Representation Theory of Symmetric Groups

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

1. Symmetric groups and symmetric functions
   1. Representations of finite groups and semisimple algebras
   2. Finite groups and their representations
   3. Characters and constructions on representations
   4. The non-commutative Fourier transform
   5. Semisimple algebras and modules
   6. The double commutant theory
   7. Symmetric functions and the Frobenius-Schur isomorphism
   8. Conjugacy classes of the symmetric groups
   9. The five bases of the algebra of symmetric functions
   10. The structure of graded self-adjoint Hopf algebra
   11. The Frobenius-Schur isomorphism
   12. The Schur-Weyl duality
   13. Combinatorics of partitions and tableaux
   14. Pieri rules and Murnaghan-Nakayama formula
   15. The Robinson-Schensted-Knuth algorithm
   16. Construction of the irreducible representations
   17. The hook-length formula
   18. II Hecke algebras and their representations
   19. Hecke algebras and the Brauer-Cartan theory
   20. Coxeter presentation of symmetric groups
   21. Representation theory of algebras
   22. Brauer-Cartan deformation theory
   23. Structure of generic and specialised Hecke algebras
   24. Polynomial construction of the q-Specht modules
2. Characters and dualities for Hecke algebras
   1. Quantum groups and their Hopf algebra structure
   2. Representation theory of the quantum groups
   3. Jimbo-Schur-Weyl duality
   4. Iwahori-Hecke duality
   5. Hall-Littlewood polynomials and characters of Hecke algebras
3. Representations of the Hecke algebras specialised at q = 0
   1. Non-commutative symmetric functions
   2. Quasi-symmetric functions
   3. The Hecke-Frobenius-Schur isomorphisms
   4. III Observables of partitions
   5. The Ivanov-Kerov algebra of observables
   6. The algebra of partial permutations
   7. Coordinates of Young diagrams and their moments
   8. Change of basis in the algebra of observables
   9. Observables and topology of Young diagrams
   10. The Jucys-Murphy elements
   11. The Gelfand-Tsetlin subalgebra of the symmetric group algebra
   12. Jucys-Murphy elements acting on the Gelfand-Tsetlin basis
   13. Observables as symmetric functions of the contents
   14. Symmetric groups and free probability
   15. Introduction to free probability
   16. Free cumulants of Young diagrams
   17. Transition measures and Jucys-Murphy elements
   18. The algebra of admissible set partitions
   19. The Stanley-Féray formula and Kerov polynomials
   20. New observables of Young diagrams
   21. The Stanley-Féray formula for characters of symmetric groups
   22. Combinatorics of the Kerov polynomials
   23. IV Models of random Young diagrams
   24. Representations of the infinite symmetric group
   25. Harmonic analysis on the Young graph and extremal characters
   26. The bi-infinite symmetric group and the Olshanski semigroup
   27. Classification of the admissible representations
   28. Spherical representations and the GNS construction
   29. Asymptotics of central measures
   30. Free quasi-symmetric functions
   31. Combinatorics of central measures
   32. Gaussian behavior of the observables
   33. Asymptotics of Plancherel and Schur-Weyl measures
   34. The Plancherel and Schur-Weyl models
   35. Limit shapes of large random Young diagrams
   36. Kerov’s central limit theorem for characters
4. Appendix
   1. A Representation theory of semisimple Lie algebras
   2. Nilpotent, solvable and semisimple algebras
   3. Root system of a semisimple complex algebra
   4. The highest weight theory

Reviews

“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