Representation Theory of Symmetric Groups

Meliot, Pierre-Loic

I Symmetric groups and symmetric functions



Representations of finite groups and semisimple algebras



Finite groups and their representations



Characters and constructions on representations



The non-commutative Fourier transform



Semisimple algebras and modules



The double commutant theory

Symmetric functions and the Frobenius-Schur isomorphism



Conjugacy classes of the symmetric groups



The five bases of the algebra of symmetric functions



The structure of graded self-adjoint Hopf algebra



The Frobenius-Schur isomorphism



The Schur-Weyl duality

Combinatorics of partitions and tableaux



Pieri rules and Murnaghan-Nakayama formula



The Robinson-Schensted-Knuth algorithm



Construction of the irreducible representations



The hook-length formula

II Hecke algebras and their representations



Hecke algebras and the Brauer-Cartan theory



Coxeter presentation of symmetric groups



Representation theory of algebras



Brauer-Cartan deformation theory



Structure of generic and specialised Hecke algebras



Polynomial construction of the q-Specht modules



Characters and dualities for Hecke algebras



Quantum groups and their Hopf algebra structure



Representation theory of the quantum groups



Jimbo-Schur-Weyl duality



Iwahori-Hecke duality



Hall-Littlewood polynomials and characters of Hecke algebras

Representations of the Hecke algebras specialised at q = 0



Non-commutative symmetric functions



Quasi-symmetric functions



The Hecke-Frobenius-Schur isomorphisms

III Observables of partitions



The Ivanov-Kerov algebra of observables



The algebra of partial permutations



Coordinates of Young diagrams and their moments



Change of basis in the algebra of observables



Observables and topology of Young diagrams

The Jucys-Murphy elements



The Gelfand-Tsetlin subalgebra of the symmetric group algebra



Jucys-Murphy elements acting on the Gelfand-Tsetlin basis



Observables as symmetric functions of the contents

Symmetric groups and free probability



Introduction to free probability



Free cumulants of Young diagrams



Transition measures and Jucys-Murphy elements



The algebra of admissible set partitions

The Stanley-Féray formula and Kerov polynomials



New observables of Young diagrams



The Stanley-Féray formula for characters of symmetric groups



Combinatorics of the Kerov polynomials

IV Models of random Young diagrams



Representations of the infinite symmetric group



Harmonic analysis on the Young graph and extremal characters



The bi-infinite symmetric group and the Olshanski semigroup



Classification of the admissible representations



Spherical representations and the GNS construction

Asymptotics of central measures



Free quasi-symmetric functions



Combinatorics of central measures



Gaussian behavior of the observables

Asymptotics of Plancherel and Schur-Weyl measures



The Plancherel and Schur-Weyl models