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Introduction to Representation Theory STUDENT MATHEMATICAL LIBRARY Volume 59 Introduction to Representation Theory Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina with historical interludes by Slava Gerovitch http://dx.doi.org/10.1090/stml/059 STUDENT MATHEMATICAL LIBRARY Volume 59 Introduction to Representation Theory Pavel Etingof Oleg Golberg Sebastian Hensel Tiankai Liu Alex Schwendner Dmitry Vaintrob Elena Yudovina with historical interludes by Slava Gerovitch American Mathematical Society Providence, Rhode Island Editorial Board Gerald B. Folland Brad G. Osgood (Chair) Robin Forman John Stillwell 2010 Mathematics Subject Classification. Primary 16Gxx, 20Gxx. For additional information and updates on this book, visit www.ams.org/bookpages/stml-59 Library of Congress Cataloging-in-Publication Data Introduction to representation theory / Pavel Etingof ...[et al.] ; with historical interludes by Slava Gerovitch. p. cm. — (Student mathematical library ; v. 59) Includes bibliographical references and index. ISBN 978-0-8218-5351-1 (alk. paper) 1. Representations of algebras. 2. Representations of groups. I. Etingof, P. I. (Pavel I.), 1969– QA155.I586 2011 512.46—dc22 2011004787 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 2011 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 161514131211 Contents Chapter 1. Introduction 1 Chapter 2. Basic notions of representation theory 5 §2.1. What is representation theory? 5 §2.2. Algebras 8 §2.3. Representations 9 §2.4. Ideals 15 §2.5. Quotients 15 §2.6. Algebras defined by generators and relations 16 §2.7. Examples of algebras 17 §2.8. Quivers 19 §2.9. Lie algebras 22 §2.10. Historical interlude: Sophus Lie’s trials and transformations 26 §2.11. Tensor products 30 §2.12. The tensor algebra 35 §2.13. Hilbert’s third problem 36 §2.14. Tensor products and duals of representations of Lie algebras 36 §2.15. Representations of sl(2) 37 iii iv Contents §2.16. Problems on Lie algebras 39 Chapter 3. General results of representation theory 41 §3.1. Subrepresentations in semisimple representations 41 §3.2. The density theorem 43 §3.3. Representations of direct sums of matrix algebras 44 §3.4. Filtrations 45 §3.5. Finite dimensional algebras 46 §3.6. Characters of representations 48 §3.7. The Jordan-H¨older theorem 50 §3.8. The Krull-Schmidt theorem 51 §3.9. Problems 53 §3.10. Representations of tensor products 56 Chapter 4. Representations of finite groups: Basic results 59 §4.1. Maschke’s theorem 59 §4.2. Characters 61 §4.3. Examples 62 §4.4. Duals and tensor products of representations 65 §4.5. Orthogonality of characters 65 §4.6. Unitary representations. Another proof of Maschke’s theorem for complex representations 68 §4.7. Orthogonality of matrix elements 70 §4.8. Character tables, examples 71 §4.9. Computing tensor product multiplicities using character tables 74 §4.10. Frobenius determinant 75 §4.11. Historical interlude: Georg Frobenius’s “Principle of Horse Trade” 77 §4.12. Problems 81 §4.13. Historical interlude: William Rowan Hamilton’s quaternion of geometry, algebra, metaphysics, and poetry 86 Contents v Chapter 5. Representations of finite groups: Further results 91 §5.1. Frobenius-Schur indicator 91 §5.2. Algebraic numbers and algebraic integers 93 §5.3. Frobenius divisibility 96 §5.4. Burnside’s theorem 98 §5.5. Historical interlude: William Burnside and intellectual harmony in mathematics 100 §5.6. Representations of products 104 §5.7. Virtual representations 105 §5.8. Induced representations 105 §5.9. The Frobenius formula for the character of an induced representation 106 §5.10. Frobenius reciprocity 107 §5.11. Examples 110 §5.12. Representations of Sn 110 §5.13. Proof of the classification theorem for representations of Sn 112 §5.14. Induced representations for Sn 114 §5.15. The Frobenius character formula 115 §5.16. Problems 118 §5.17. The hook length formula 118 §5.18. Schur-Weyl duality for gl(V ) 119 §5.19. Schur-Weyl duality for GL(V ) 122 §5.20. Historical interlude: Hermann Weyl at the intersection of limitation and freedom 122 §5.21. Schur polynomials 128 §5.22. The characters of Lλ 129 §5.23. Algebraic representations of GL(V ) 130 §5.24. Problems 131 §5.25. Representations of GL2(Fq) 132 §5.26. Artin’s theorem 141 vi Contents §5.27. Representations of semidirect products 142 Chapter 6. Quiver representations 145 §6.1. Problems 145 §6.2. Indecomposable representations of the quivers A1,A2,A3 150 §6.3. Indecomposable representations of the quiver D4 154 §6.4. Roots 160 §6.5. Gabriel’s theorem 163 §6.6. Reflection functors 164 §6.7. Coxeter elements 169 §6.8. Proof of Gabriel’s theorem 170 §6.9. Problems 173 Chapter 7. Introduction to categories 177 §7.1. The definition of a category 177 §7.2. Functors 179 §7.3. Morphisms of functors 181 §7.4. Equivalence of categories 182 §7.5. Representable functors 183 §7.6. Adjoint functors 184 §7.7. Abelian categories 186 §7.8. Complexes and cohomology 187 §7.9. Exact functors 190 §7.10. Historical interlude: Eilenberg, Mac Lane, and “general abstract nonsense” 192 Chapter 8. Homological algebra 201 §8.1. Projective and injective modules 201 §8.2. Tor and Ext functors 203 Chapter 9. Structure of finite dimensional algebras 209 §9.1. Lifting of idempotents 209 §9.2. Projective covers 210 Contents vii §9.3. The Cartan matrix of a finite dimensional algebra 211 §9.4. Homological dimension 212 §9.5. Blocks 213 §9.6. Finite abelian categories 214 §9.7. Morita equivalence 215 References for historical interludes 217 Mathematical references 223 Index 225 References for historical interludes [1] Adelmann, Clemens, and Eberhard H.-A. Gerbracht. “Letters from William Burnside to Robert Fricke: Automorphic Functions, and the Emergence of the Burnside Problem”. Archive for History of Exact Sciences 63:1 (2009): 33–50. [2] Arnold, Vladimir. “Matematika s chelovecheskim litsom” (Mathe- matics with a Human Face). Priroda, no. 3 (1988): 117–119. [3] Atiyah, Michael. “Hermann Weyl, 1885–1955: A Biographical Mem- oir”. In Biographical Memoirs, vol. 82 (Washington, DC: National Academy of Sciences, 2003), 321–332. [4] Baumslag, Gilbert, comp. Reviews on Infinite Groups: As Printed in Mathematical Reviews 1940 through 1970, vol. 2 (Providence, RI: American Mathematical Society, 1974). [5] Beaulieu, Liliane. “Bourbaki’s Art of Memory”. Osiris,2ndseries,14 (1999): 219–251. [6] Bass, Hyman, et al. “Samuel Eilenberg (1913–1998)”. Notices of the American Mathematical Society 45, no. 10 (1998): 1344–1352. [7] Bass, Hyman. “Eilenberg, Samuel”. New Dictionary of Scientific Bi- ography, vol. 20 (Detroit, MI: Charles Scribner’s Sons, 2008), pp. 360–363. [8] Burnside, William. “On the Theory of Groups of Finite Order”. Proc. London Math. Soc. (1909): 1–7. [9] Corry, Leo. Modern Algebra and the Rise of Mathematical Structures, 2nd ed. (Basel: Birkh¨auser, 2004). 217 218 References for historical interludes [10] Crowe, Michael J. A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (New York: Dover, 1985). [11] Curtis, Charles W. Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, vol. 15 (Prov- idence, RI: American Mathematical Society, 1999). [12] Eilenberg, Samuel, and Saunders Mac Lane. “General Theory of Nat- ural Equivalences”. Transactions of the American Mathematical So- ciety 58 (1945): 231–294. [13] Everett, Martin G., A. J. S. Mann, and Kathy Young. “Notes of Burnside’s life”. In The Collected Papers of William Burnside,eds. Everett, Mann, and Young, vol. 1 (New York: Oxford University Press, 2004), pp. 89–110. [14] Feit, Walter. “Foreword”. In The Collected Papers of William Burn- side, vol. 1, pp. 3–6. [15] Forsyth, A. R. “William Burnside”. J. London Math. Soc. s1-3:1 (Jan- uary 1928): 64–80. [16] Freudenthal, Hans. “Lie, Marius Sophus”. In Dictionary of Scientific Biography, vol. 8 (New York: Scribner, 1973), pp. 323–327. [17] Graves, Robert P. Life of Sir William Rowan Hamilton,3vols. (Dublin: Dublin University Press, 1882–1889). [18] Hankins, Thomas L. “Hamilton, William Rowan”. Complete Dictio- nary of Scientific Biography, ed. Charles C. Gillespie, vol. 6 (Detroit: Charles Scribner’s Sons, 2008), pp. 85–93 (Gale Virtual Reference Library). [19] Hankins, Thomas L. Sir William Rowan Hamilton (Baltimore, MD: Johns Hopkins University Press, 1980). [20] Hankins, Thomas L. “Triplets and Triads: Sir William Rowan Hamil- ton on the Metaphysics of Mathematics”. Isis 68 (1977): 175–193. [21] Haubrich, Ralf. “Frobenius, Schur, and the Berlin algebraic tradi- tion”. In Mathematics in Berlin, eds. Heinrich Begehr et al. (Berlin: Birkh¨auser, 1998), pp. 83–96. [22] Hawkins, Thomas. Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics, 1869–1926 (Berlin: Springer, 2000). [23] Hawkins, Thomas. “Hypercomplex Numbers, Lie Groups, and the Creation of Group Representation Theory”. Archive for History of Exact Sciences 8 (1971): 243–287. [24] Hawkins, Thomas.
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