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

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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. “New light on Frobenius’ creation of the theory of group characters”. Archive for History of Exact Sciences 12:3 (1974): 217–243. [25] Hawkins, Thomas. “The origins of the theory of group characters”. Archive for History of Exact Sciences 7:2 (1971): 142–170. References for historical interludes 219

[26] Hermann, Robert. “Preface”. In Lie Groups: History, Frontiers and Applications, vol. 3 (Brookline, MA: Math Sci Press, 1976), pp. iii–iv. [27] Kimberling, Clark. “Dedekind letters”; http://faculty.evansville.edu/ck6/bstud/dedek.html. [28] Kr¨omer, Ralf. Tool and Object: A History and Philosophy of Category Theory (Basel: Birkh¨auser, 2007). [29] Lam, T. Y. “Representations of finite groups: A hundred years: Part I”. Notices of the AMS 45:3 (1998): 361–372. [30] Lam, T. Y. “Representations of Finite Groups: A Hundred Years: Part II”. Notices of the AMS 45:4 (1998): 465–474. [31] Landry, Elaine, and Jean-Pierre Marquis. “Categories in Context: Historical, Foundational, and Philosophical”. Philosophia Mathemat- ica 13 (2005): 1–43. [32] Lawvere, F. W., and Colin McLarty. “Mac Lane, Saunders”. New Dic- tionary of Scientific Biography, vol. 23 (Detroit, MI: Charles Scrib- ner’s Sons, 2008), pp. 1–5. [33] Mac Lane, Saunders. A Mathematical Autobiography (Wellesley, MA: A.K. Peters, 2005). [34] Mac Lane, Saunders. Mathematics, Form and Function (New York: Springer, 1986). [35] Mac Lane, Saunders. “Concepts and Categories in Perspective”. In A Century of Mathematics in America, ed. Peter Duren, Part I (Provi- dence, RI: American Mathematical Society, 1988), pp. 323–365. [36] Mac Lane, Saunders. “Mathematics at the University of Chicago: A Brief History”. In A Century of Mathematics in America,ed.Pe- ter Duren, Part II (Providence, RI: American Mathematical Society, 1989), pp. 127–154. [37] Mac Lane, Saunders. “The PNAS way back then”. Proc. Natl. Acad. Sci. USA 94 (June 1997): 5983–5985. [38] Marquis, Jean-Pierre. From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory (New York: Springer, 2009). [39] McLarty, Colin. “The Last Mathematician from Hilbert’s G¨ottingen: Saunders Mac Lane as Philosopher of Mathematics”. British Journal for the Philosophy of Science 58:1 (2007): 1–36. [40] McLarty, Colin. “Saunders Mac Lane (1909–2005): His Mathematical Life and Philosophical Works”. Philosophia Mathematica 13 (2005): 237–251. [41] Neumann, Peter M. “The Context of Burnside’s Contributions to Group Theory”. In The Collected Papers of William Burnside,vol. 1, pp. 15–54. 220 References for historical interludes

[42] Pesic, Peter. “Introduction”. In Hermann Weyl, Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics (Prince- ton, NJ: Princeton University Press, 2009), pp. 1–19. [43] Pickering, Andrew. “Concepts and the Mangle of Practice: Con- structing Quaternions”. In 18 Unconventional Essays on the Nature of Mathematics, ed. Reuben Hersh (New York: Springer, 2006), pp. 250–288. [44] Pycior, Helena M. “The Role of Sir William Rowan Hamilton in the Development of British Modern Algebra”. Ph. D. dissertation, Cor- nell University, 1976. [45] Reid, Constance. Hilbert-Courant (New York: Springer, 1986). [46] Rowe, David E. “Hermann Weyl, the Reluctant Revolutionary”. Mathematical Intelligencer 25:1 (2003): 61–70. [47] Rowe, David E. “Lie, Marius Sophus”. In New Dictionary of Scientific Biography, vol. 22 (Detroit, MI: Charles Scribner’s Sons, 2008), pp. 307–310. [48] Rowe, David E. “Three letters from Sophus Lie to Felix Klein on Parisian mathematics during the early 1880s”. Mathematical Intelli- gencer 7:3 (1985): 74–77. [49] Sigurdsson, Skuli. “Physics, Life, and Contingency: Born, Schr¨o- dinger and Weyl in Exile”. In Forced Migration and Scientific Change: Emigre´ German-Speaking Scientists and Scholars after 1933,eds. Mitchell G. Ash and Alfons S¨ollner (Cambridge, UK: Cambridge Uni- versity Press, 1996), pp. 48–70. [50] Sigurdsson, Skuli. “On the Road”. In Positioning the History of Sci- ence, eds. Kostas Gavroglu and J¨urgen Renn (Dordrecht: Springer, 2007), pp. 149–157. [51] Sigurdsson, Skuli. “Journeys in Spacetime”. In Hermann Weyl’s Raum-Zeit-Materie and a General Introduction to His Scientific Work, ed. Erhard Scholz (Basel: Birkh¨auser Verlag, 2001), pp. 15–47. [52] Sigurdsson, Skuli. “Hermann Weyl, Mathematics and Physics, 1900– 1927”. PhD dissertation, Harvard University, 1991. [53] Sigurdsson, Skuli. “Unification, Geometry and Ambivalence: Hilbert, Weyl, and the G¨ottingen Community”. In Trends in the Histori- ography of Science, eds. Kostas Gavroglu, Jean Christianidis, and Efthymios Nicolaidis (Dordrecht: Kluwer, 1994), pp. 355–367. [54] Solomon, Ronald M. “Burnside and Finite Simple Groups”. In The Collected Papers of William Burnside, vol. 1, pp. 45–54. [55] Sternberg, Shlomo. Group Theory and Physics (Cambridge, UK: Cambridge University Press, 1995). References for historical interludes 221

[56] Stubhaug, Arild. The Mathematician Sophus Lie: It Was the Au- dacity of My Thinking, transl. Richard H. Daly (Berlin: Springer, 2002). [57] Weyl, Hermann. The Classical Groups, Their Invariants and Repre- sentations (Princeton, NJ: Princeton University Press, 1946). [58] Weyl, Hermann. “David Hilbert. 1862–1943”. Obituary Notices of Fellows of the Royal Society 4:13 (1944): 547–553. [59] Weyl, Hermann. “The Open World: Three Lectures on the Metaphys- ical Implications of Science”. (1932). In Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics (Princeton, NJ: Princeton University Press, 2009), pp. 34–82. [60] Weyl, Hermann. The Theory of Groups and Quantum Mechanics,2nd ed. (New York: Dover, 1950). [61] Winterbourne, Anthony T. “Algebra and Pure Time: Hamilton’s Affinity with Kant”. Historia Mathematica 9 (1982): 195–200.

Mathematical references

[BGP] J. Bernstein, I. Gelfand, V. Ponomarev, Coxeter functors and Gabriel’s theorem, Russian Math. Surveys 28 (1973), no. 2, 17–32. [Cu] C. Curtis, Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, Amer. Math. Soc., 1999. [CR] C. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Amer. Math. Soc., 2006. [FH] W. Fulton and J. Harris, Representation Theory, A First Course, Springer, New York, 1991. [Fr] P. J. Freyd, Abelian Categories, an Introduction to the Theory of Functors, Harper and Row, 1964. [Ki] A. A. Kirillov Jr., An Introduction to Lie Groups and Lie Algebras, Cambridge University Press, 2008. [McL] S. Mac Lane, Categories for a Working Mathematician: 2nd ed., Graduate Texts in Mathematics 5, Springer, 1998. [S] J.-P. Serre, Linear Representations of Finite Groups, Springer-Verlag, 1997. [W] C. Weibel, An Introduction to Homological Algebra, Cambridge Univ. Press, 1994.

223

Index

Abelian category, 186 Central function, 61 Acyclic complex, 188 Character group, 62 Additive functor, 190 Character of a group Adjacency matrix, 146 representation, 61 Adjoint functors, 184 Character of a representation, 48 Ado’s theorem, 22 Character table, 71 Affine Dynkin diagram, 148 Class function, 61 Algebra of dual numbers, 55 Clifford algebra, 55 Algebra of endomorphisms of a Coboundary, 188 vector space, 9 Cocycle, 54, 188 Algebra of polynomials, 9 Cohomology, 187 Algebraic conjugate, 95 Cohomology class, 188 Algebraic integer, 93 Commutant of a Lie algebra, 39 Algebraic number, 93 Commutative algebra, 9 Algebraic representation, 130 Companion matrix, 94 Algebras defined by generators and Complete system of orthogonal relations, 16 idempotents, 210 Artin’s theorem, 141 Completely reducible Associative algebra, 5, 9 representation, 41 Complex, 187 Basic algebra, 216 Complex conjugate representation, Block, 213 69 Burnside’s theorem, 98 Compression modulus, 86 Connecting homomorphism, 189 Cartan matrix, 160 Content of a Young diagram, 118 Cartan matrix of a finite Coxeter element, 169 dimensional algebra, 212 Coxeter functor, 164 Casimir operator, 38 Cyclic quiver, 173 Category, 178 Cyclic representation, 16 Central character, 14 Cyclic vector, 16

225 226 Index

De Rham complex, 187 Highest weight, 130 Defining relations, 16 Hilbert syzygies theorem, 207 Deformation tensor, 85 Hilbert’s third problem, 36 Dehn invariant, 36 Homological dimension, 213 Density theorem, 43 Homomorphism of algebras, 9 Differential, 187 Homomorphism of Lie algebras, 23 Dimension vector, 163 Homomorphism of representations, Direct sum of representations, 5, 11 11 Double centralizer theorem, 119 Hook, 119 Dual group, 62 Hook length formula, 119 Dual representation of a Lie algebra, 37 Ideal, 15 Dual representation of an Ideal generated by a subset, 15 associative algebra, 44 Idempotent, 67 Dynkin diagram, 8, 146 Indecomposable representation, 6, 11 Einstein summation, 32 Induced representation, 105 Enriched category, 179 Intertwining operator, 11 Equivalence of categories, 182 Irreducible representation Exact functor, 191 (module), 6, 10 Exact sequence, 188 Isomorphism of representations, 11 Ext functor, 203 Extensions of representations, 53 Jordan-H¨older series, 51 Exterior algebra, 35 Jordan-H¨older theorem, 50 Exterior power, 33 Kostka numbers, 114 Faithful representation, 18 Koszul resolution, 207 Filtration, 45 Krull-Schmidt theorem, 51 Finite abelian category, 214 K¨unneth formula, 190 Flat module, 202 Left adjoint functor, 184 Formal deformation, 55 Left exact functor, 191 Free algebra, 9 Left ideal, 15 Frobenius character formula, 115 Left module, 5, 10 Frobenius determinant, 75 Length of a representation, 51 Frobenius divisibility theorem, 96 Lexicographic ordering on Frobenius formula, 106 partitions, 112 Frobenius reciprocity, 107 Lie algebra, 22 Frobenius-Schur indicator, 92 Lie group, 24 Full subcategory, 179 Lie subalgebra, 22 Functor, 179 Lie’s theorem, 39 Functorial morphism, 181 Linear abelian category (over a Gabriel’s theorem, 163 field), 187 General linear group, 13 Linear algebraic group, 25 General linear Lie algebra, 22 Linear functor (over a field), 190 Group algebra, 9 Linked modules, 213 Group determinant, 75 Locally small category, 179 Long exact sequence of Heisenberg group, 81 cohomology, 189 Index 227

Manifold, 25 Representation of a quiver, 7, 19 Maschke’s theorem, 59 Representation of complex type, 91 Maximalideal,15 Representation of quaternionic McKay graph, 149 type, 91 Minimal polynomial, 95 Representation of real type, 91 Module of finite length, 53 Representations of the symmetric Monoidal category, 179 group, 110 Morita equivalence, 215 Restriction of a representation, 105 Morita equivalent rings, 187 Right adjoint functor, 184 Morphism of complexes, 188 Right exact functor, 191 Right ideal, 15 Natural transformation, 181 Right module, 10 Root, 161 Orthogonality of characters, 65 Orthogonality of matrix elements, Schur functor, 180 70 Schur polynomials, 128 Schur’s lemma, 11 Partition, 110 Schur-Weyl duality, 119, 122 Path algebra of a quiver, 20 Semidirect product, 142 Polynomial representation, 130 Semisimple abelian category, 191 Positive root, 162 Semisimple algebra, 48 Progenerator, 214 Semisimple representation, 41 Projective cover, 210 Shearing modulus, 86 Projective dimension of a module, Short exact sequence, 188 212 Simple algebra, 15 Projective generator, 214 Projective module, 202, 203 Simple reflection, 163 Projective object, 203 Simple representation (module), 10 Projective resolution, 204 Simple root, 161 Pure tensor, 31 Sink, 164 Solvable group, 98 q-Weyl algebra, 17, 18 Solvable Lie algebra, 39 Quasi-inverse functor, 183 Source, 164 Quaternion algebra, 83 Specht module, 111 Quaternion group, 63 Split short exact sequence, 188 Quaternions, 83 Stress tensor, 85 Quiver, 7, 19 Subrepresentation, 5, 10 Quiver of finite type, 7, 148 Sum of squares formula, 60 Quotient algebra, 15 Sylvester criterion, 147 Quotient representation, 15 Symmetric algebra, 35 Symmetric power, 33 Radical, 46 Rational representation, 130 Tensor algebra, 35 Reflection functor, 164 Tensor product of linear maps, 32 Regular representation, 10 Tensor product of representations Representable functor, 183 of a Lie algebra, 37 Representation, 5, 10 Tensor product of vector spaces, 31 Representation of a group, 59 Tensors of type (m, n), 31 Representation of a Lie algebra, 24 Tor functor, 203 228 Index

Two-sided ideal, 15

Unit, 9 Unit morphism, 178 Unitary representation of a group, 69 Universal enveloping algebra of a Lie algebra, 24, 35

Virtual representation, 105

Weyl algebra, 17 Weyl group, 163

Young diagram, 110 Young projector, 111 Young tableau, 110

Zariski dense subset, 145 Titles in This Series

59 Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina, with historical interludes by Slava Gerovitch, Introduction to representation theory, 2011 58 Alvaro´ Lozano-Robledo, Elliptic curves, modular forms, and their L-functions, 2011 57 Charles M. Grinstead, William P. Peterson, and J. Laurie Snell, Probability Tales, 2011 56 Julia Garibaldi, Alex Iosevich, and Steven Senger, The Erd˝os distance problem, 2011 55 Gregory F. Lawler, Random walk and the heat equation, 2010 54 Alex Kasman, Glimpses of soliton theory: The algebra and geometry of nonlinear PDEs, 2010 53 Jiˇr´ıMatouˇsek, Thirty-three miniatures: Mathematical and algorithmic applications of linear algebra, 2010 52 Yakov Pesin and Vaughn Climenhaga, Lectures on fractal geometry and dynamical systems, 2009 51 Richard S. Palais and Robert A. Palais, Differential equations, mechanics, and computation, 2009 50 Mike Mesterton-Gibbons, A primer on the calculus of variations and optimal control theory, 2009 49 Francis Bonahon, Low-dimensional geometry: From euclidean surfaces to hyperbolic knots, 2009 48 John Franks, A (terse) introduction to Lebesgue integration, 2009 47 L. D. Faddeev and O. A. Yakubovski˘ı, Lectures on quantum mechanics for mathematics students, 2009 46 Anatole Katok and Vaughn Climenhaga, Lectures on surfaces: (Almost) everything you wanted to know about them, 2008 45 Harold M. Edwards, Higher arithmetic: An algorithmic introduction to number theory, 2008 44 Yitzhak Katznelson and Yonatan R. Katznelson, A (terse) introduction to linear algebra, 2008 43 Ilka Agricola and Thomas Friedrich, Elementary geometry, 2008 42 C. E. Silva, Invitation to ergodic theory, 2007 41 Gary L. Mullen and Carl Mummert, Finite fields and applications, 2007 40 Deguang Han, Keri Kornelson, David Larson, and Eric Weber, Frames for undergraduates, 2007 39 Alex Iosevich, A view from the top: Analysis, combinatorics and number theory, 2007 38 B. Fristedt, N. Jain, and N. Krylov, Filtering and prediction: A primer, 2007 TITLES IN THIS SERIES

37 Svetlana Katok, p-adic analysis compared with real, 2007 36 Mara D. Neusel, Invariant theory, 2007 35 J¨org Bewersdorff, Galois theory for beginners: A historical perspective, 2006 34 Bruce C. Berndt, Number theory in the spirit of Ramanujan, 2006 33 Rekha R. Thomas, Lectures in geometric combinatorics, 2006 32 Sheldon Katz, Enumerative geometry and string theory, 2006 31 John McCleary, A first course in topology: Continuity and dimension, 2006 30 Serge Tabachnikov, Geometry and billiards, 2005 29 Kristopher Tapp, Matrix groups for undergraduates, 2005 28 Emmanuel Lesigne, Heads or tails: An introduction to limit theorems in probability, 2005 27 Reinhard Illner, C. Sean Bohun, Samantha McCollum, and Thea van Roode, Mathematical modelling: A case studies approach, 2005 26 Robert Hardt, Editor, Six themes on variation, 2004 25 S. V. Duzhin and B. D. Chebotarevsky, Transformation groups for beginners, 2004 24 Bruce M. Landman and Aaron Robertson, Ramsey theory on the integers, 2004 23 S. K. Lando, Lectures on generating functions, 2003 22 Andreas Arvanitoyeorgos, An introduction to Lie groups and the geometry of homogeneous spaces, 2003 21 W. J. Kaczor and M. T. Nowak, Problems in mathematical analysis III: Integration, 2003 20 Klaus Hulek, Elementary algebraic geometry, 2003 19 A. Shen and N. K. Vereshchagin, Computable functions, 2003 18 V. V. Yaschenko, Editor, Cryptography: An introduction, 2002 17 A. Shen and N. K. Vereshchagin, Basic set theory, 2002 16 Wolfgang Kuhnel, ¨ Differential geometry: curves – surfaces – manifolds, second edition, 2006 15 Gerd Fischer, Plane algebraic curves, 2001 14 V. A. Vassiliev, Introduction to topology, 2001 13 Frederick J. Almgren, Jr., Plateau’s problem: An invitation to varifold geometry, 2001 12 W. J. Kaczor and M. T. Nowak, Problems in mathematical analysis II: Continuity and differentiation, 2001

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum fi eld theory. The goal of this book is to give a “holistic” introduction to representation theory, presenting it as a unifi ed subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the repre- sentation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more diffi cult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra.

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