preface
Ⅰ Symmetric groups and symmetric functions
1 Representations of finite groups and semisimple algebras
1.1 Finite groups and their representations
1.2 Characters and constructions on representations
1.3 The non-commutative Fourier transform
1.4 Semisimple algebras and modules
1.5 The double commutant theory
2 Symmetric functions and the Frobenius-Schur isomorphism
2.1 Conjugacy classes of the symmetric groups
2.2 The five bases of the algebra of symmetric functions
2.3 The structure of graded self-adjoint Hopf algebra
2.4 The Frobenius-Schur isomorphism
2.5 The Schur-Weyl duality
3 Combinatorics of partitions and tableaux
3.1 Pieri rules and Murnaghan-Nakayama formula
3.2 The Robinson-Schensted-Knuth algorithm
3.3 Construction of the irreducible representations
3.4 The hook-length formula
Ⅱ Hecke algebras and their representations
4 Hecke algebras and the Brauer-Cartan theory
4.1 Coxeter presentation of symmetric groups
4.2 Representation theory of algebras
4.3 Brauer-Cartan deformation theory
4.4 Structure of generic and specialized Hecke algebras
4.5 Polynomial construction of the q-Specht modules
5 Characters and dualities for Hecke algebras
5.1 Quantum groups and their Hopf algebra structure
5.2 Representation theory of the quantum groups
5.3 Jimbo-Schur-Weyl duality
5.4 Iwahori-Hecke duality
5.5 Hall-Littlewood polynomials and characters of Hecke algebras
6 Representations of the Hecke algebras specialized at q = 0
6.1 Non-commutative symmetric functions
6.2 Quasi-symmetric functions
6.3 The Hecke-Frobenius-Schur isomorphisms
Ⅲ Observables of partitions
7 The Ivanov-Kerov algebra of observables
7.1 The algebra of partial permutations
7.2 Coordinates of Young diagrams and their moments
7.3 Change of basis in the algebra of observables
7.4 Observables and topology of Young diagrams
8 The Jucys-Murphy elements
8.1 The Gelfand-Tsetlin subalgebra of the symmetric group algebra
8.2 Jucys-Murphy elements acting on the Gelfand-Tsetlin basis . .
8.3 Observables as symmetric functions of the contents
9 Symmetric groups and free probability
9.1 Introduction to free probability
9.2 Free cumulants of Young diagrams
9.3 Transition measures and Jucys-Murphy elements
9.4 The algebra of admissible set partitions
10 The Stanley-Feray formula for characters and Kerov polynomials
10.1 New observables of Young diagrams
10.2 The Stanley-Feray formula for characters of symmetric groups
10.3 Combinatorics of the Kerov polynomials
Ⅳ Models of random Young diagrams
11 Representations of the infinite symmetric group
11.1 Harmonic analysis on the Young graph and extremal characters
11.2 The bi-infinite symmetric group and the Olshanski semigroup
11.3 Classification of the admissible representations
11.4 Spherical representations and the Gelfand-Naimark-Segal cons-truction
12 Asymptoties of central measures
12.1 Free quasi-symmetric functions
12.2 Combinatorics of central measures
12.3 Gaussian behavior of the observables
13 Asymptotics of Plancherel and Schur-Weyl measures
13.1 The Plancherel and Schur-Weyl models
13.2 Limit shapes of large random Young diagrams
13.3 Kerov's central limit theorem for characters
Appendix
Appendix A Representation theory of semisimple Lie algebras
A.1 Nilpotent, solvable and semisimple algebras
A.2 Root system of a semisimple complex algebra
A.3 The highest weight theory
References
Index
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