is not the commonly used one; but the immediate applications to the irreducible representations of the symmetric group more than justify such deviations as occur.

As a whole, the book incorporates much recent research work (the solvability of groups of odd order is, unfortunately, too recent), calculates many examples, both easy and intricate, and suggests, here and there, lines of inquiry that look promising for fruitful investigation. These features make it well suited as a text for the serious graduate student and as a reference work for the professional group representation theorist.

1.7. Review by: Olga T Todd.
American Scientist 51 (4) (1963), 425A-426A.

Although the authors claim that this book is not an encyclopaedia on the subject of its title it will be used by many as if it were. For the book contains an enormous amount of information. However, the book is more than an encyclopaedia, it contains new ideas and results. It includes three chapters of background work on groups, modules and algebraic number theory which can be used as a text in modern algebra by themselves. The book is less concerned with the historic development of the subject, then with giving, as fully as possible, an account of the present day state of knowledge and interest.

Although abstract group theory is in itself a fascinating subject, like all abstract theories it arose from something not abstract and finds one of its ultimate goals in the characterisation of groups by objects formed from complex numbers. However, with group theory there seems to be more involved. Many facts in abstract group theory are not only easier understood via representation theory, but some are not so far accessible without it.

A great deal of work has gone previously into finding the representations of special groups. It is not the primary aim of the book to report on this. The book aims mainly at the study of the general methods. Modern representation theory studies finite groups from the point of view of representations of associative algebras.

One of the most important questions is to find in what rings the representations can be written. Great progress has been made here in recent years. Representations by rational integers are of special interest and the book devotes much space to it.

The authors were fortunate in the fact that the book was not yet finished when the news about Feit and Thompson's result concerning simple groups of odd order came along. They include a review of the work of other authors in this area as well.

1.8. Review of the 2006 Reprint by: Fernando Q Gouvêa.
Mathematical Association of America (29 April 2006).
https://old.maa.org/press/maa-reviews/representation-theory-of-finite-groups-and-associative-algebras

I guess I'm officially an old fogey now that a book that was recommended to me in graduate school as a standard reference has been republished by AMS/Chelsea. After all, this is the publisher (now taken over by AMS) who made its name by publishing old classics in (fairly) inexpensive hardcover editions. Curtis and Reiner's famous account of representation theory certainly deserves to be considered a classic, and its return should be greeted with delight by everyone.

As I was working on this review, my group theorist colleague Tom Berger saw it on my desk, and said "That's the book! That's the book I memorised in graduate school." I think a lot of people will react in the same way. For a very long time, "Curtis and Reiner" was where you went to learn this subject. It was well written and contained almost all that was important to know about the basics of representation theory.

It's not a perfect book. The section on modular representation theory at the end is too short, for example, and very classical, very close to Brauer's original work. There are no infinite groups at all, so that those whose interest in representation theory comes from physics or from the Langlands program will have to go elsewhere.

The authors themselves recognised some of these problems, and went on to write a massive second book, Methods of Representation Theory: With Applications to Finite Groups and Orders, which filled two volumes and never quite caught on like their first book. It was last reprinted in the "Wiley Classics Library", but seems now to be out of print.

Nevertheless, while today's students may well want to start with, say, Fulton and Harris's Representation Theory : A First Course, they will still find much useful material in this book. I'm certainly glad to have it back.

2.1. From the Preface.

This book gives a quick introduction to the basic ideas and calculations of matrix theory. It may be used as supplementary material in the second or third semester of a standard calculus course, and can be covered in 15 to 20 class hours. Optional topics and optional proofs may be omitted if desired without affecting the treatment of most of the material following such topics. Optional material will be enclosed in brackets *[...]* throughout the book.

Rather than stressing proofs of theorems, I have tried to emphasise basic concepts and manipulative skills. Proofs are often given in special cases, working sometimes in two or three-dimensional space for simplicity. For more detailed accounts of matrix theory and linear algebra, the reader may consult Linear Algebra by C W Curtis, Elementary Matrix Algebra by F E Hohn, or Theory of Matrices by S Perlis. These are three of many texts on these subjects.

The student should be slightly familiar with determinants, although all of the necessary results on determinants are reviewed in Section 5. Hopefully the student knows what is meant by the sum of two vectors and by a scalar multiple of a vector. These ideas are convenient for motivating certain matrix definitions and are reviewed briefly when first introduced. Some results from calculus are needed in Section 8, and partial derivatives occur in Section 15. It would be possible to use this book in a precalculus course by omitting part of Section 8 and all of Section 15.

The numbers and scalar quantities which occur in the text are usually assumed to be real numbers, although almost all of the results obtained are equally valid when complex numbers are used. In Sections 11 to 15, the student's attention is restricted to vectors and matrices whose entries are real numbers. Such additional restrictions are always stated explicitly when they are needed.

Starred problems are more difficult than average. Solutions of some of the exercises are given at the end of the book.

2.2 Review by: F M Hall.
The Mathematical Gazette 57 (401) (1973), 242.

This book is written at an essentially elementary level, and emphasises basic concepts and manipulative skills rather than stressing proofs of theorems. It deals with the basic manipulation of matrices and determinants, and follows this with a brief discussion of vector spaces and linear transformations, defining a vector rather naively as a matrix with a single row or column and a scalar as a 'number'. The book finishes with chapters on characteristic vectors, orthogonal vectors and matrices, symmetric matrices and Jacobians. Not only are few proofs of theorems given, but there is practically no motivation: definitions are stated without any introductory preamble or justification, and there are very few applications mentioned of the work that is done. As a reference book of techniques for a beginner in the subject the book would have some use, but would give very little understanding.

2.3. Review by: Irving Allen Dodes.
The Mathematics Teacher 65 (1) (1972), 52.

A little book designed as a supplement to a regular elementary calculus course, requiring 15 to 20 hours according to the author. The standard topics of linear algebra are included, except that the author stresses concepts and skills rather than proofs; where proofs are omitted, references are given to the sources where proofs may be found. There are many worked examples and exercises, some with answers. Nicely written.

2.4. Review by: Editors.
Mathematical Reviews MR0404282 (53 #8084).

This is a concise introduction to elementary matrix theory, including systems of linear equations, eigenvalue problems and canonical forms but excluding the Jordan form. Vector spaces are discussed and there is some geometrical material in the sections on orthogonal and symmetric matrices and principal axes but the emphasis is on matrices and matrix calculations. There are many exercises, some of them rather challenging and many of them with solutions.

2.5. Review by: D E Christie.
The American Mathematical Monthly 80 (6) (1973), 702-704.

Reiner's compact treatment is designed for a fifteen to twenty-hour supplement to a calculus sequence. It is divided into fifteen sections ranging in length (aside from exercises) from one page ("Other Notations for Matrices") to seventeen pages ("Symmetric Matrices and Principal Axes"). In the first forty-five pages (comprising eight sections), the author introduces matrix operations, shows two ways of computing matrix inverses and two or three ways of solving sets of linear equations, and discusses geometrically linear transformations in two dimensions (including the idea of area magnification). This initial presentation is concise, but remarkably clear. Specific examples motivate broader conclusions, computational procedures (e.g., evaluation of determinants, calculations of matrix inverses) are given step by step labelling, geometric and algebraic considerations reinforce each other.

The next half dozen sections of Reiner's book deal with structure: vector spaces (based on objects in an n-dimensional space), linear transformations (defined in terms of matrices), characteristic vectors, orthogonality, and principal axes. Even when theoretical conclusions are drawn from rather restricted cases, the argument is hand-led so aptly that an average student will be convinced while a strong student will sense the broader scheme of things: "The same argument works in more general circumstances..." Some students (and perhaps more instructors) may feel uneasy because some crucial definitions change as the course develops. The rank of a matrix is first defined in terms of a set of linear equations as the number of variables remaining after an echelon elimination procedure; later it is the dimension of a row space. A general definition of a vector space is given only in an optional section on applications to calculus (where the previous restriction to n dimensions is dropped). Similarly a general definition of a linear transformation is reserved for an optional section where spaces are infinite dimensional. New definitions are properly keyed to old ones, so no permanent injury is sustained.

One might expect a short treatment emphasising manipulations to be quite superficial. Reiner avoids this danger brilliantly. Key concepts repeatedly come alive in carefully selected examples. There is much to be said for a presentation of theory which focuses on phenomena rather than deduction. While proofs are often omitted, the deductive flavour is by no means lost; in fact skill and good taste in proofs is a feature of the book.

On the whole the author treats his readers very well. The theory is always motivated and modelled by elementary examples. There are numerous applications to geometry and analysis. Section 15, devoted to Jacobians, picks up again the area-magnification theme and also discusses chain rules and the implicit-function theorem.

Other applications to analysis include some attention to the vector space of all real-valued functions as well as to important subspaces. Beginners will be pleased to discover that they already can characterise the kernel of the differentiation operator for differentiable functions over [0,1]. There are no explicit applications outside of mathematics, but the i j k treatment of vector geometry, the work on principal axes, and the section on Jacobians come very close to, say, mechanics, elasticity, and thermodynamics. Most sections have a good set of exercises of varying types, theoretical as well as manipulative. About half of the 170 exercises have solutions, often in considerable detail. The index is good but could easily be improved. The reviewer noted only a few minor misprints in this carefully produced book.

3.1. From the Publisher of 2002 reprint.

This is a reissue of a classic text, which includes the author's own corrections and provides a very accessible, self contained introduction to the classical theory of orders and maximal orders over a Dedekind ring. It starts with a long chapter that provides the algebraic prerequisites for this theory, covering basic material on Dedekind domains, localisations and completions, as well as semisimple rings and separable algebras. This is followed by an introduction to the basic tools in studying orders, such as reduced norms and traces, discriminants, and localisation of orders. The theory of maximal orders is then developed in the local case, first in a complete setting, and then over any discrete valuation ring. This paves the way to a chapter on the ideal theory in global maximal orders, with detailed expositions on ideal classes, the Jordan-Zassenhaus Theorem, and genera. This is followed by a chapter on Brauer groups and crossed product algebras, where Hasse's theory of cyclic algebras over local fields is presented in a clear and self-contained fashion.

Assuming a couple of facts from class field theory, the book goes on to present the theory of simple algebras over global fields, covering in particular Eichler's Theorem on the ideal classes in a maximal order, as well as various results on the $KO$ group and Picard group of orders. The rest of the book is devoted to a discussion of non-maximal orders, with particular emphasis on hereditary orders and group rings.

The ideas collected in this book have found important applications in the smooth representation theory of reductive $p$-adic groups. This text provides a useful introduction to this wide range of topics. It is written at a level suitable for beginning postgraduate students, is highly suited to class teaching and provides a wealth of exercises.

3.2. From the Foreword by M J Taylor of 2002 reprint.

Irv Reiner saw the theory of maximal orders as 'non-commutative arithmetic'. His initial motivation came principally from arithmetic applications, and in particular the study of rings of integers as modules over Galois groups. The book still serves as an excellent introduction to the basic methods and theory in this area.

Many mathematics books have an initial impact, and then their influence wanes, as they are left behind by the tides of mathematical development. To my mind the exact opposite is the case with this book. The importance of many of the themes and techniques dealt with in this book has grown with the passage of time: maximal orders are an important tool in a number of currently very active areas, including non-commutative geometry and non-abelian Iwasawa theory. When the book first appeared it was clear that the treatment of local and global division algebras was one of its finest features; such arithmetic division algebras now enjoy a central role in Langlands theory.

Tsit-Yuen Lam pointed out to me that, when the book first appeared in 1975, the review by H Jacobinski concluded with the remark: 'The book fills a gap in the mathematical literature, since no book on maximal orders has been available. The author has succeeded very well in giving a clear and easily accessible presentation of the subject.' This remark is as true today as it was then, and Maximal Orders remains the reference of choice.

Irv was a real mathematical scholar: he was both a fine researcher and a talented writer of mathematics. He was a craftsman who worked tirelessly to present mathematics in a clear and well-motivated manner. Maximal Orders is one of his finest pieces of work.

3.3. From the Preface.

The theory of maximal orders originates in the work of Dedekind, who studied the factorisation properties of ideals of $R$, where $R$ is a ring of algebraic integers in an algebraic number field $K$. As shown by Dedekind, the factorisation theory is especially simple in the extreme case where $R$ is the ring of all algebraic integers in $K$. This ring is in fact the unique maximal $\mathbb{Z}$-order in $K$.

In this book we shall consider the generalisation of Dedekind's ideal theory to the case of maximal $R$-orders in separable $K$-algebras. The entire theory reduces almost immediately to the case of a maximal $R$-order $\Lambda$ in a central simple $K$-algebra $A$. It turns out that there are two ideal theories, one concerned with two-sided ideals, the other with one-sided ideals. The two-sided theory is easier, since the group of two-sided fractional $\Lambda$-ideals in $A$ is the free abelian group generated by the prime ideals of $\Lambda$. The difficulty in the one-sided theory is that a left $\Lambda$-ideal in $A$ is also a right $\Lambda '$-ideal, where $\Lambda '$ is some other maximal order in $A$. It is therefore necessary to consider the set of normal ideals in $A$, namely all one-sided ideals relative to the various possible maximal orders in $A$. These normal ideals need not form a group, and instead constitute the Brandt groupoid of $A$.

The deeper aspects of the theory of maximal orders depend on properties of central simple algebras over local and global fields. Many of these deeper results, such as splitting theorems, the theory of the different and discriminant, reduced norms, and so on, become almost trivial in the commutative case.

The theory of maximal orders is of interest in its own right, and is essentially the study of "noncommutative arithmetic". The beauty of the subject stems from the fascinating interplay between the arithmetical properties of orders, and the algebraic properties of the algebras containing them. Apart from aesthetic considerations, however, this theory provides an excellent introduction to the general theory of orders (maximal or not). Questions involving integral representations of groups, and those concerned with matrices with integer entries, often reduce to the study of non-maximal orders. Many problems can be handled by embedding such orders in maximal orders, and then using known facts on maximal orders.

One aim of this book is to present, in as self-contained a fashion as practicable, most of the basic algebraic techniques needed for the study of orders, maximal or not. The subject matter is arranged in the form of a textbook, on the level of a second-year graduate course. The exercises at the end of each section are an integral part of the book. In many cases, the results of the exercises will be needed later in the book, and occasionally are needed within the section itself. For this reason, detailed hints are given for many exercises.

The reader is expected to be familiar with basic facts from module theory and algebraic number theory, such as those given in the introductory chapters of Curtis-Reiner [Representation theory of finite groups and associative algebras]. We have included, for the convenience of the reader, a lengthy chapter on algebraic preliminaries. This chapter provides brief surveys of topics needed later in the book, and may be skimmed quickly in a first reading. Proofs are often omitted in sections 1-5, especially when long or computational. In the rest of the book, from section 6 on, proofs are given in detail. The only exceptions are the Hasse Norm Theorem and the Grunwald-Wang Theorem; these are stated without proof, since otherwise several additional chapters would become necessary.

This book is intended as an introduction to maximal orders, and no attempt has been made to compile an encyclopaedic treatise, or to provide the historical background of each result. Our approach draws heavily from that in Deuring, and benefits from the numerous simplifications in Swan-Evans. We have not covered any of the analytic theory, which is readily available in the excellent book by Weil. We have also omitted the theory of Asano orders, which would require a book of its own. In many ways, the theory of orders merges into the vast topic of algebras over commutative rings. Among the many references on this subject, we may mention the fine books by Bass, DeMeyer-Ingraham, Kaplansky, and Matsumura.

Sections are numbered consecutively throughout the book. A list of permanent notation precedes Chapter 1. Boldface "Theorem" indicates that the theorem is one of the major results proved in the book. We have distributed such honours lavishly - there are about 50 such results in the book!

This book divides naturally into three parts. The first part consists of the preliminary material in Chapter 1, which may be skimmed in a first reading, together with some generalities on orders in Chapter 2. The second part, Chapters 3-6, deals mainly with the ideal theory of maximal orders. In Chapter 3 we consider such orders in skewfields, in the complete local case. The Morita correspondence, explained in Chapter 4, is used in Chapter 5 to study maximal orders in central simple algebras in the local case. The local results are then applied in Chapter 6 to obtain the global theory. Many of the techniques developed in this book are useful for the theory of non-maximal orders. Thus, for example, Chapter 6 contains a proof of the Jordan-Zassenhaus Theorem and a discussion of genus for arbitrary orders. We may also mention the proof of the Krull-Schmidt Theorem in section 6 for algebras over local rings.

The final third of the book covers the deeper theory of central simple algebras over global fields, and maximal orders in such algebras. Chapter 7 treats Brauer groups, crossed-product and cyclic algebras. The results of Chapter 7 are combined with the Hasse Norm Theorem and Grunwald-Wang Theorem in Chapter 8, to derive some of the major theorems on simple algebras over global fields. In particular, Eichler's Theorem is proved in Chapter 8, and is used to calculate the ideal class group of a maximal order.

Chapter 8 also contains an introduction to Fröhlich's theory of Picard groups of orders, and a discussion of locally free class groups of non-maximal orders. The last chapter deals with hereditary orders, which are in a sense not much harder to handle than maximal orders. Chapter 9 also includes some miscellaneous facts about group rings.

This book is based on class notes for courses given at the University of Illinois in 1969 and 1973. I would like to thank Janet Largent and Melody Armstrong for their excellent work in typing these class notes. My thanks also go to the members of the classes who helped with the proofreading, and corrected errors in the notes. I especially thank Robert L Long, who read the entire manuscript with great care and attention; he deserves credit for catching innumerable mistakes in printing and in the mathematical content. His suggestions have helped clarify the presentations. Finally I am glad to thank my wife Irma, not only for her advice on the contents and style of the book, and her help in its preparation, but also for her constant encouragement and support.

It is also a pleasure to acknowledge with thanks the financial support I received from the National Science Foundation and the Science Research Council, during part of the time when the book was being written.

3.4. Review by: H Jacobinski.
Mathematical Reviews MR0393100 (52 #13910).

This book is about the classical theory of maximal orders over a Dedekind ring in a separable algebra. The presentation and methods of proof are essentially the classical ones in a modernised version. The book has developed from a series of lectures for graduate students, and the author's intention has been to make the book - and the subject - easily accessible to a large variety of readers.

The contents are briefly as follows. The book starts with a long introductory chapter, containing prerequisites mainly from algebraic number theory, such as integral closure, localisation, Dedekind domains, etc. Then follow the basic facts about orders, in particular, the existence of maximal orders. Chapters 3-5 contain the local theory of maximal orders, the basic fact being the unicity of the maximal order in a skew-field over a complete valuation ring. Chapter 6 presents the ideal theory of global maximal orders, for which both the direct proof and the one using the local theory are given. Chapter 7 contains standard material on the Brauer group and cyclic algebras, which is needed in the next chapter, in which the ground field is a global field. Here a couple of theorems from class field theory (the norm theorem and the Grunewald-Wang existence theorem) are assumed without proof. This is used to develop the theory of simple algebras over global fields and to deduce Eichler's characterisation of the ideal classes in a maximal order. The book concludes with three shorter sections, one on Picard groups, one giving some applications to non-maximal orders and finally a detailed study of hereditary orders.

The text is very well written and complete; all proofs are given in full detail. There are well-chosen exercises at the end of each chapter.

The book certainly fills a gap in the mathematical literature, since no modern textbook on maximal orders has been available. The author has succeeded very well in giving a clear and easily accessible presentation of the subject.

4.1. From the Preface.

These notes arose from series of lectures delivered by the authors at the Fourth School of Algebra, Sao Paulo, July 12-30, 1976, organised by Professor Alfredo Jones and Professor Cesar Polcino Miliés.

4.2. Review by: Ken-Ichi.
Mathematical Reviews MR0549035 (80k:20010).

Two lectures entitled "Topics in integral representation theory" by the first author and "Integral representations and presentations of finite groups" by the second author are bound together in this book.

 I. Topics in integral representation theory: Recently the first author [Irving Reiner] determined all integral representations of a cyclic group of order $p^{2}$ for a prime $p$. The aim of this lecture is to develop more general theory in integral representations of groups.

 II. Integral representations and presentations of finite groups: K W Gruenberg studied group theory with many cohomological methods twenty years ago. Since then, integral representation theory of groups has grown. The aim of this lecture is to develop group theory with integral representation theory of groups and presentations of finite groups.

5.1. From the Preface.

In the past 20 years, representation theory of finite groups and associative algebras has enjoyed a period of vigorous development. The foundations have been strengthened and reorganised from new points of view. The applications and connections with other parts of mathematics, while already substantial, have grown in depth and variety and now include powerful results in directions only dimly perceived 20 years ago. It therefore seemed worthwhile and challenging to attempt a survey of the developments since the appearance of our first book (Curtis and Reiner, Representation theory of finite groups and associative algebras), in the hope that such an effort might encourage the continuation of work already begun, and might lead to further applications.

Representation theory is concerned with the following kind of situation. We are given an action of a finite group on some object, such as a set, a vector space or module over a commutative ring, a simplicial complex, or an algebraic variety. We then introduce a ring (usually a group algebra or a twisted group algebra over an appropriate commutative ring), and we consider modules over the ring, which capture some features of the group action. The next step is to classify the modules so obtained, in terms of the structure of the group algebra, the endomorphism algebras of the modules, the extension problem for the modules, and their character theory. The aim of these efforts is to obtain algebraic or number-theoretical information which may yield new insight about the structure of the finite groups considered, or about their actions on the objects studied initially. The most extensively developed part of the subject has been the application of representation theory and character theory to the structure of finite groups.

This area has been dominated, during the period of our survey, by the work of Richard Brauer (1901-1977). The strength and originality of his own work, combined with the friendly encouragement and inspiration he provided to younger colleagues and students, make his place in the subject a special one. He lived to see the near completion of the work that he and his predecessors Frobenius, Burnside, and Schur all viewed as the grand task to which character theory could make a central contribution, namely, the complete classification of finite simple groups. It is perhaps not too much to hope that new ideas from representation theory, combined with the extraordinary achievements of the past decade on the structure of finite groups, will lead to simplifications of proofs and a better understanding of this classification problem.

Another subject, algebraic $K$-theory, has recently approached maturity, and exerts a strong influence on integral representation theory. This area of research, guided by parallels with certain constructions in topology, has suggested new problems of major importance in representation theory. Their solution, in turn, has led to fresh applications to topology and algebraic number theory.

A second interaction between representation theory and geometry has occurred in the representation theory of finite groups of Lie type. In the examples where character tables were known, there were certain representations that were elusive and difficult to construct by standard methods. In a dramatic breakthrough, Deligne and Lusztig discovered general methods for constructing these and other representations through a systematic study of actions of the groups on algebraic varieties.

Our objective in this volume, and in Volume II, is to give an essentially self-contained account of the three main branches of representation theory: ordinary, modular, and integral representation theory. We exhibit here numerous interrelationships among these three subjects, thereby obtaining deeper understanding of each of them. Our approach is not intended to be encyclopaedic, but each topic is considered in sufficient depth that the reader may obtain a clear idea of some of the major results in the area. Our selection of topics, from what has now become a vast subject, was guided by our aim of preparing the reader for further work in representation theory and its applications as described above.

As in our first book (hereafter referred to as CR), we have concentrated on general methods. However, the present book also contains considerably sharper and more powerful computational techniques, some of which have been developed in the past few decades. The reader will also find a greater emphasis on methods from homological algebra and commutative ring theory, and somewhat less on purely ring-theoretic considerations.

We have attempted to make this book relatively self-contained, and do not presuppose familiarity with all of the contents of our previous book CR. On the other hand, we occasionally omit details of some of the more elementary results and refer instead to material in CR for background and further information. While it has not been possible to include in this work all of the results in CR, the reader will find that most of the contents of CR are treated here in greater depth and generality. Furthermore, in many areas we go far beyond the contents of CR.

In a few of the sections in this book, we have also inserted references to the book on Maximal Orders (see Reiner [75]), which we cite as MO for brevity. In particular, some of the introductory material in Chapter 3, especially in
Generally speaking, we assume that the reader has a good general background in algebra, with no more familiarity with representation theory than what is contained in the usual first year graduate course in algebra. Where this approach results in overlapping between this book and CR, we have tried always to give either new proofs or a revised presentation.
...
In most cases, we have tried to carry each topic in this volume to the point where further research and extensions of the methods can be considered.

The material in this volume has been used by us as a basis for full-year courses on ordinary and modular representation theory, and integral representation theory. We have included exercises after almost every section to increase the book's usefulness in connection with graduate courses or for self-study, and to present examples and auxiliary results that should prove useful to the researcher

The bibliography lists only the books and articles referred to in the text of this volume. References are listed according to the last two digits of the year in which they appeared. More extensive bibliographies, and surveys of current research, are available in CR and also in the books listed on the first page of the bibliography.

5.2. Review by: D G Higman.
American Scientist 71 (4) (1983), 425.

This is the first of two proposed volumes in which the authors undertake to provide a largely self-contained treatment of a broad range of topics in the theory of representations of finite groups and orders. An extensive introductory chapter offers background material from noncommutative and commutative ring theory, group theory, homological algebra, and number theory. It is followed by a chapter on ordinary representations of groups, stressing applications to group theory, particularly via character theory, and to classification of finite simple groups, and a chapter on modular theory motivated by applications to the ordinary theory. Two chapters deal with integral representations of finite groups and orders, in which the representations themselves (in the guise of lattices) take over the central role.

The book is set apart by the wide range of material covered, much of it developed in the past 20 years or so, and by the attention paid to interaction between areas.

A nice balance has been struck between breadth and depth of coverage. Already in the first volume, the reader is brought to a take-off point for further research in some areas.

To borrow an adjective from the world of computers, this is a user-friendly book. The reader can open it to a topic of interest with the expectation of being able to track down the needed notation, definitions, and results with reasonable effort. The authors have taken great care in providing motivation, clear and accurate exposition, and examples and exercises. Readers of this volume will look forward to the authors' treatment of major topics promised for the second volume, including representations of finite groups of the Lie type and algebraic K-theory.

5.3. Review by: J L Alperin.
Mathematical Reviews MR0632548 (82i:20001).

This is the first volume of a two-volume treatise on representation theory of groups and orders. After a long introduction with background material the book divides into two separate but related parts: two chapters on ordinary and modular representation theory of finite groups; two chapters on integral representations.

The introduction covers a variety of topics: homological algebra including Hom, tensor products, Ext, Tor, projective modules and covers; semisimple rings, Morita equivalence, radicals and separable algebras; Dedekind domains.

The first chapter is devoted to the basic results and applications of ordinary representation theory and character theory. There are also two sections with material on tensor algebras, Adams operations, symmetric and skew-symmetric squares of induced modules, tensor induction and transfer.

The second chapter is an introduction to modular representations and so deals with reduction modulo p, Brauer characters, and the Green correspondence. It also has a section on induction theorems generalising Brauer's theorem to more general fields.

The last two chapters cover a vast amount of results on integral representations with the third chapter devoted to orders and lattices and the fourth to local and global theories of integral representations. The last chapter concludes with sections on finite representation type, examples, invertible ideals, uniqueness of decomposition in discrete valuation rings, Bass and Gorenstein orders.

6.1. Review by: Jon F Carlson.
Bulletin of the American Mathematical Society 19 (2) (1988), 484-488.

The most unusual feature of the books by Curtis and Reiner is that they give the reader a picture of the vast array of topics that can be labelled under the heading of representation theory. The second volume contains long chapters on K-theory and on class groups for group rings and orders, as well as somewhat shorter expositions of block theory and of the representations of finite groups of Lie type. Briefer, though still lengthy, chapters discuss rationality questions such as Schur multipliers, contributions from the representation theory of finite-dimensional algebras such as quivers and Auslander-Reiten sequences, and results on Burnside rings and representation rings of finite groups. No other text comes close to covering so much ground. However, even with two volumes of almost 1800 pages the books are nowhere near complete. Some of the most exciting areas of current research, such as cohomological methods and the connections with the representation theory of algebraic groups, are given only introductions or barely mentioned. The field has become too big.

When Curtis's and Reiner's first book [Representation theory of finite groups and associative algebras] was published in 1962 it immediately became everybody's standard reference. It was by far the best and most complete of the few texts that were available. By contrast, there are many books on representation theory in print today, but they are all somewhat specialised, highlighting only one or only a few aspects of the modern theory. At the same time most of us who work in group representations are also specialised, and for us the new volumes by Curtis and Reiner will provide access to many of those topics that are outside of our areas. The text is well written and organised, and the approach is generally up to date. The books should be easily readable by students with any sort of reasonable background in group theory and abstract algebra. As introductions and references to the numerous topics that make up modern representation theory, the volumes by Curtis and Reiner are to be highly recommended.

It is unfortunate that Irving Reiner did not live to see the publication of the second volume. He died in October of 1986 after a long illness. Reiner was a helpful and personal friend to many of us and a good friend of the mathematical community in general. We will miss him.

6.2. Review by: I M Isaacs.
American Scientist 76 (6) (1988), 628.

This book constitutes volume 2 of an expanded and updated version of the now classic book by the same authors, Representation Theory of Finite Groups and Associative Algebras, published in 1962. As with the earlier work, and as is suggested by the subtitle, several threads run through this book. These are intertwined, but they are not completely inter woven. Two main themes, corresponding to the research interests of the two authors, are the representation theory of finite groups over fields and the integral representation theory of finite groups (and its generalisation, the theory of orders). While to the non-specialist these may sound similar and, indeed, they are closely related, there are many differences in the techniques used, the type of results proved, and the mathematical flavour of these subjects. The integral representation theory is largely number theoretic and homological in spirit, while the "classical" representation theory over fields is more group theoretic in its methods and results.

Volume two consists of seven chapters. The first two are concerned mainly with various groups associated with integral group rings and orders: Grothendieck groups, Whitehead groups, K-theory groups, class groups, and Picard groups. Next, there is a chapter on block theory. This subject, pioneered by Brauer, yields some of the deepest and most powerful applications of representation theory to the understanding of abstract finite groups. Unlike Brauer, whose approach was character theoretic, Curtis and Reiner use the more modern and powerful module and ring theoretic approach to develop this theory to the point where they can present many of the deep theorems and major applications. The fourth chapter treats an area of intense recent research: the representation theory of groups of Lie type (which include most finite simple groups). The remaining three chapters are shorter. They discuss a variety of topics including Schur indices, Artin exponents, Burnside rings, Auslander-Reiten almost-split exact sequences, and representations of symmetric groups.

6.3. Review by: J L Alperin.
Mathematical Reviews MR0892316 (88f:20002).

This second volume, comprising Chapters 5 through 11, concludes a massive treatise on the representation theory of finite groups and related orders and algebras. Chapter 5 is devoted to the low-dimensional algebraic $K$-groups, $K_{0},K_{1}$, and $K_{2}$, and covers Grothendieck groups, Whitehead groups and calculations for the case of group algebras. Chapter 6 is the high point of the integral representation theory developed in the first volume and deals with class groups of integral group rings and orders, and, in particular, the Swan subgroup and Picard groups. Chapter 7 is an introduction to the theory of blocks, covering the main theorems and concepts, and uses the variety of different methods now available. It concludes with a thorough beginning for the theory of blocks with cyclic defect groups, the high point of this branch of representation theory, and applications to the structure of groups. The next chapter is an introduction to the representation theory of finite groups of Lie type, almost all in characteristic zero, except for the last section where the natural characteristic is studied at least as far as giving a version of the theory of weights that is applicable there. After a study of the structure of such groups of Lie type, the chapter investigates homology representations, Hecke algebras, generic degrees and cuspidal characters. Chapter 9 looks at rationality questions including real representations and the Schur index while examining the case of the symmetric groups. The next chapter is devoted to indecomposable modules for algebras including Gabriel's theorem and the almost split sequences of Auslander and Reiten. The final chapter studies the Burnside ring and other representation rings such as the Green algebra.

6.4. Review by: Dave Benson.
Bulletin of the London Mathematical Society 20 (5) (1988), 535-540.

This is the third and last 1 volume to come from the collaborative pens of Curtis and Reiner. The first was Representation theory of finite groups and associative algebras [4], which in the twenty-five years since its first appearance has been a standard text in representation theory. Indeed, as a student learning representation theory in the seventies, this volume and Part B of Dornhoff 's book [Group representation theory (1972)] were virtually the only texts I could find of any depth. Quite recently, say over the last half dozen years, the situation has improved dramatically with the appearance of Volume I of this work in 1981, Feit's book [The representation theory of finite groups] in 1982, Landrock's book [Finite group algebras and their modules] in 1983, Carter's book [Finite groups of Lie type, conjugacy classes and complex characters] in 1985, Alperin's book [Local representation theory] in 1986, and now Volume II of this work in 1987. I find it quite surprising to what extent these volumes do not overlap in their treatment of material or point of view.

The book consists of seven chapters (numbered 5 to 11 to follow Volume I), the first four of which are quite long, averaging 180 pages each, while the last three are quite a lot shorter at about 70 pages each. They are intended to be readable independently of each other; however, any but the most knowledgeable reader will wish to have Volume I by his side for the necessary background material. One could say that each chapter of Volume II is a specialised topic resting on quite a lot of the general representation theoretic material set up in Volume I.
...
To summarise, the reader should not be put off by the extreme length of this book, because of the extent to which the different chapters are written so as to be independently readable. The writing is up to the usual high standard of these authors, and I am sure this book will be treasured both by graduate students and experienced representation theorists alike. Even in a book of this size, one can't say everything, but the choice of material is good. To a large extent this choice reflects the tastes of the authors, so that this can be regarded as a record of journeys through mathematics travelled by the authors. The main directions in current research on representation theory of finite groups are all represented here, and I think the authors are justified in the hope expressed in their introduction that this will be a useful guide, as their first volume was in its time, during a period of continuing activity in representation theory and its applications.

It is sad that Reiner did not live to see this beautiful volume in print.

7.1. From the Publisher.

The papers of Irving Reiner reproduced here represent a part of the life's work of a dedicated and influential mathematician. His early papers were concerned with groups of invertible matrices over integral domains and their automorphisms. These studies led naturally to the study of integral representations of groups and orders with attempts to classify their representations using Grothendieck rings and their relative versions, Picard groups, and class groups. His later papers dealing with zeta-functions bring quantitative methods and new ideas to the representation theory. The influence of Reiner's work is most strongly felt by research mathematicians interested in problems related to integral representation theory.

7.2. Review by: B Fein.
Mathematical Reviews MR1008246 (90g:01074).

Irving Reiner(1924-1986) played a major role in the development of integral representation theory. This volume contains photocopy reproductions of 48 of Reiner's most influential papers as well as abstracts, prepared by the editor, of 14 others. The abstracts are quite detailed and include statements of main results, comments on the relationship with other results in the literature, and ideas behind the proofs. The volume also includes the editor's moving obituary of Reiner which originally appeared in a special issue of the Illinois Journal of Mathematics, as well as complete lists of Reiner's students and Reiner's publications.