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An Introduction to the Representation Theory of Groups An Introduction to the Representation Theory of Groups Emmanuel Kowalski Graduate Studies in Mathematics Volume 155 American Mathematical Society An Introduction to the Representation Theory of Groups https://doi.org/10.1090//gsm/155 An Introduction to the Representation Theory of Groups Emmanuel Kowalski Graduate Studies in Mathematics Volume 155 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 20-01, 20Cxx, 22A25. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-155 Library of Congress Cataloging-in-Publication Data Kowalski, Emmanuel, 1969– An introduction to the representation theory of groups / Emmanuel Kowalski. pages cm. — (Graduate studies in mathematics ; volume 155) Includes bibliographical references and index. ISBN 978-1-4704-0966-1 (alk. paper) 1. Lie groups. 2. Representations of groups. 3. Group algebras. I. Title. QA387.K69 2014 515.7223—dc23 2014012974 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 Contents Chapter 1. Introduction and motivation 1 §1.1. Presentation 3 §1.2. Four motivating statements 4 §1.3. Prerequisites and notation 8 Chapter 2. The language of representation theory 13 §2.1. Basic language 13 §2.2. Formalism: changing the space 21 §2.3. Formalism: changing the group 42 §2.4. Formalism: changing the field 65 §2.5. Matrix representations 68 §2.6. Examples 70 §2.7. Some general results 80 §2.8. Some Clifford theory 121 §2.9. Conclusion 124 Chapter 3. Variants 127 §3.1. Representations of algebras 127 §3.2. Representations of Lie algebras 132 §3.3. Topological groups 139 §3.4. Unitary representations 145 Chapter 4. Linear representations of finite groups 159 §4.1. Maschke’s Theorem 159 v vi Contents §4.2. Applications of Maschke’s Theorem 163 §4.3. Decomposition of representations 169 §4.4. Harmonic analysis on finite groups 190 §4.5. Finite abelian groups 200 §4.6. The character table 208 §4.7. Applications 240 §4.8. Further topics 262 Chapter 5. Abstract representation theory of compact groups 269 §5.1. An example: the circle group 269 §5.2. The Haar measure and the regular representation of a locally compact group 272 §5.3. The analogue of the group algebra 288 §5.4. The Peter–Weyl Theorem 294 §5.5. Characters and matrix coefficients for compact groups 304 §5.6. Some first examples 310 Chapter 6. Applications of representations of compact groups 319 §6.1. Compact Lie groups are matrix groups 319 §6.2. The Frobenius–Schur indicator 324 §6.3. The Larsen alternative 332 §6.4. The hydrogen atom 344 Chapter 7. Other groups: a few examples 355 §7.1. Algebraic groups 355 §7.2. Locally compact groups: general remarks 369 §7.3. Locally compact abelian groups 371 §7.4. A non-abelian example: SL2pRq 376 Appendix A. Some useful facts 409 §A.1. Algebraic integers 409 §A.2. The spectral theorem 414 §A.3. The Stone–Weierstrass Theorem 420 Bibliography 421 Index 425 Bibliography [1] J. Baez and J. Huerta: The algebra of Grand Unified Theories, Bulletin of the American Mathematical Society 47 (2010), 483–552. [2]B.Bekka,P.delaHarpe,andA.Valette: Kazhdan’s Property pT q,NewMath. Monographs 11, Cambridge University Press, 2008. [3] J. Bernstein and S. Gelbart (eds): An introduction to the Langlands program, Birkha¨user, 2003. [4]J.BernsteinandA.Reznikov: Analytic continuation of representations and es- timates of automorphic forms, Annals of Math. 150 (1999), 329–352. [5] N. Berry, A. Dubickas, N. Elkies, B. Poonen, and C. J. Smyth: The conjugate dimension of algebraic numbers, Quart. J. Math. 55 (2004), 237–252. [6]E.D.Bolker: The spinor spanner, American Math. Monthly 80 (1973), 977–984. [7] A. Borel: Linear algebraic groups, 2nd edition, Graduate Texts in Mathematics 126, Springer, 1991. [8] W. Bosma, J. Cannon, and C. Playoust: The Magma algebra system, I. The user language J. Symbolic Comput. 24 (1997), 235–265; also http://magma.maths. usyd.edu.au/magma/ [9] N. Bourbaki: Algebra I, Chapter 2, Springer, 1989. [10] A. E. Brouwer and M. Popoviciu: The invariants of the binary nonic,J.Symb. Computation 45 (2010) 709–720. [11] D. Bump: Automorphic forms and representations, Cambridge Studies in Adv. Math. 55, Cambridge University Press, 1998. [12] C. W. Curtis and I. Reiner: Representation theory of finite groups and associative algebras, AMS Chelsea Publishing, 1962. [13] G. Davidoff, P. Sarnak, and A. Valette: Elementary number theory, group theory and Ramanujan graphs, London Mathematical Society Student Text 55, Cam- bridge University Press, 2003. [14] A. Deitmar and S. Echterhoff: Principles in harmonic analysis,Universitext, Springer, 2009. [15] P. Deligne: La conjecture de Weil, II, Publ. Math. I.H.E.S´ 52 (1980), 137–252. 421 422 Bibliography [16] P. Diaconis: Group representations in probability and statistics, Inst. Math. Stat. Lecture Notes 11, Institute of Math. Statistics, 1988. [17] N. Dunford and J. T. Schwarz: Linear operators, part II: spectral theory, Wiley, 1963. [18] P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, and E. Yudovina: Introduction to representation theory, Student Math. Library 59, American Mathematical Society, 2011; also arXiv:0901.0827. [19] G. Folland: Real Analysis, Wiley, 1984. [20] W. Fulton and J. Harris: Representation theory, a first course,Universitext, Springer, 1991. [21] The GAP Group: GAP – Groups, Algorithms, and Programming, Version 4.4.9, 2007; also www.gap-system.org [22] K. Girstmair: Linear relations between roots of polynomials, Acta Arithmetica 89 (1999), 53–96. [23] W. T. Gowers: Quasirandom groups, Comb. Probab. Comp. 17 (2008), 363–387. [24] W. T. Gowers (editor): The Princeton companion to mathematics,Princeton University Press, 2008. [25] I. S. Gradshteyn and I. M. Ryzhik: Table of integrals, series, and products,6th Ed., Academic Press, 2000. [26] P. de la Harpe: Topics in geometric group theory, Chicago Lectures in Mathe- matics, Chicago University Press, 2000. [27] R. Hartshorne: Algebraic Geometry, Grad. Texts in Math. 52, Springer Verlag, 1977. [28] I. M. Isaacs: Character theory of finite groups, Academic Press, 1976. [29] I. M. Isaacs: Finite group theory, Graduate Studies in Math. 92, American Math- ematical Society, 2008. [30] H. Iwaniec and E. Kowalski: Analytic number theory, Coll. Publ. 53, American Mathematical Society, 2004. [31] F. Jouve, E. Kowalski and D. Zywina: An explicit integral polynomial whose split- ting field has Galois group W pE8q, Journal de ThEorie´ des Nombres de Bordeaux 20 (2008), 761–782. [32] N. M. Katz: Larsen’s alternative, moments and the monodromy of Lefschetz pencils, in “Contributions to automorphic forms, geometry, and number theory (collection in honor of J. Shalika’s 60th birthday)”, Johns Hopkins University Press (2004), 521–560. [33] N. M. Katz: Gauss sums, Kloosterman sums and monodromy groups, Annals of Math. Studies, 116, Princeton University Press, 1988. [34] N. M. Katz: Rigid local systems, Annals of Math. Studies, 139, Princeton Uni- versity Press, 1996. [35] A. Kirillov: Introduction to Lie groups and Lie algebras, Cambridge Studies in Advanced Math. 113, 2008. [36] A. W. Knapp: Representation theory of semisimple groups, Princeton Landmarks in Math., Princeton University Press, 2001. [37] A. W. Knapp: Lie groups beyond an introduction, Progress in Math. 140, Birkha¨user, 2002. Bibliography 423 [38] E. Kowalski: The large sieve, monodromy, and zeta functions of algebraic curves, II: independence of the zeros, International Math. Res. Notices 2008, doi:10. 1093/imrn/rnn091. [39] E. Kowalski: Spectral theory in Hilbert spaces, Lectures notes for ETH course (2009), www.math.ethz.ch/~kowalski/spectral-theory.pdf [40] S. Lang: Algebra, 3rd edition, Graduate Texts in Mathematics 211, Springer, 2002. [41] S. Lang: SL2pRq, Addison Wesley, 1974. [42] M. Larsen: The normal distribution as a limit of generalized Sato-Tate measures, preprint (http://mlarsen.math.indiana.edu/~larsen/papers/gauss.pdf). [43] M. W. Liebeck, E. A. O’Brien, A. Shalev, and P. H. Tiep: The Ore conjecture, J. Eur. Math. Soc. (JEMS) 12 (2010), 939–1008. [44] G. W. Mackey: The theory of unitary group representations, Chicago Lectures in Math., Chicago University Press, 1976. [45] W. Magnus: Residually finite groups, Bull. Amer. Math. Soc. 75 (1969), 305–316. [46] B. Nica: Linear groups: Malcev’s theorem and Selberg’s lemma, arXiv:1306. 2385. [47] N. Nikolov and L. Pyber: Product decompositions of quasirandom groups and a Jordan-type theorem, J. European Math. Soc., to appear. [48] I. Piatetstki-Shapiro: Complex representations of GLp2,Kq for finite fields K, Contemporary Math., Vol. 16, American Mathematical Society, 1983. [49] M. Reed and B. Simon: Methods of modern mathematical physics, I: Functional analysis, Academic Press, 1980. [50] M. Reed and B. Simon: Methods of modern mathematical physics, II: Self- adjointness and Fourier theoretic techniques, Academic Press, 1980.
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