1st Edition

Representation Theory of Symmetric Groups

By Pierre-Loic Meliot Copyright 2017
    682 Pages 134 B/W Illustrations
    by CRC Press

    682 Pages 134 B/W Illustrations
    by Chapman & Hall

    682 Pages 134 B/W Illustrations
    by Chapman & Hall

    Representation Theory of Symmetric Groups is the most up-to-date abstract algebra book on the subject of symmetric groups and representation theory. Utilizing new research and results, this book can be studied from a combinatorial, algorithmic or algebraic viewpoint.

    This book is an excellent way of introducing today’s students to representation theory of the symmetric groups, namely classical theory. From there, the book explains how the theory can be extended to other related combinatorial algebras like the Iwahori-Hecke algebra.

    In a clear and concise manner, the author presents the case that most calculations on symmetric group can be performed by utilizing appropriate algebras of functions. Thus, the book explains how some Hopf algebras (symmetric functions and generalizations) can be used to encode most of the combinatorial properties of the representations of symmetric groups.

    Overall, the book is an innovative introduction to representation theory of symmetric groups for graduate students and researchers seeking new ways of thought.

    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


    Meliot, Pierre-Loic