Description
Book SynopsisRepresentation 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 innovati
Trade Review
"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
Table of ContentsI 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
Appendix
A Representation theory of semisimple Lie algebras
Nilpotent, solvable and semisimple algebras
Root system of a semisimple complex algebra
The highest weight theory
