Matrix Representations of Groups

Matrix Representations of Groups

by Morris Newman

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Recognizing that the theory of group representations is fundamental to several areas of science and mathematics — including particle physics, crystallography, and group theory — the National Bureau of Standards published this basic but complete exposition of the subject in 1968 in their Applied Mathematics Series. The most significant facts about group representation are developed in an accessible manner, requiring only a familiarity with classical matrix theory. The treatment is rendered self-contained with a series of concise Appendixes that explore elements of the theory of algebraic numbers.
Subjects include representations of arbitrary groups, representations of finite groups, multiplication of representations, and bounded representations and Weyl's theorem. All of the important elementary results are featured, a number of advanced topics are discussed, and several special representations are worked out in detail. 1968 edition.

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Product Details

ISBN-13: 9780486841557
Publisher: Dover Publications
Publication date: 07/17/2019
Series: Dover Books on Mathematics
Sold by: Barnes & Noble
Format: NOOK Book
Pages: 112
File size: 13 MB
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About the Author

Morris Newman (1924–2007) received his Ph.D. from the University of Pennsylvania and was a research mathematician at the National Bureau of Standards from 1952–77. From 1977 until his 1993 retirement, he was Professor of Mathematics at the University of California, Santa Barbara, where he continued working with students for many years after his retirement. He is the author of Integral Matrices.

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In this volume the most important facts about group representations are developed, entirely along the original matrix-theoretic lines set down by Burnside, Frobenius, and Schur in their fundamental memoirs on this subject. In the writer's opinion the approach through matrices is the one most easily grasped by the beginning student and is also the one which is most easily applied to other parts of mathematics and other disciplines. The more general approach through modules is not discussed.

Very little is presupposed about groups, but the reader should certainly be familiar with classical matrix theory: a good reference source containing all necessary material is the tract by MacDuffee [24]. Another is the volume by Marcus [25] in this Series. Any nonstandard results from group theory or matrix theory are proved in the text.

An appendix on the elements of the theory of algebraic numbers has been included, so that the volume is self-contained in this respect. This appendix (which parallels the introductory material in Hecke's book [19] quite closely) serves by itself as a complete introduction to the classical theory of algebraic numbers, up to and including the unique factorization theorem for ideals. There is also an appendix on the roots of unity, containing an elementary proof of the irreducibility of the cyclotomic polynomial.

The entire development has arbitrarily been limited to the case when the ground field is the field of complex numbers (so that modular representations are not even discussed). The experienced reader will see that many of the proofs are valid for arbitrary fields, or can be modified slightly to be so. Furthermore many important topics, such as projective representations, are not mentioned. On the other hand all of the important elementary results have been included, a number of advanced topics are treated, and a considerable number of special representations have been worked out in detail.

The volume follows in spirit the lectures given by Schur and prepared by Stiefel at Zurich in 1936 [27]. Also the original papers of Burnside, Frobenius, and Schur were consulted frequently, and many of the discussions follow these papers quite closely.

Without doubt the most important modern book on this subject is the book by Curtis and Reiner [9], and this has been consulted as well. The ultimate selection of subjects was of course a matter of taste and reflects the writer's personal likes and dislikes.

The writer's principal objective was to make available the results and techniques of the subject of group representations to an audience with at most a standard mathematical background, and to indicate some interesting applications of this subject. The volume should be accessible to a serious reader with some knowledge of matrix theory, who is prepared to make an effort to understand it. The writer has lectured from this volume to mathematically unsophisticated audiences with good results.

The writer thanks R. C. Thompson for his critical reading of the manuscript, which disclosed a number of gaps and inaccuracies. He also thanks Doris Burrell for her painstaking efforts in preparing the typed manuscript.


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Table of Contents

Representations of Arbitrary Groups 
Representations of Finite Groups
Kinds of Representations
The Principal Results
Some Theorems of Burnside
Multiplication of Representations 
Bounded Representations and Weyl’s Theorem
Appendix A. The Elements of the Theory of Algebraic Numbers 
Appendix B. The Roots of Unity

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