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Groups and Symmetries: From Finite Groups to Lie Groups presents an introduction to the theory of group representations and its applications in quantum mechanics. Accessible to advanced undergraduates in mathematics and physics as well as beginning graduate students, the text deals with the theory of representations of finite groups, compact groups, linear Lie groups and their Lie algebras, concisely and in one volume. Prerequisites include calculus and linear algebra. This new edition contains an additional chapter that deals with Clifford algebras, spin groups, and the theory of spinors, as well as new sections entitled "Topics in history" comprising notes on the history of the material treated within each chapter. (Taken together, they constitute an account of the development of the theory of groups from its inception in the 18th century to the mid-20th.) References for additional resources and further study are provided in each chapter. All chapters end with exercises of varying degree of difficulty, some of which introduce new definitions and results. The text concludes with a collection of problems with complete solutions making it ideal for both course work and independent study. Key Topics include: Brisk review of the basic definitions of group theory, with examples Representation theory of finite groups: character theory Representations of compact groups using the Haar measure Lie algebras and linear Lie groups Detailed study of SO(3) and SU(2), and their representations Spherical harmonics Representations of SU(3), roots and weights, with quark theory as a consequence of the mathematical properties of this symmetry group Spin groups and spinors.

Group theory --- Algebra --- Mathematics --- Mathematical physics --- Quantum mechanics. Quantumfield theory --- Chemical and physical crystallography --- Rocks. Minerals --- algebra --- kristallografie --- quantumfysica --- toegepaste wiskunde --- theoretische fysica --- wiskunde --- fysica --- kristallen --- Finite groups. --- Representations of groups. --- Lie groups. --- Representacions de grups --- Grups finits --- Grups de Lie

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Modules (Algebra) --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra) --- Anells (Àlgebra) --- Monoides --- Mòduls (Àlgebra) --- Sistemes modulars (Àlgebra) --- Anells artinians --- Mòduls de Banach (Àlgebra) --- Mòduls de Cohen-Macaulay --- Mòduls injectius (Àlgebra) --- Mòduls projectius (Àlgebra) --- Teoria de divisors --- Teoria de torsió (Àlgebra) --- Grups finits --- Semigrups --- Anells algebraics --- Anells algèbrics --- Àlgebres no associatives --- Àlgebres topològiques --- Anells associatius --- Anells commutatius --- Anells de grup --- Anells no commutatius --- Ideals (Àlgebra) --- Representacions d'anells (Àlgebra) --- Teoria de la mesura

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Teoria de grups --- Substitucions (Matemàtica) --- Àlgebra --- Anells de grup --- Automorfismes --- Categories (Matemàtica) --- Cristal·lografia matemàtica --- Endomorfismes (Teoria de grups) --- Esquemes de grups (Matemàtica) --- Grupoides --- Grups abelians --- Grups algebraics diferencials --- Grups algebraics lineals --- Grups continus --- Grups de permutacions --- Grups de transformacions --- Grups discontinus --- Grups d'homotopia --- Grups espacials --- Grups finits --- Grups fonamentals (Matemàtica) --- Grups infinits --- Grups modulars --- Grups ordenats --- Grups quàntics --- Grups resolubles --- Jocs d'estratègia (Matemàtica) --- Representacions de grups --- Semigrups --- Simetria (Matemàtica) --- Subgrups maximals --- Teoria dels reticles --- Teoria geomètrica de grups --- Group theory. --- Algebra --- Groups, Theory of --- Substitutions (Mathematics)

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Geometry. --- Group theory. --- Thurston, William P., --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Mathematics --- Euclid's Elements --- Thurston, W. P. --- Geometria --- Teoria de grups --- Substitucions (Matemàtica) --- Àlgebra --- Anells de grup --- Automorfismes --- Categories (Matemàtica) --- Cristal·lografia matemàtica --- Endomorfismes (Teoria de grups) --- Esquemes de grups (Matemàtica) --- Grupoides --- Grups abelians --- Grups algebraics diferencials --- Grups algebraics lineals --- Grups continus --- Grups de permutacions --- Grups de transformacions --- Grups discontinus --- Grups d'homotopia --- Grups espacials --- Grups finits --- Grups fonamentals (Matemàtica) --- Grups infinits --- Grups modulars --- Grups ordenats --- Grups quàntics --- Grups resolubles --- Jocs d'estratègia (Matemàtica) --- Representacions de grups --- Semigrups --- Simetria (Matemàtica) --- Subgrups maximals --- Teoria dels reticles --- Teoria geomètrica de grups --- Matemàtica --- Congruències (Geometria) --- Dibuix lineal --- Envolupants (Geometria) --- Esfera --- Geometria algebraica --- Geometria computacional --- Geometria conforme --- Geometria convexa --- Geometria de l'espai --- Geometria diferencial --- Geometria euclidiana --- Porismes --- Programació geomètrica --- Ràtio i proporció --- Similitud (Geometria) --- Teorema de Pitàgores --- Transformacions (Matemàtica) --- Trigonometria --- Geometria en l'art

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This monograph adopts an operational and functional analytic approach to the following problem: given a short exact sequence (group extension) 1 N G H 1 of finite groups, describe the irreducible representations of G by means of the structure of the group extension. This problem has attracted many mathematicians, including I. Schur, A.H. Clifford, and G. Mackey and, more recently, M. Isaacs, B. Huppert, Y.G. Berkovich & E.M. Zhmud, and J.M.G. Fell & R.S. Doran. The main topics are, on the one hand, Clifford Theory and the Little Group Method (of Mackey and Wigner) for induced representations, and, on the other hand, Kirillov’s Orbit Method (for step-2 nilpotent groups of odd order) which establishes a natural and powerful correspondence between Lie rings and nilpotent groups. As an application, a detailed description is given of the representation theory of the alternating groups, of metacyclic, quaternionic, dihedral groups, and of the (finite) Heisenberg group. The Little Group Method may be applied if and only if a suitable unitary 2-cocycle (the Mackey obstruction) is trivial. To overcome this obstacle, (unitary) projective representations are introduced and corresponding Mackey and Clifford theories are developed. The commutant of an induced representation and the relative Hecke algebra is also examined. Finally, there is a comprehensive exposition of the theory of projective representations for finite Abelian groups which is applied to obtain a complete description of the irreducible representations of finite metabelian groups of odd order.

Finite groups. --- Group extensions (Mathematics) --- Group theory. --- Grups finits --- Teoria de grups --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Extensions, Group (Mathematics) --- Group theory --- Groups, Finite --- Modules (Algebra) --- Substitucions (Matemàtica) --- Àlgebra --- Anells de grup --- Automorfismes --- Categories (Matemàtica) --- Cristal·lografia matemàtica --- Endomorfismes (Teoria de grups) --- Esquemes de grups (Matemàtica) --- Grupoides --- Grups abelians --- Grups algebraics diferencials --- Grups algebraics lineals --- Grups continus --- Grups de permutacions --- Grups de transformacions --- Grups discontinus --- Grups d'homotopia --- Grups espacials --- Grups fonamentals (Matemàtica) --- Grups infinits --- Grups modulars --- Grups ordenats --- Grups quàntics --- Grups resolubles --- Jocs d'estratègia (Matemàtica) --- Representacions de grups --- Semigrups --- Simetria (Matemàtica) --- Subgrups maximals --- Teoria dels reticles --- Teoria geomètrica de grups --- Grups esporàdics (Matemàtica) --- Subgrups de Sylow --- Mòduls (Àlgebra) --- Group Theory and Generalizations.