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## Book Description

The representation theory of the symmetric groups is a classical topic that, since the pioneering work of Frobenius, Schur and Young, has grown into a huge body of theory, with many important connections to other areas of mathematics and physics. This self-contained book provides a detailed introduction to the subject, covering classical topics such as the Littlewood–Richardson rule and the Schur–Weyl duality. Importantly the authors also present many recent advances in the area, including Lassalle's character formulas, the theory of partition algebras, and an exhaustive exposition of the approach developed by A. M. Vershik and A. Okounkov. A wealth of examples and exercises makes this an ideal textbook for graduate students. It will also serve as a useful reference for more experienced researchers across a range of areas, including algebra, computer science, statistical mechanics and theoretical physics.

1. Cover
2. Half Title
3. Title Page
5. Dedication
6. Contents
7. Preface
8. 1. Representation theory of finite groups
1. 1.1 Basic facts
2. 1.2 Schur’s lemma and the commutant
3. 1.3 Characters and the projection formula
4. 1.4 Permutation representations
5. 1.5 The group algebra and the Fourier transform
6. 1.6 Induced representations
9. 2. The theory of Gelfand–Tsetlin bases
1. 2.1 Algebras of conjugacy invariant functions
2. 2.2 Gelfand–Tsetlin bases
10. 3. The Okounkov–Vershik approach
1. 3.1 The Young poset
2. 3.2 The Young–Jucys–Murphy elements and a Gelfand–Tsetlin basis for Gn
3. 3.3 The spectrum of the Young–Jucys–Murphy elements and the branching graph of Gn
4. 3.4 The irreducible representations of Gn
5. 3.5 Skew representations and the Murnhagan–Nakayama rule
6. 3.6 The Frobenius–Young correspondence
7. 3.7 The Young rule
11. 4. Symmetric functions
1. 4.1 Symmetric polynomials
2. 4.2 The Frobenius character formula
3. 4.3 Schur polynomials
4. 4.4 The Theorem of Jucys and Murphy
12. 5. Content evaluation and character theory of the symmetric group
1. 5.1 Binomial coefficients
2. 5.2 Taylor series for the Frobenius quotient
3. 5.3 Lassalle’s explicit formulas for the characters of the symmetric group
4. 5.4 Central characters and class symmetric functions
13. 6. Radon transforms, Specht modules and the Littlewood–Richardson rule
1. 6.1 The combinatorics of pairs of partitions and the Littlewood–Richardson rule
2. 6.2 Randon transforms, Specht modules and orthogonal decompositions of Young modules
14. 7. Finite dimensional *-algebras
1. 7.1 Finite dimensional algebras of operators
2. 7.2 Schur’s lemma and the commutant
3. 7.3 The double commutant theorem and the structure of a finite dimensional *-algebra
4. 7.4 Ideals and representation theory of a finite dimensional *-algebra
5. 7.5 Subalgebras and reciprocity laws
15. 8. Schur–Weyl dualities and the partition algebra
1. 8.1 Symmetric and antisymmetric tensors
2. 8.2 Classical Schur–Weyl duality
3. 8.3 The partition algebra
16. References
17. Index