Symmetry, Representations, and Invariants
Symmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Li
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Symmetry, Representations, and Invariants
13
Graduate Texts in Mathematics
255
Editorial Board S. Axler K.A. Ribet
For other titles published in this series, go to http://www.springer.com/series/136
Roe Goodman Nolan R. Wallach Symmetry, Representations, and Invariants
123
Roe Goodman Department of Mathematics Rutgers University Piscataway, NJ 08854-8019 USA [email protected]
Nolan R. Wallach Department of Mathematics University of California, San Diego La Jolla, CA 92093 USA [email protected]
Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA [email protected]
ISBN 978-0-387-79851-6 e-ISBN 978-0-387-79852-3 DOI 10.1007/978-0-387-79852-3 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009927015 Mathematics Subject Classification (2000): 20G05, 14L35, 14M17, 17B10, 20C30, 20G20, 22E10, 22E46, 53B20, 53C35, 57M27 © Roe Goodman and Nolan R. Wallach 2009 Based on Representations and Invariants of the Classical Groups, Roe Goodman and Nolan R. Wallach, Cambridge University Press, 1998, third corrected printing 2003. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Organization and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix 1
Lie Groups and Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 General and Special Linear Groups . . . . . . . . . . . . . . . . . . . . . 1.1.2 Isometry Groups of Bilinear Forms . . . . . . . . . . . . . . . . . . . . . 1.1.3 Unitary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Quaternionic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Classical Lie Algebras . . . . . . . . . . . . . . . . . . . . .
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