Representation Theory of Symmetric Groups  book cover
1st Edition

Representation Theory of Symmetric Groups

ISBN 9781498719124
Published March 21, 2017 by Chapman and Hall/CRC
682 Pages 134 B/W Illustrations

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Book Description

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.

Table of Contents

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

Limit shapes of large random Young diagrams

Kerov’s central limit theorem for characters


A Representation theory of semisimple Lie algebras

Nilpotent, solvable and semisimple algebras

Root system of a semisimple complex algebra

The highest weight theory

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Meliot, Pierre-Loic



"The book will be most useful as a reference for researchers...I believe it is a valuable contribution to the literature on

the symmetric group and related algebras."

~Mark J. Wildon, Mathematical Reviews, March 2018