Cover; Half Title; Title Page; Copyright Page; Table of Contents; 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 algebra2.4 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 algebras4.5 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 observables7.1 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 elements9.4 The algebra of admissible set partitions; 10: The Stanley-Féray formula and Kerov polynomials; 10.1 New observables of Young diagrams; 10.2 The Stanley-Féray 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
