Representation Theory of Symmetric Groups

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 Appendix A Representation theory of semisimple Lie algebras Nilpotent, solvable and semisimple algebras Root system of a semisimple complex algebra The highest weight theory