Representation Theory of the Symmetric Groups: The Okounkov-Vershik Approach, Character Formulas, and Partition Algebras (Cambridge Studies in Advanced Mathematics, Series Number 121)

"The representation theory of the symmetric groups is a classical topic that, since the pioneering work of Frobenius, Schur and Young, has grown into a huge body of theory, with many important connections to other areas of mathematics and physics. This self-contained book provides a detailed introduction to the subject, covering classical topics such as the Littlewood-Richardson rule and the Schur-Weyl duality. Importantly the authors also present many recent advances in the area, including Lassalle's character formulas, the theory of partition algebras, and an exhaustive exposition of the approach developed by A. M. Vershik and A. Okounkov. A wealth of examples and exercises makes this an ideal textbook for graduate students. It will also serve as a useful reference for more experienced researchers across a range of areas, including algebra, computer science, statistical mechanics and theoretical physics"--Provided by publisher. Cover 1 Half-title 3 Title 5 Copyright 6 Dedication 7 Contents 9 Preface 15 1 Representation theory of finite groups 19 1.1 Basic facts 19 1.1.1 Representations 19 1.1.2 Examples 20 1.1.3 Intertwining operators 22 1.1.4 Direct sums and complete reducibility 23 1.1.5 The adjoint representation 24 1.1.6 Matrix coefficients 25 1.1.7 Tensor products 26 1.1.8 Cyclic and invariant vectors 28 1.2 Schur's lemma and the commutant 29 1.2.1 Schur's lemma 29 1.2.2 Multiplicities and isotypic components 30 1.2.3 Finite dimensional algebras 32 1.2.4 The structure of the commutant 34 1.2.5 Another description of HomG(W, V) 36 1.3 Characters and the projection formula 37 1.3.1 The trace 37 1.3.2 Central functions and characters 38 1.3.3 Central projection formulas 40 1.4 Permutation representations 45 1.4.1 Wielandt's lemma 45 1.4.2 Symmetric actions and Gelfand's lemma 48 1.4.3 Frobenius reciprocity for a permutation representation 48 1.4.4 The structure of the commutant ofa permutation representation 53 1.5 The group algebra and the Fourier transform 55 1.5.1 L(G) and the convolution 55 1.5.2 The Fourier transform 60 1.5.3 Algebras of bi-K-invariant functions 64 1.6 Induced representations 69 1.6.1 Definitions and examples 69 1.6.2 First properties of induced representations 71 1.6.3 Frobenius reciprocity 73 1.6.4 Mackey's lemma and the intertwining number theorem 75 2 The theory of Gelfand-Tsetlin bases 77 2.1 Algebras of conjugacy invariant functions 77 2.1.1 Conjugacy invariant functions 77 2.1.2 Multiplicity-free subgroups 82 2.1.3 Greenhalgebras 83 2.2 Gelfand-Tsetlin bases 87 2.2.1 Branching graphs and Gelfand-Tsetlin bases 87 2.2.2 Gelfand-Tsetlin algebras 89 2.2.3 Gelfand-Tsetlin bases for permutation representations 93 3 The Okounkov-Vershik approach 97 3.1 The Young poset 97 3.1.1 Partitions and conjugacy classes in Sn 97 3.1.2 Young frames 99 3.1.3 Young tableaux 99 3.1.4 Coxeter generators 101 3.1.5 The content of a tableau 103 3.1.6 The Young poset 107 3.2 The Young–Jucys–Murphy elements and a Gelfand-Tsetlin basis for Sn 109 3.2.1 The Young-Jucys-Murphy elements 110 3.2.2 Marked permutations 110 3.2.3 Olshanskii's theorem 113 3.2.4 A characterization of the YJM elements 116 3.3 The spectrum of the Young–Jucys–Murphy elements and the branching graph of Sn 118 3.3.1 The weight of a Young basis vector 118 3.3.2 The spectrum of the YJM elements 120 3.3.3 Spec(n) = Cont(n) 122 3.4 The irreducible representations of Sn 128 3.4.1 Young's seminormal form 128 3.4.2 Young's orthogonal form 130 3.4.3 The Murnaghan-Nakayama rule for a cycle 134 3.4.4 The Young seminormal units 136 3.5 Skew representations and the Murnhagan-Nakayama rule 139 3.5.1 Skew shapes 139 3.5.2 Skew representations of the symmetric group 141 3.5.3 Basic properties of the skew representations and Pieri's rule 144 3.5.4 Skew hooks 148 3.5.5 The Murnaghan-Nakayama rule 150 3.6 The Frobenius-Young correspondence 153 3.6.1 The dominance and the lexicographic orders for partitions 153 3.6.2 The Young modules 156 3.6.3 The Frobenius-Young correspondence 158 3.6.4 Radon transforms between Young's modules 162 3.7 The Young rule 163 3.7.1 Semistandard Young tableaux 163 3.7.2 The reduced Young poset 166 3.7.3 The Young rule 168 3.7.4 A Greenhalgebra with the symmetric group 171 4 Symmetric functions 174 4.1 Symmetric polynomials 174 4.1.1 More notation and results on partitions 174 4.1.2 Monomial symmetric polynomials 175 4.1.3 Elementary, complete and power sumssymmetric polynomials 177 4.1.4 The fundamental theorem on symmetric polynomials 183 4.1.5 An involutive map 185 4.1.6 Antisymmetric polynomials 186 4.1.7 The algebra of symmetric functions 188 4.2 The Frobenius character formula 189 4.2.1 On the characters of the Young modules 189 4.2.2 Cauchy's formula 191 4.2.3 Frobenius character formula 192 4.2.4 Applications of Frobenius character formula 197 4.3 Schur polynomials 203 4.3.1 Definition of Schur polynomials 203 4.3.2 A scalar product 206 4.3.3 The characteristic map 207 4.3.4 Determinantal identities 211 4.4 The Theorem of Jucys and Murphy 217 4.4.1 Minimal decompositions of permutations asproducts of transpositions 217 4.4.2 The Theorem of Jucys and Murphy 222 4.4.3 Bernoulli and Stirling numbers 226 4.4.4 Garsia's expression for chilambda 231 5 Content evaluation and character theory of the symmetric group 239 5.1 Binomial coefficients 239 5.1.1 Ordinary binomial coefficients: basic identities 239 5.1.2 Binomial coefficients: some technical results 242 5.1.3 Lassalle's coefficients 246 5.1.4 Binomial coefficients associated with partitions 251 5.1.5 Lassalle's symmetric function 253 5.2 Taylor series for the Frobenius quotient 256 5.2.1 The Frobenius function 256 5.2.2 Lagrange interpolation formula 260 5.2.3 The Taylor series at infinity for the Frobenius quotient 263 5.2.4 Some explicit formulas for the coefficients... 268 5.3 Lassalle’s explicit formulas for the characters of the symmetric group 270 5.3.1 Conjugacy classes with one nontrivial cycle 270 5.3.2 Conjugacy classes with two nontrivial cycles 272 5.3.3 The explicit formula for an arbitrary conjugacy class 276 5.4 Central characters and class symmetric functions 281 5.4.1 Central characters 282 5.4.2 Class symmetric functions 285 5.4.3 Kerov-Vershik asymptotics 289 6 Radon transforms, Specht modules and the Littlewood–Richardson rule 291 6.1 The combinatorics of pairs of partitions and the Littlewood–Richardson rule 292 6.1.1 Words and lattice permutations 292 6.1.2 Pairs of partitions 295 6.1.3 James' combinatorial theorem 299 6.1.4 Littlewood-Richardson tableaux 302 6.1.5 The Littlewood-Richardson rule 308 6.2 Radon transforms, Specht modules and orthogonal decompositions of Young modules 311 6.2.1 Generalized Specht modules 311 6.2.2 A family of Radon transforms 316 6.2.3 Decomposition theorems 321 6.2.4 The Gelfand-Tsetlin bases for Ma revisited 325 7 Finite dimensional *-algebras 332 7.1 Finite dimensional algebras of operators 332 7.1.1 Finite dimensional *-algebras 332 7.1.2 Burnside's theorem 334 7.2 Schur's lemma and the commutant 336 7.2.1 Schur's lemma for a linear algebra 336 7.2.2 The commutant of a *-algebra 338 7.3 The double commutant theorem and the structure of a finite dimensional *-algebra 341 7.3.1 Tensor product of algebras 341 7.3.2 The double commutant theorem 343 7.3.3 Structure of finite dimensional *-algebras 345 7.3.4 Matrix units and central elements 349 7.4 Ideals and representation theory of a finite dimensional *-algebra 350 7.4.1 Representation theory of End(V) 350 7.4.2 Representation theory of finite dimensional *-algebras 352 7.4.3 The Fourier transform 354 7.4.4 Complete reducibility of finite dimensional *-algebras 354 7.4.5 The regular representation of a *-algebra 356 7.4.6 Representation theory of finite groups revisited 357 7.5 Subalgebras and reciprocity laws 359 7.5.1 Subalgebras and Bratteli diagrams 359 7.5.2 The centralizer of a subalgebra 361 7.5.3 A reciprocity law for restriction 363 7.5.4 A reciprocity law for induction 365 7.5.5 Iterated tensor product of permutation representations 369 8 Schur-Weyl dualities and the partition algebra 375 8.1 Symmetric and antisymmetric tensors 375 8.1.1 Iterated tensor product 376 8.1.2 The action of Sk on... 378 8.1.3 Symmetric tensors 379 8.1.4 Antisymmetric tensors 383 8.2 Classical Schur-Weyl duality 386 8.2.1 The general linear group GL(n, C) 386 8.2.2 Duality between GL(n,C) and Sk 392 8.2.3 Clebsch-Gordan decomposition and branching formulas 396 8.3 The partition algebra 402 8.3.1 The partition monoid 403 8.3.2 The partition algebra 409 8.3.3 Schur-Weyl duality for the partition algebra 411 References 420 Index 427 Cover......Page 1 Half-title......Page 3 Title......Page 5 Copyright......Page 6 Dedication......Page 7 Contents......Page 9 Preface......Page 15 1.1.1 Representations......Page 19 1.1.2 Examples......Page 20 1.1.3 Intertwining operators......Page 22 1.1.4 Direct sums and complete reducibility......Page 23 1.1.5 The adjoint representation......Page 24 1.1.6 Matrix coefficients......Page 25 1.1.7 Tensor products......Page 26 1.1.8 Cyclic and invariant vectors......Page 28 1.2.1 Schur's lemma......Page 29 1.2.2 Multiplicities and isotypic components......Page 30 1.2.3 Finite dimensional algebras......Page 32 1.2.4 The structure of the commutant......Page 34 1.2.5 Another description of HomG(W, V)......Page 36 1.3.1 The trace......Page 37 1.3.2 Central functions and characters......Page 38 1.3.3 Central projection formulas......Page 40 1.4.1 Wielandt's lemma......Page 45 1.4.3 Frobenius reciprocity for a permutation representation......Page 48 1.4.4 The structure of the commutant ofa permutation representation......Page 53 1.5.1 L(G) and the convolution......Page 55 1.5.2 The Fourier transform......Page 60 1.5.3 Algebras of bi-K-invariant functions......Page 64 1.6.1 Definitions and examples......Page 69 1.6.2 First properties of induced representations......Page 71 1.6.3 Frobenius reciprocity......Page 73 1.6.4 Mackey's lemma and the intertwining number theorem......Page 75 2.1.1 Conjugacy invariant functions......Page 77 2.1.2 Multiplicity-free subgroups......Page 82 2.1.3 Greenhalgebras......Page 83 2.2.1 Branching graphs and Gelfand-Tsetlin bases......Page 87 2.2.2 Gelfand-Tsetlin algebras......Page 89 2.2.3 Gelfand-Tsetlin bases for permutation representations......Page 93 3.1.1 Partitions and conjugacy classes in Sn......Page 97 3.1.3 Young tableaux......Page 99 3.1.4 Coxeter generators......Page 101 3.1.5 The content of a tableau......Page 103 3.1.6 The Young poset......Page 107 3.2 The Young–Jucys–Murphy elements and a Gelfand-Tsetlin basis for Sn......Page 109 3.2.2 Marked permutations......Page 110 3.2.3 Olshanskii's theorem......Page 113 3.2.4 A characterization of the YJM elements......Page 116 3.3.1 The weight of a Young basis vector......Page 118 3.3.2 The spectrum of the YJM elements......Page 120 3.3.3 Spec(n) = Cont(n)......Page 122 3.4.1 Young's seminormal form......Page 128 3.4.2 Young's orthogonal form......Page 130 3.4.3 The Murnaghan-Nakayama rule for a cycle......Page 134 3.4.4 The Young seminormal units......Page 136 3.5.1 Skew shapes......Page 139 3.5.2 Skew representations of the symmetric group......Page 141 3.5.3 Basic properties of the skew representations and Pieri's rule......Page 144 3.5.4 Skew hooks......Page 148 3.5.5 The Murnaghan-Nakayama rule......Page 150 3.6.1 The dominance and the lexicographic orders for partitions......Page 153 3.6.2 The Young modules......Page 156 3.6.3 The Frobenius-Young correspondence......Page 158 3.6.4 Radon transforms between Young's modules......Page 162 3.7.1 Semistandard Young tableaux......Page 163 3.7.2 The reduced Young poset......Page 166 3.7.3 The Young rule......Page 168 3.7.4 A Greenhalgebra with the symmetric group......Page 171 4.1.1 More notation and results on partitions......Page 174 4.1.2 Monomial symmetric polynomials......Page 175 4.1.3 Elementary, complete and power sumssymmetric polynomials......Page 177 4.1.4 The fundamental theorem on symmetric polynomials......Page 183 4.1.5 An involutive map......Page 185 4.1.6 Antisymmetric polynomials......Page 186 4.1.7 The algebra of symmetric functions......Page 188 4.2.1 On the characters of the Young modules......Page 189 4.2.2 Cauchy's formula......Page 191 4.2.3 Frobenius character formula......Page 192 4.2.4 Applications of Frobenius character formula......Page 197 4.3.1 Definition of Schur polynomials......Page 203 4.3.2 A scalar product......Page 206 4.3.3 The characteristic map......Page 207 4.3.4 Determinantal identities......Page 211 4.4.1 Minimal decompositions of permutations asproducts of transpositions......Page 217 4.4.2 The Theorem of Jucys and Murphy......Page 222 4.4.3 Bernoulli and Stirling numbers......Page 226 4.4.4 Garsia's expression for chilambda......Page 231 5.1.1 Ordinary binomial coefficients: basic identities......Page 239 5.1.2 Binomial coefficients: some technical results......Page 242 5.1.3 Lassalle's coefficients......Page 246 5.1.4 Binomial coefficients associated with partitions......Page 251 5.1.5 Lassalle's symmetric function......Page 253 5.2.1 The Frobenius function......Page 256 5.2.2 Lagrange interpolation formula......Page 260 5.2.3 The Taylor series at infinity for the Frobenius quotient......Page 263 5.2.4 Some explicit formulas for the coefficients.........Page 268 5.3.1 Conjugacy classes with one nontrivial cycle......Page 270 5.3.2 Conjugacy classes with two nontrivial cycles......Page 272 5.3.3 The explicit formula for an arbitrary conjugacy class......Page 276 5.4 Central characters and class symmetric functions......Page 281 5.4.1 Central characters......Page 282 5.4.2 Class symmetric functions......Page 285 5.4.3 Kerov-Vershik asymptotics......Page 289 6 Radon transforms, Specht modules and the Littlewood–Richardson rule......Page 291 6.1.1 Words and lattice permutations......Page 292 6.1.2 Pairs of partitions......Page 295 6.1.3 James' combinatorial theorem......Page 299 6.1.4 Littlewood-Richardson tableaux......Page 302 6.1.5 The Littlewood-Richardson rule......Page 308 6.2.1 Generalized Specht modules......Page 311 6.2.2 A family of Radon transforms......Page 316 6.2.3 Decomposition theorems......Page 321 6.2.4 The Gelfand-Tsetlin bases for Ma revisited......Page 325 7.1.1 Finite dimensional *-algebras......Page 332 7.1.2 Burnside's theorem......Page 334 7.2.1 Schur's lemma for a linear algebra......Page 336 7.2.2 The commutant of a *-algebra......Page 338 7.3.1 Tensor product of algebras......Page 341 7.3.2 The double commutant theorem......Page 343 7.3.3 Structure of finite dimensional *-algebras......Page 345 7.3.4 Matrix units and central elements......Page 349 7.4.1 Representation theory of End(V)......Page 350 7.4.2 Representation theory of finite dimensional *-algebras......Page 352 7.4.4 Complete reducibility of finite dimensional *-algebras......Page 354 7.4.5 The regular representation of a *-algebra......Page 356 7.4.6 Representation theory of finite groups revisited......Page 357 7.5.1 Subalgebras and Bratteli diagrams......Page 359 7.5.2 The centralizer of a subalgebra......Page 361 7.5.3 A reciprocity law for restriction......Page 363 7.5.4 A reciprocity law for induction......Page 365 7.5.5 Iterated tensor product of permutation representations......Page 369 8.1 Symmetric and antisymmetric tensors......Page 375 8.1.1 Iterated tensor product......Page 376 8.1.2 The action of Sk on.........Page 378 8.1.3 Symmetric tensors......Page 379 8.1.4 Antisymmetric tensors......Page 383 8.2.1 The general linear group GL(n, C)......Page 386 8.2.2 Duality between GL(n,C) and Sk......Page 392 8.2.3 Clebsch-Gordan decomposition and branching formulas......Page 396 8.3 The partition algebra......Page 402 8.3.1 The partition monoid......Page 403 8.3.2 The partition algebra......Page 409 8.3.3 Schur-Weyl duality for the partition algebra......Page 411 References......Page 420 Index......Page 427 The Representation Theory Of The Symmetric Groups Is A Classical Topic That, Since The Pioneering Work Of Frobenius, Schur And Young, Has Grown Into A Huge Body Of Theory, With Many Important Connections To Other Areas Of Mathematics And Physics. This Self-contained Book Provides A Detailed Introduction To The Subject, Covering Classical Topics Such As The Littlewood-richardson Rule And The Schur-weyl Duality. Importantly The Authors Also Present Many Recent Advances In The Area, Including Lassalle's Character Formulas, The Theory Of Partition Algebras, And An Exhaustive Exposition Of The Approach Developed By A.m. Vershik And A. Okounkov. A Wealth Of Examples And Exercises Makes This An Ideal Textbook For Graduate Students. It Will Also Serve As A Useful Reference For More Experienced Researchers Across A Range Of Areas, Including Algebra, Computer Science, Statistical Mechanics And Theoretical Physics--provided By Publisher. Representation Theory Of Finite Groups -- The Theory Of Gelfand-tsetlin Bases -- The Okounkov-vershik Approach -- Symmetric Functions -- Content Evaluation And Character Theory Of The Symmetric Group -- Radon Transforms, Specht Modules And The Littlewood-richardson Rule -- Finite Dimensional *-algebras -- Schur-weyl Dualities And The Partition Algebra. Tullio Ceccherini-silberstein, Fabio Scarabotti, Filippo Tolli. Includes Bibliographical References (p. 402-408) And Index. Machine generated contents note: Preface; 1. Representation theory of finite groups; 2. The theory of Gelfand-Tsetlin bases; 3. The Okounkov-Vershik approach; 4. Symmetric functions; 5. Content evaluation and character theory; 6. The Littlewood-Richardson rule; 7. Finite dimensional *-algebras; 8. Schur-Weyl dualities and the partition algebra; Bibliography; Index. A self-contained introduction to the representation theory of the symmetric groups, including an exhaustive exposition of the Okounkov-Vershik approach.