Symmetry, representations, and invariants / Roe Goodman, Nolan R. Wallach.
Series Graduate texts in mathematics ; 255Editor: Dordrecht [Netherlands] ; New York : Springer, c2009Descripción: xx, 716 p. : il. ; 24 cmISBN: 9780387798516 (acidfree paper)Tema(s): Representations of groups | Invariants | Symmetry (Mathematics) | Lie groups | Algebra | Darstellungstheorie | Invariante | Lineare algebraische GruppeOtra clasificación: 20G05 (14L35 17B10 20C30 20G20 22E46)Preface xv Organization and Notation xix 1 Lie Groups and Algebraic Groups [1] 1.1 The Classical Groups [1] 1.1.1 General and Special Linear Groups [1] 1.1.2 Isometry Groups of Bilinear Forms [3] 1.1.3 Unitary Groups [7] 1.1.4 Quaternionic Groups [8] 1.1.5 Exercises [11] 1.2 The Classical Lie Algebras [13] 1.2.1 General and Special Linear Lie Algebras [13] 1.2.2 Lie Algebras Associated with Bilinear Forms [14] 1.2.3 Unitary Lie Algebras [15] 1.2.4 Quaternionic Lie Algebras [16] 1.2.5 Lie Algebras Associated with Classical Groups [17] 1.2.6 Exercises [17] 1.3 Closed Subgroups of GL(n, R) [18] 1.3.1 Topological Groups [18] 1.3.2 Exponential Map [19] 1.3.3 Lie Algebra of a Closed Subgroup of GL(n,R) [23] 1.3.4 Lie Algebras of the Classical Groups [25] 1.3.5 Exponential Coordinates on Closed Subgroups [28] 1.3.6 Differentials of Homomorphisms [30] 1.3.7 Lie Algebras and Vector Fields [31] 1.3.8 Exercises [34] 1.4 Linear Algebraic Groups [35] 1.4.1 Definitions and Examples [35] 1.4.2 Regular Functions [36] 1.4.3 Lie Algebra of an Algebraic Group [39] 1.4.4 Algebraic Groups as Lie Groups [43] 1.4.5 Exercises [45] 1.5 Rational Representations [47] 1.5.1 Definitions and Examples [47] 1.5.2 Differential of a Rational Representation [49] 1.5.3 The Adjoint Representation [53] 1.5.4 Exercises [54] 1.6 Jordan Decomposition [55] 1.6.1 Rational Representations of C [55] 1.6.2 Rational Representations of Cx [57] 1.6.3 Jordan-Chevalley Decomposition [58] 1.6.4 Exercises [61] 1.7 Real Forms of Complex Algebraic Groups [62] 1.7.1 Real Forms and Complex Conjugations [62] 1.7.2 Real Forms of the Classical Groups [65] 1.7.3 Exercises [67] 1.8 Notes [68] 2 Structure of Classical Groups [69] 2.1 Semisimple Elements [69] 2.1.1 Toral Groups [70] 2.1.2 Maximal Torus in a Classical Group [72] 2.1.3 Exercises [76] 2.2 Unipotent Elements [77] 2.2.1 Low-Rank Examples [77] 2.2.2 Unipotent Generation of Classical Groups [79] 2.2.3 Connected Groups [81] 2.2.4 Exercises [83] 2.3 Regular Representations of SL(2,C) [83] 2.3.1 Irreducible Representations of sl(2, C) [84] 2.3.2 Irreducible Regular Representations of SL(2, C) [86] 2.3.3 Complete Reducibility of SL (2, C) [88] 2.3.4 Exercises [90] 2.4 The Adjoint Representation [91] 2.4.1 Roots with Respect to a Maximal Torus [91] 2.4.2 Commutation Relations of Root Spaces [95] 2.4.3 Structure of Classical Root Systems [99] 2.4.4 Irreducibility of the Adjoint Representation [104] 2.4.5 Exercises [106] 2.5 Semisimple Lie Algebras [108] 2.5.1 Solvable Lie Algebras [108] 2.5.2 Root Space Decomposition [114] 2.5.3 Geometry of Root Systems [118] 2.5.4 Conjugacy of Cartan Subalgebras [122] 2.5.5 Exercises [125] 2.6 Notes [125] 3 Highest-Weight Theory [127] 3.1 Roots and Weights [127] 3.1.1 Weyl Group [128] 3.1.2 Root Reflections [132] 3.1.3 Weight Lattice [138] 3.1.4 Dominant Weights [141] 3.1.5 Exercises [145] 3.2 Irreducible Representations [147] 3.2.1 Theorem of the Highest Weight [148] 3.2.2 Weights of Irreducible Representations [153] 3.2.3 Lowest Weights and Dual Representations [157] 3.2.4 Symplectic and Orthogonal Representations [158] 3.2.5 Exercises [161] 3.3 Reductivity of Classical Groups [163] 3.3.1 Reductive Groups [163] 3.3.2 Casimir Operator [166] 3.3.3 Algebraic Proof of Complete Reducibility [169] 3.3.4 The Unitarian Trick [171] 3.3.5 Exercises [173] 3.4 Notes [174] 4 Algebras and Representations [175] 4.1 Representations of Associative Algebras [175] 4.1.1 Definitions and Examples [175] 4.1.2 Schur’s Lemma [180] 4.1.3 Jacobson Density Theorem [181] 4.1.4 Complete Reducibility [182] 4.1.5 Double Commutant Theorem [184] 4.1.6 Isotypic Decomposition and Multiplicities [184] 4.1.7 Characters [187] 4.1.8 Exercises [191] 4.2 Duality for Group Representations [195] 4.2.1 General Duality Theorem [195] 4.2.2 Products of Reductive Groups [197] 4.2.3 Isotypic Decomposition of O[G] [199] 4.2.4 Schur-Weyl Duality [200] 4.2.5 Commuting Algebra and Highest-Weight Vectors [203] 4.2.6 Abstract Capelli Theorem [204] 4.2.7 Exercises [206] 4.3 Group Algebras of Finite Groups [206] 4.3.1 Structure of Group Algebras [206] 4.3.2 Schur Orthogonality Relations [208] 4.3.3 Fourier Inversion Formula [209] 4.3.4 The Algebra of Central Functions [211] 4.3.5 Exercises [214] 4.4 Representations of Finite Groups [215] 4.4.1 Induced Representations [216] 4.4.2 Characters of Induced Representations [217] 4.4.3 Standard Representation of Gn [218] 4.4.4 Representations of Gk on Tensors [221] 4.4.5 Exercises [222] 4.5 Notes [224] 5 Classical Invariant Theory [225] 5.1 Polynomial Invariants for Reductive Groups [226] 5.1.1 The Ring of Invariants [226] 5.1.2 Invariant Polynomials for Gn [228] 5.1.3 Exercises [234] 5.2 Polynomial Invariants [237] 5.2.1 First Fundamental Theorems for Classical Groups [238] 5.2.2 Proof of a Basic Case [242] 5.2.3 Exercises [246] 5.3 Tensor Invariants [246] 5.3.1 Tensor Invariants for GL(V) [247] 5.3.2 Tensor Invariants for O(V) and Sp(V) [248] 5.3.3 Exercises [254] 5.4 Polynomial FFT for Classical Groups [256] 5.4.1 Invariant Polynomials as Tensors [256] 5.4.2 Proof of Polynomial FFT for GL(V) [258] 5.4.3 Proof of Polynomial FFT for O(V) and Sp(V) [258] 5.5 Irreducible Representations of Classical Groups [259] 5.5.1 Skew Duality for Classical Groups [259] 5.5.2 Fundamental Representations [268] 5.5.3 Cartan Product [272] 5.5.4 Irreducible Representations of GL(V) [273] 5.5.5 Irreducible Representations of O(V) [275] 5.5.6 Exercises [277] 5.6 Invariant Theory and Duality [278] 5.6.1 Duality and the Weyl Algebra [278] 5.6.2 GL(n)-GL(k) Schur-Weyl Duality [282] 5.6.3 O(n)-sp(k) Howe Duality [284] 5.6.4 Spherical Harmonics [287] 5.6.5 Sp(n)-so(2k) Howe Duality [290] 5.6.6 Exercises [292] 5.7 Further Applications of Invariant Theory [293] 5.7.1 Capelli Identities [293] 5.7.2 Decomposition of S(S2(V)) under GL(V) [295] 5.7.3 Decomposition of S(Λ2(V)) under GL(V) [296] 5.7.4 Exercises [298] 5.8 Notes [298] 6 Spinors [301] 6.1 Clifford Algebras [301] 6.1.1 Construction of Cliff (V) [301] 6.1.2 Spaces of Spinors [303] 6.1.3 Structure of Cliff (V) [307] 6.1.4 Exercises [310] 6.2 Spin Representations of Orthogonal Lie Algebras [311] 6.2.1 Embedding so(V) in Cliff (V) [312] 6.2.2 Spin Representations [314] 6.2.3 Exercises [315] 6.3 Spin Groups [316] 6.3.1 Action of O(V) on Cliff (V) [316] 6.3.2 Algebraically Simply Connected Groups [321] 6.3.3 Exercises [322] 6.4 Real Forms of Spin(n, C) [323] 6.4.1 Real Forms of Vector Spaces and Algebras [323] 6.4.2 Real Forms of Clifford Algebras [325] 6.4.3 Real Forms of Pin(n) and Spin(n) [325] 6.4.4 Exercises [327] 6.5 Notes [327] 7 Character Formulas [329] 7.1 Character and Dimension Formulas [329] 7.1.1 Weyl Character Formula [329] 7.1.2 Weyl Dimension Formula [334] 7.1.3 Commutant Character Formulas [337] 7.1.4 Exercises [339] 7.2 Algebraic Group Approach to the Character Formula [342] 7.2.1 Symmetric and Skew-Symmetric Functions [342] 7.2.2 Characters and Skew-Symmetric Functions [344] 7.2.3 Characters and Invariant Functions [346] 7.2.4 Casimir Operator and Invariant Functions [347] 7.2.5 Algebraic Proof of the Weyl Character Formula [352] 7.2.6 Exercises [353] 7.3 Compact Group Approach to the Character Formula [354] 7.3.1 Compact Form and Maximal Compact Torus [354] 7.3 2 Weyl Integral Formula [356] 7.3.3 Fourier Expansions of Skew Functions [358] 7.3.4 Analytic Proof of the Weyl Character Formula [360] 7.3.5 Exercises [361] 7.4 Notes [362]
8 Branching Laws [363] 8.1 Branching for Classical Groups [363] 8.1.1 Statement of Branching Laws [364] 8.1.2 Branching Patterns and Weight Multiplicities [366] 8.1.3 Exercises [368] 8.2 Branching Laws from Weyl Character Formula [370] 8.2.1 Partition Functions [370] 8.2.2 Kostant Multiplicity Formulas [371] 8.2.3 Exercises [372] 8.3 Proofs of Classical Branching Laws [373] 8.3.1 Restriction from GL(n) to GL(n — 1) [373] 8.3.2 Restriction from Spin(2n +1) to Spin(2n) [375] 8.3.3 Restriction from Spin(2n) to Spin(2n — 1) [378] 8.3.4 Restriction from Sp(n) to Sp(n —1) [379] 8.4 Notes [384] 9 Tensor Representations of GL(V) [387] 9.1 Schur-Weyl Duality [387] 9.1.1 Duality between GL(n) and Gk [388] 9.1.2 Characters of Gk [391] 9.1.3 Frobenius Formula [394] 9.1.4 Exercises [396] 9.2 Dual Reductive Pairs [399] 9.2.1 Seesaw Pairs [399] 9.2.2 Reciprocity Laws [401] 9.2.3 Schur-Weyl Duality and GL(k)-GL(n) Duality [405] 9.2.4 Exercises [406] 9.3 Young Symmetrizers and Weyl Modules [407] 9.3.1 Tableaux and Symmetrizers [407] 9.3.2 Weyl Modules [412] 9.3.3 Standard Tableaux [414] 9.3.4 Projections onto Isotypic Components [416] 9.3.5 Littlewood-Richardson Rule [418] 9.3.6 Exercises [421] 9.4 Notes [423] 10 Tensor Representations of O(V) and Sp(V) [425] 10.1 Commuting Algebras on Tensor Spaces [425] 10.1.1 Centralizer Algebra [426] 10.1.2 Generators and Relations [432] 10.1.3 Exercises [434] 10.2 Decomposition of Harmonic Tensors [435] 10.2.1 Harmonic Tensors [435] 10.2.2 Harmonic Extreme Tensors [436] 10.2.3 Decomposition of Harmonics for Sp(V) [440] 10.2.4 Decomposition of Harmonics for O(2l +1) [442] 10.2.5 Decomposition of Harmonics for O(2l) [446] 10.2.6 Exercises [451] 10.3 Riemannian Curvature Tensors [451] 10.3.1 The Space of Curvature Tensors [453] 10.3.2 Orthogonal Decomposition of Curvature Tensors [455] 10.3.3 The Space of Weyl Curvature Tensors [458] 10.3.4 Exercises [460] 10.4 Invariant Theory and Knot Polynomials [461] 10.4.1 The Braid Relations [461] 10.4.2 Orthogonal Invariants and the Yang-Baxter Equation [463] 10.4.3 The Braid Group [464] 10.4.4 The Jones Polynomial [469] 10.4.5 Exercises [475] 10.5 Notes [476] 11 Algebraic Groups and Homogeneous Spaces [479] 11.1 General Properties of Linear Algebraic Groups [479] 11.1.1 Algebraic Groups as Affine Varieties [479] 11.1.2 Subgroups and Homomorphisms [481] 11.1.3 Group Structures on Affine Varieties [484] 11.1.4 Quotient Groups [485] 11.1.5 Exercises [490] 11.2 Structure of Algebraic Groups [491] 11.2.1 Commutative Algebraic Groups [491] 11.2.2 Unipotent Radical [493] 11.2.3 Connected Algebraic Groups and Lie Groups [496] 11.2.4 Simply Connected Semisimple Groups [497] 11.2.5 Exercises [500] 11.3 Homogeneous Spaces [500] 11.3.1 G-Spaces and Orbits [500] 11.3.2 Flag Manifolds [501] 11.3.3 Involutions and Symmetric Spaces [506] 11.3.4 Involutions of Classical Groups [507] 11.3.5 Classical Symmetric Spaces [510] 11.3.6 Exercises [516] 11.4 Borel Subgroups [519] 11.4.1 Solvable Groups [519] 11.4.2 Lie-Kolchin Theorem [520] 11.4.3 Structure of Connected Solvable Groups [522] 11.4.4 Conjugacy of Borel Subgroups [524] 11.4.5 Centralizer of a Torus [525] 11.4.6 Weyl Group and Regular Semisimple Conjugacy Classes [526] 11.4.7 Exercises [530] 11.5 Further Properties of Real Forms [531] 11.5.1 Groups with a Compact Real Form [531] 11.5.2 Polar Decomposition by a Compact Form [536] 11.6 Gauss Decomposition [538] 11.6.1 Gauss Decomposition of GL(n, C) [538] 11.6.2 Gauss Decomposition of an Algebraic Group [540] 11.6.3 Gauss Decomposition for Real Forms [541] 11.6.4 Exercises [543] 11.7 Notes [543] 12 Representations on Spaces of Regular Functions [545] 12.1 Some General Results [545] 12.1.1 Isotypic Decomposition of O [X] [546] 12.1.2 Frobenius Reciprocity [548] 12.1.3 Function Models for Irreducible Representations [549] 12.1.4 Exercises [550] 12.2 Multiplicity-Free Spaces [551] 12.2.1 Multiplicities and B-Orbits [552] 12.2.2 B-Eigenfunctions for Linear Actions [553] 12.2.3 Branching from GL(n) to GL(n — 1) [554] 12.2.4 Second Fundamental Theorems [557] 12.2.5 Exercises [563] 12.3 Regular Functions on Symmetric Spaces [566] 12.3.1 Iwasawa Decomposition [566] 12.3.2 Examples of Iwasawa Decompositions [575] 12.3.3 Spherical Representations [585] 12.3.4 Exercises [587] 12.4 Isotropy Representations of Symmetric Spaces [588] 12.4.1 A Theorem of Kostant and Rallis [589] 12.4.2 Invariant Theory of Reflection Groups [590] 12.4.3 Classical Examples [593] 12.4.4 Some Results from Algebraic Geometry [597] 12.4.5 Proof of the Kostant-Rallis Theorem [601] 12.4.6 Some Remarks on the Proof [605] 12.4.7 Exercises [607] 12.5 Notes [609] A Algebraic Geometry [611] A.1 Affine Algebraic Sets [611] A. 1.1 Basic Properties [611] A. 1.2 Zariski Topology [615] A. 1.3 Products of Affine Sets [616] A. 1.4 Principal Open Sets [617] A. 1.5 Irreducible Components [617] A. 1.6 Transcendence Degree and Dimension [619] A. 1.7 Exercises [621] A.2 Maps of Algebraic Sets [622] A.2.1 Rational Maps [622] A.2.2 Extensions of Homomorphisms [623] A.2.3 Image of a Dominant Map [626] A.2.4 Factorization of a Regular Map [626] A.2.5 Exercises [627] A.3 Tangent Spaces [628] A.3.1 Tangent Space and Differentials of Maps [628] A.3.2 Vector Fields [630] A.3.3 Dimension [630] A.3.4 Differential Criterion for Dominance [632] A. 4 Projective and Quasiprojective Sets [635] A.4.1 Basic Definitions [635] A.4.2 Products of Projective Sets [637] A. 4.3 Regular Functions and Maps [638] B Linear and Multilinear Algebra [643] B. l Jordan Decomposition [643] B. 1.1 Primary Projections [643] B.1.2 Additive Jordan Decomposition [644] B.1.3 Multiplicative Jordan Decomposition [645] B. 2 Multilinear Algebra [645] B.2.1 Bilinear Forms [646] B.2.2 Tensor Products [647] B.2.3 Symmetric Tensors [650] B.2.4 Alternating Tensors [653] B.2.5 Determinants and Gauss Decomposition [654] B.2.6 Pfaffians and Skew-Symmetric Matrices [656] B. 2.7 Irreducibility of Determinants and Pfaffians [659] C Associative Algebras and Lie Algebras [661] C. l Some Associative Algebras [661] C. 1.1 Filtered and Graded Algebras [661] C.l.2 Tensor Algebra [663] C.1.3 Symmetric Algebra [663] C.1.4 Exterior Algebra [666] C.l.5 Exercises [668] C.2 Universal Enveloping Algebras [668] C.2.1 Lie Algebras [668] C.2.2 Poincaré-Birkhoff-Witt Theorem [670] C.2.3 Adjoint Representation of Enveloping Algebra [672] D Manifolds and Lie Groups [675] D.l C°° Manifolds [675] D. 1.1 Basic Definitions [675] D. 1.2 Tangent Space [680] D.1.3 Differential Forms and Integration [683] D.1.4 Exercises [686] D.2 Lie Groups [687] D.2.1 Basic Definitions [687] D.2.2 Lie Algebra of a Lie Group [688] D.2.3 Homogeneous Spaces [691] D.2.4 Integration on Lie Groups and Homogeneous Spaces [692] D.2.5 Exercises [696] References [697] Index of Symbols [705] Subject Index [709]
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 20 G653 (Browse shelf) | Available | A-9290 |
"Symmetry, in the title of this book, should be understood as the geometry of Lie (and algebraic) group actions"--P. xv.
"Based on 'Representations and invariants of the classical groups', Roe Goodman and Nolan R. Wallach, Cambridge University Press, 1998, third corrected printing 2003"--P. [iv].
Incluye referencias bibliográficas (p. 697-703) e índices.
Preface xv --
Organization and Notation xix --
1 Lie Groups and Algebraic Groups [1] --
1.1 The Classical Groups [1] --
1.1.1 General and Special Linear Groups [1] --
1.1.2 Isometry Groups of Bilinear Forms [3] --
1.1.3 Unitary Groups [7] --
1.1.4 Quaternionic Groups [8] --
1.1.5 Exercises [11] --
1.2 The Classical Lie Algebras [13] --
1.2.1 General and Special Linear Lie Algebras [13] --
1.2.2 Lie Algebras Associated with Bilinear Forms [14] --
1.2.3 Unitary Lie Algebras [15] --
1.2.4 Quaternionic Lie Algebras [16] --
1.2.5 Lie Algebras Associated with Classical Groups [17] --
1.2.6 Exercises [17] --
1.3 Closed Subgroups of GL(n, R) [18] --
1.3.1 Topological Groups [18] --
1.3.2 Exponential Map [19] --
1.3.3 Lie Algebra of a Closed Subgroup of GL(n,R) [23] --
1.3.4 Lie Algebras of the Classical Groups [25] --
1.3.5 Exponential Coordinates on Closed Subgroups [28] --
1.3.6 Differentials of Homomorphisms [30] --
1.3.7 Lie Algebras and Vector Fields [31] --
1.3.8 Exercises [34] --
1.4 Linear Algebraic Groups [35] --
1.4.1 Definitions and Examples [35] --
1.4.2 Regular Functions [36] --
1.4.3 Lie Algebra of an Algebraic Group [39] --
1.4.4 Algebraic Groups as Lie Groups [43] --
1.4.5 Exercises [45] --
1.5 Rational Representations [47] --
1.5.1 Definitions and Examples [47] --
1.5.2 Differential of a Rational Representation [49] --
1.5.3 The Adjoint Representation [53] --
1.5.4 Exercises [54] --
1.6 Jordan Decomposition [55] --
1.6.1 Rational Representations of C [55] --
1.6.2 Rational Representations of Cx [57] --
1.6.3 Jordan-Chevalley Decomposition [58] --
1.6.4 Exercises [61] --
1.7 Real Forms of Complex Algebraic Groups [62] --
1.7.1 Real Forms and Complex Conjugations [62] --
1.7.2 Real Forms of the Classical Groups [65] --
1.7.3 Exercises [67] --
1.8 Notes [68] --
2 Structure of Classical Groups [69] --
2.1 Semisimple Elements [69] --
2.1.1 Toral Groups [70] --
2.1.2 Maximal Torus in a Classical Group [72] --
2.1.3 Exercises [76] --
2.2 Unipotent Elements [77] --
2.2.1 Low-Rank Examples [77] --
2.2.2 Unipotent Generation of Classical Groups [79] --
2.2.3 Connected Groups [81] --
2.2.4 Exercises [83] --
2.3 Regular Representations of SL(2,C) [83] --
2.3.1 Irreducible Representations of sl(2, C) [84] --
2.3.2 Irreducible Regular Representations of SL(2, C) [86] --
2.3.3 Complete Reducibility of SL (2, C) [88] --
2.3.4 Exercises [90] --
2.4 The Adjoint Representation [91] --
2.4.1 Roots with Respect to a Maximal Torus [91] --
2.4.2 Commutation Relations of Root Spaces [95] --
2.4.3 Structure of Classical Root Systems [99] --
2.4.4 Irreducibility of the Adjoint Representation [104] --
2.4.5 Exercises [106] --
2.5 Semisimple Lie Algebras [108] --
2.5.1 Solvable Lie Algebras [108] --
2.5.2 Root Space Decomposition [114] --
2.5.3 Geometry of Root Systems [118] --
2.5.4 Conjugacy of Cartan Subalgebras [122] --
2.5.5 Exercises [125] --
2.6 Notes [125] --
3 Highest-Weight Theory [127] --
3.1 Roots and Weights [127] --
3.1.1 Weyl Group [128] --
3.1.2 Root Reflections [132] --
3.1.3 Weight Lattice [138] --
3.1.4 Dominant Weights [141] --
3.1.5 Exercises [145] --
3.2 Irreducible Representations [147] --
3.2.1 Theorem of the Highest Weight [148] --
3.2.2 Weights of Irreducible Representations [153] --
3.2.3 Lowest Weights and Dual Representations [157] --
3.2.4 Symplectic and Orthogonal Representations [158] --
3.2.5 Exercises [161] --
3.3 Reductivity of Classical Groups [163] --
3.3.1 Reductive Groups [163] --
3.3.2 Casimir Operator [166] --
3.3.3 Algebraic Proof of Complete Reducibility [169] --
3.3.4 The Unitarian Trick [171] --
3.3.5 Exercises [173] --
3.4 Notes [174] --
4 Algebras and Representations [175] --
4.1 Representations of Associative Algebras [175] --
4.1.1 Definitions and Examples [175] --
4.1.2 Schur’s Lemma [180] --
4.1.3 Jacobson Density Theorem [181] --
4.1.4 Complete Reducibility [182] --
4.1.5 Double Commutant Theorem [184] --
4.1.6 Isotypic Decomposition and Multiplicities [184] --
4.1.7 Characters [187] --
4.1.8 Exercises [191] --
4.2 Duality for Group Representations [195] --
4.2.1 General Duality Theorem [195] --
4.2.2 Products of Reductive Groups [197] --
4.2.3 Isotypic Decomposition of O[G] [199] --
4.2.4 Schur-Weyl Duality [200] --
4.2.5 Commuting Algebra and Highest-Weight Vectors [203] --
4.2.6 Abstract Capelli Theorem [204] --
4.2.7 Exercises [206] --
4.3 Group Algebras of Finite Groups [206] --
4.3.1 Structure of Group Algebras [206] --
4.3.2 Schur Orthogonality Relations [208] --
4.3.3 Fourier Inversion Formula [209] --
4.3.4 The Algebra of Central Functions [211] --
4.3.5 Exercises [214] --
4.4 Representations of Finite Groups [215] --
4.4.1 Induced Representations [216] --
4.4.2 Characters of Induced Representations [217] --
4.4.3 Standard Representation of Gn [218] --
4.4.4 Representations of Gk on Tensors [221] --
4.4.5 Exercises [222] --
4.5 Notes [224] --
5 Classical Invariant Theory [225] --
5.1 Polynomial Invariants for Reductive Groups [226] --
5.1.1 The Ring of Invariants [226] --
5.1.2 Invariant Polynomials for Gn [228] --
5.1.3 Exercises [234] --
5.2 Polynomial Invariants [237] --
5.2.1 First Fundamental Theorems for Classical Groups [238] --
5.2.2 Proof of a Basic Case [242] --
5.2.3 Exercises [246] --
5.3 Tensor Invariants [246] --
5.3.1 Tensor Invariants for GL(V) [247] --
5.3.2 Tensor Invariants for O(V) and Sp(V) [248] --
5.3.3 Exercises [254] --
5.4 Polynomial FFT for Classical Groups [256] --
5.4.1 Invariant Polynomials as Tensors [256] --
5.4.2 Proof of Polynomial FFT for GL(V) [258] --
5.4.3 Proof of Polynomial FFT for O(V) and Sp(V) [258] --
5.5 Irreducible Representations of Classical Groups [259] --
5.5.1 Skew Duality for Classical Groups [259] --
5.5.2 Fundamental Representations [268] --
5.5.3 Cartan Product [272] --
5.5.4 Irreducible Representations of GL(V) [273] --
5.5.5 Irreducible Representations of O(V) [275] --
5.5.6 Exercises [277] --
5.6 Invariant Theory and Duality [278] --
5.6.1 Duality and the Weyl Algebra [278] --
5.6.2 GL(n)-GL(k) Schur-Weyl Duality [282] --
5.6.3 O(n)-sp(k) Howe Duality [284] --
5.6.4 Spherical Harmonics [287] --
5.6.5 Sp(n)-so(2k) Howe Duality [290] --
5.6.6 Exercises [292] --
5.7 Further Applications of Invariant Theory [293] --
5.7.1 Capelli Identities [293] --
5.7.2 Decomposition of S(S2(V)) under GL(V) [295] --
5.7.3 Decomposition of S(Λ2(V)) under GL(V) [296] --
5.7.4 Exercises [298] --
5.8 Notes [298] --
6 Spinors [301] --
6.1 Clifford Algebras [301] --
6.1.1 Construction of Cliff (V) [301] --
6.1.2 Spaces of Spinors [303] --
6.1.3 Structure of Cliff (V) [307] --
6.1.4 Exercises [310] --
6.2 Spin Representations of Orthogonal Lie Algebras [311] --
6.2.1 Embedding so(V) in Cliff (V) [312] --
6.2.2 Spin Representations [314] --
6.2.3 Exercises [315] --
6.3 Spin Groups [316] --
6.3.1 Action of O(V) on Cliff (V) [316] --
6.3.2 Algebraically Simply Connected Groups [321] --
6.3.3 Exercises [322] --
6.4 Real Forms of Spin(n, C) [323] --
6.4.1 Real Forms of Vector Spaces and Algebras [323] --
6.4.2 Real Forms of Clifford Algebras [325] --
6.4.3 Real Forms of Pin(n) and Spin(n) [325] --
6.4.4 Exercises [327] --
6.5 Notes [327] --
7 Character Formulas [329] --
7.1 Character and Dimension Formulas [329] --
7.1.1 Weyl Character Formula [329] --
7.1.2 Weyl Dimension Formula [334] --
7.1.3 Commutant Character Formulas [337] --
7.1.4 Exercises [339] --
7.2 Algebraic Group Approach to the Character Formula [342] --
7.2.1 Symmetric and Skew-Symmetric Functions [342] --
7.2.2 Characters and Skew-Symmetric Functions [344] --
7.2.3 Characters and Invariant Functions [346] --
7.2.4 Casimir Operator and Invariant Functions [347] --
7.2.5 Algebraic Proof of the Weyl Character Formula [352] --
7.2.6 Exercises [353] --
7.3 Compact Group Approach to the Character Formula [354] --
7.3.1 Compact Form and Maximal Compact Torus [354] --
7.3 2 Weyl Integral Formula [356] --
7.3.3 Fourier Expansions of Skew Functions [358] --
7.3.4 Analytic Proof of the Weyl Character Formula [360] --
7.3.5 Exercises [361] --
7.4 Notes [362] --
8 Branching Laws [363] --
8.1 Branching for Classical Groups [363] --
8.1.1 Statement of Branching Laws [364] --
8.1.2 Branching Patterns and Weight Multiplicities [366] --
8.1.3 Exercises [368] --
8.2 Branching Laws from Weyl Character Formula [370] --
8.2.1 Partition Functions [370] --
8.2.2 Kostant Multiplicity Formulas [371] --
8.2.3 Exercises [372] --
8.3 Proofs of Classical Branching Laws [373] --
8.3.1 Restriction from GL(n) to GL(n — 1) [373] --
8.3.2 Restriction from Spin(2n +1) to Spin(2n) [375] --
8.3.3 Restriction from Spin(2n) to Spin(2n — 1) [378] --
8.3.4 Restriction from Sp(n) to Sp(n —1) [379] --
8.4 Notes [384] --
9 Tensor Representations of GL(V) [387] --
9.1 Schur-Weyl Duality [387] --
9.1.1 Duality between GL(n) and Gk [388] --
9.1.2 Characters of Gk [391] --
9.1.3 Frobenius Formula [394] --
9.1.4 Exercises [396] --
9.2 Dual Reductive Pairs [399] --
9.2.1 Seesaw Pairs [399] --
9.2.2 Reciprocity Laws [401] --
9.2.3 Schur-Weyl Duality and GL(k)-GL(n) Duality [405] --
9.2.4 Exercises [406] --
9.3 Young Symmetrizers and Weyl Modules [407] --
9.3.1 Tableaux and Symmetrizers [407] --
9.3.2 Weyl Modules [412] --
9.3.3 Standard Tableaux [414] --
9.3.4 Projections onto Isotypic Components [416] --
9.3.5 Littlewood-Richardson Rule [418] --
9.3.6 Exercises [421] --
9.4 Notes [423] --
10 Tensor Representations of O(V) and Sp(V) [425] --
10.1 Commuting Algebras on Tensor Spaces [425] --
10.1.1 Centralizer Algebra [426] --
10.1.2 Generators and Relations [432] --
10.1.3 Exercises [434] --
10.2 Decomposition of Harmonic Tensors [435] --
10.2.1 Harmonic Tensors [435] --
10.2.2 Harmonic Extreme Tensors [436] --
10.2.3 Decomposition of Harmonics for Sp(V) [440] --
10.2.4 Decomposition of Harmonics for O(2l +1) [442] --
10.2.5 Decomposition of Harmonics for O(2l) [446] --
10.2.6 Exercises [451] --
10.3 Riemannian Curvature Tensors [451] --
10.3.1 The Space of Curvature Tensors [453] --
10.3.2 Orthogonal Decomposition of Curvature Tensors [455] --
10.3.3 The Space of Weyl Curvature Tensors [458] --
10.3.4 Exercises [460] --
10.4 Invariant Theory and Knot Polynomials [461] --
10.4.1 The Braid Relations [461] --
10.4.2 Orthogonal Invariants and the Yang-Baxter Equation [463] --
10.4.3 The Braid Group [464] --
10.4.4 The Jones Polynomial [469] --
10.4.5 Exercises [475] --
10.5 Notes [476] --
11 Algebraic Groups and Homogeneous Spaces [479] --
11.1 General Properties of Linear Algebraic Groups [479] --
11.1.1 Algebraic Groups as Affine Varieties [479] --
11.1.2 Subgroups and Homomorphisms [481] --
11.1.3 Group Structures on Affine Varieties [484] --
11.1.4 Quotient Groups [485] --
11.1.5 Exercises [490] --
11.2 Structure of Algebraic Groups [491] --
11.2.1 Commutative Algebraic Groups [491] --
11.2.2 Unipotent Radical [493] --
11.2.3 Connected Algebraic Groups and Lie Groups [496] --
11.2.4 Simply Connected Semisimple Groups [497] --
11.2.5 Exercises [500] --
11.3 Homogeneous Spaces [500] --
11.3.1 G-Spaces and Orbits [500] --
11.3.2 Flag Manifolds [501] --
11.3.3 Involutions and Symmetric Spaces [506] --
11.3.4 Involutions of Classical Groups [507] --
11.3.5 Classical Symmetric Spaces [510] --
11.3.6 Exercises [516] --
11.4 Borel Subgroups [519] --
11.4.1 Solvable Groups [519] --
11.4.2 Lie-Kolchin Theorem [520] --
11.4.3 Structure of Connected Solvable Groups [522] --
11.4.4 Conjugacy of Borel Subgroups [524] --
11.4.5 Centralizer of a Torus [525] --
11.4.6 Weyl Group and Regular Semisimple Conjugacy Classes [526] --
11.4.7 Exercises [530] --
11.5 Further Properties of Real Forms [531] --
11.5.1 Groups with a Compact Real Form [531] --
11.5.2 Polar Decomposition by a Compact Form [536] --
11.6 Gauss Decomposition [538] --
11.6.1 Gauss Decomposition of GL(n, C) [538] --
11.6.2 Gauss Decomposition of an Algebraic Group [540] --
11.6.3 Gauss Decomposition for Real Forms [541] --
11.6.4 Exercises [543] --
11.7 Notes [543] --
12 Representations on Spaces of Regular Functions [545] --
12.1 Some General Results [545] --
12.1.1 Isotypic Decomposition of O [X] [546] --
12.1.2 Frobenius Reciprocity [548] --
12.1.3 Function Models for Irreducible Representations [549] --
12.1.4 Exercises [550] --
12.2 Multiplicity-Free Spaces [551] --
12.2.1 Multiplicities and B-Orbits [552] --
12.2.2 B-Eigenfunctions for Linear Actions [553] --
12.2.3 Branching from GL(n) to GL(n — 1) [554] --
12.2.4 Second Fundamental Theorems [557] --
12.2.5 Exercises [563] --
12.3 Regular Functions on Symmetric Spaces [566] --
12.3.1 Iwasawa Decomposition [566] --
12.3.2 Examples of Iwasawa Decompositions [575] --
12.3.3 Spherical Representations [585] --
12.3.4 Exercises [587] --
12.4 Isotropy Representations of Symmetric Spaces [588] --
12.4.1 A Theorem of Kostant and Rallis [589] --
12.4.2 Invariant Theory of Reflection Groups [590] --
12.4.3 Classical Examples [593] --
12.4.4 Some Results from Algebraic Geometry [597] --
12.4.5 Proof of the Kostant-Rallis Theorem [601] --
12.4.6 Some Remarks on the Proof [605] --
12.4.7 Exercises [607] --
12.5 Notes [609] --
A Algebraic Geometry [611] --
A.1 Affine Algebraic Sets [611] --
A. 1.1 Basic Properties [611] --
A. 1.2 Zariski Topology [615] --
A. 1.3 Products of Affine Sets [616] --
A. 1.4 Principal Open Sets [617] --
A. 1.5 Irreducible Components [617] --
A. 1.6 Transcendence Degree and Dimension [619] --
A. 1.7 Exercises [621] --
A.2 Maps of Algebraic Sets [622] --
A.2.1 Rational Maps [622] --
A.2.2 Extensions of Homomorphisms [623] --
A.2.3 Image of a Dominant Map [626] --
A.2.4 Factorization of a Regular Map [626] --
A.2.5 Exercises [627] --
A.3 Tangent Spaces [628] --
A.3.1 Tangent Space and Differentials of Maps [628] --
A.3.2 Vector Fields [630] --
A.3.3 Dimension [630] --
A.3.4 Differential Criterion for Dominance [632] --
A. 4 Projective and Quasiprojective Sets [635] --
A.4.1 Basic Definitions [635] --
A.4.2 Products of Projective Sets [637] --
A. 4.3 Regular Functions and Maps [638] --
B Linear and Multilinear Algebra [643] --
B. l Jordan Decomposition [643] --
B. 1.1 Primary Projections [643] --
B.1.2 Additive Jordan Decomposition [644] --
B.1.3 Multiplicative Jordan Decomposition [645] --
B. 2 Multilinear Algebra [645] --
B.2.1 Bilinear Forms [646] --
B.2.2 Tensor Products [647] --
B.2.3 Symmetric Tensors [650] --
B.2.4 Alternating Tensors [653] --
B.2.5 Determinants and Gauss Decomposition [654] --
B.2.6 Pfaffians and Skew-Symmetric Matrices [656] --
B. 2.7 Irreducibility of Determinants and Pfaffians [659] --
C Associative Algebras and Lie Algebras [661] --
C. l Some Associative Algebras [661] --
C. 1.1 Filtered and Graded Algebras [661] --
C.l.2 Tensor Algebra [663] --
C.1.3 Symmetric Algebra [663] --
C.1.4 Exterior Algebra [666] --
C.l.5 Exercises [668] --
C.2 Universal Enveloping Algebras [668] --
C.2.1 Lie Algebras [668] --
C.2.2 Poincaré-Birkhoff-Witt Theorem [670] --
C.2.3 Adjoint Representation of Enveloping Algebra [672] --
D Manifolds and Lie Groups [675] --
D.l C°° Manifolds [675] --
D. 1.1 Basic Definitions [675] --
D. 1.2 Tangent Space [680] --
D.1.3 Differential Forms and Integration [683] --
D.1.4 Exercises [686] --
D.2 Lie Groups [687] --
D.2.1 Basic Definitions [687] --
D.2.2 Lie Algebra of a Lie Group [688] --
D.2.3 Homogeneous Spaces [691] --
D.2.4 Integration on Lie Groups and Homogeneous Spaces [692] --
D.2.5 Exercises [696] --
References [697] --
Index of Symbols [705] --
Subject Index [709] --
MR, MR2522486
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