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    Matrix Representations of Groups
    Matrix Representations of Groups
    Matrix Representations of Groups
    Ebook167 pages1 hour

    Matrix Representations of Groups

    By Morris Newman

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    About this ebook

    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.
    LanguageEnglish
    PublisherDover Publications
    Release dateJul 17, 2019
    ISBN9780486841557
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      Book preview

      Matrix Representations of Groups - Morris Newman

      Bibliographical Note

      This Dover edition, first published in 2019, is an unabridged and newly reset republication of the work originally published in 1968 as Volume 60 in the Applied Mathematics Series issued by the National Bureau of Standards, Washington, D.C.

      Library of Congress Cataloging-in-Publication Data

      Names: Newman, Morris, 1924–2007, author.

      Title: Matrix representations of groups / Morris Newman.

      Description: Dover edition. | Mineola, New York : Dover Publications, Inc., 2019. | Originally published: Washington, D.C. : National Bureau of Standards, 1968 (Applied mathematics series ; volume 60). | Includes bibliographical references.

      Identifiers: LCCN 2018059409 | ISBN 9780486832456 | ISBN 0486832457

      Subjects: LCSH: Representations of groups. | Matrices.

      Classification: LCC QA176 .N49 2019 | DDC 512/.22—dc23

      LC record available at https://lccn.loc.gov/2018059409

      Manufactured in the United States by LSC Communications

      83245701 2019

      www.doverpublications.com

      Contents

      Foreword

      Introduction

      I Representations of Arbitrary Groups

      1 Reducibility

      Examples

      2 Schur’s Lemma

      3 Representations

      Examples

      4 Character of a Representation

      5 Theorems of Burnside, Frobenius, and Schur

      6 An Application to Matrix Groups

      7 Further Consequences

      II Representations of Finite Groups

      8 Theorems of Maschke and Schur

      9 Characters of Finite Groups

      10 A Divisibility Theorem

      III Kinds of Representations

      11 Classification of Representations

      Examples

      IV The Principal Results

      12 Further Formulas for Characters

      13 The Matrix Associated With a Representation

      14 The Regular Representation

      15 The Principal Formulas

      16 Further Results

      Examples

      Examples

      17 The Symmetric Group

      Examples

      V Some Theorems of Burnside

      18 Conjugacy Classes

      Examples and Applications

      VI Multiplication of Representations

      19 Kronecker Products

      20 Powers of a Representation

      Examples

      21 Direct Products

      Examples

      VII Bounded Representations and Weyl’s Theorem

      22 Hermitian and Bounded Representations

      Appendix A. The Elements of the Theory of Algebraic Numbers

      1 Fields. Polynomials Over Fields. Irreducibility

      2 Algebraic Numbers Over k

      3 Algebraic Extensions of k

      4 Generators of Fields. Fundamental Systems. Subfields of k ( θ )

      5 Algebraic Integers

      6 Ideals

      Appendix B. The Roots of Unity

      References

      Index

      Foreword

      The theory of group representations is of fundamental importance in such disciplines as particle physics and crystallography and has been a major force in group theory. Thus the celebrated theorem of J. Thompson and W. Feit, that every group of odd order is solvable, depends heavily for its proof on this theory; and the new classifications of particles are described by means of certain special continuous groups and their representations.

      For some time there has been a need for a simple but complete exposition of this subject and the present volume should meet this need. It should be of value to the worker in the field who has occasion to use the subject or who must understand it. The many special representations worked out in detail should also prove quite useful.

      A. V. Astin, Director

      National Bureau of Standards

      Introduction

      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].¹ 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.

      ¹Figures in brackets indicate the literature references at the end of this paper.

      Chapter I

      Representations of Arbitrary Groups

      1 Reducibility

      Let A = {A} be a set (finite or infinite) of n × n matrices over the complex field C. Then A is said to be reducible if fixed positive integers, p, q and a fixed nonsingular matrix S exist such that for each A ∈ A,

      where A11 is p × p, A21 q × p, and A22 q × q. Otherwise A is said to be irreducible. If the form (1) can be achieved with A21 = 0 as well for all A ∈ A, then A is said to be fully reducible. Thus if T is any n × n nonsingular matrix, then T−1AT = {T−1AT} is reducible if and only if A is reducible, and fully reducible if and only if A is fully reducible.

      Examples

      (a) The set consisting of a single n × n matrix A alone, n > 1, is reducible. In fact there is a nonsingular matrix S such that S−1 AS is lower triangular.

      (c) Let A be a set of n ×n column stochastic matrices (having all column sums 1). Then A is reducible. To see this, choose

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