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Representation theory of symmetric groups

Author: Pierre-Loic Méliot
Publisher: Boca Raton, FL : CRC Press, Taylor & Francis Group, [2017]
Series: CRC Press series on discrete mathematics and its applications.
Edition/Format:   Print book : EnglishView all editions and formats
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Document Type: Book
All Authors / Contributors: Pierre-Loic Méliot
ISBN: 9781498719124 1498719120
OCLC Number: 962025611
Description: xvi, 666 pages ; 24 cm.
Contents: Machine generated contents note: I. Symmetric groups and symmetric functions --
1. Representations of finite groups and semisimple algebras --
1.1. Finite groups and their representations --
1.2. Characters and constructions on representations --
1.3. The non-commutative Fourier transform --
1.4. Semisimple algebras and modules --
1.5. The double commutant theory --
2. Symmetric functions and the Frobenius-Schur isomorphism --
2.1. Conjugacy classes of the symmetric groups --
2.2. The five bases of the algebra of symmetric functions --
2.3. The structure of graded self-adjoint Hopf algebra --
2.4. The Frobenius-Schur isomorphism --
2.5. The Schur-Weyl duality --
3. Combinatorics of partitions and tableaux --
3.1. Pieri rules and Murnaghan-Nakayama formula --
3.2. The Robinson-Schensted-Knuth algorithm --
3.3. Construction of the irreducible representations --
3.4. The hook-length formula --
II. Hecke algebras and their representations --
4. Hecke algebras and the Brauer-Cartan theory --
4.1. Coxeter presentation of symmetric groups --
4.2. Representation theory of algebras --
4.3. Brauer-Cartan deformation theory --
4.4. Structure of generic and specialized Hecke algebras --
4.5. Polynomial construction of the q-Specht modules --
5. Characters and dualities for Hecke algebras --
5.1. Quantum groups and their Hopf algebra structure --
5.2. Representation theory of the quantum groups --
5.3. Jimbo-Schur-Weyl duality --
5.4. Iwahori-Hecke duality --
5.5. Hall-Littlewood polynomials and characters of Hecke algebras --
6. Representations of the Hecke algebras specialized at q = 0 --
6.1. Non-commutative symmetric functions --
6.2. Quasi-symmetric functions --
6.3. The Hecke-Frobenius-Schur isomorphisms --
III. Observables of partitions --
7. The Ivanov-Kerov algebra of observables --
7.1. The algebra of partial permutations --
7.2. Coordinates of Young diagrams and their moments --
7.3. Change of basis in the algebra of observables --
7.4. Observables and topology of Young diagrams --
8. The Jucys-Murphy elements --
8.1. The Gelfand-Tsetlin subalgebra of the symmetric group algebra --
8.2. Jucys-Murphy elements acting on the Gelfand-Tsetlin basis --
8.3. Observables as symmetric functions of the contents --
9. Symmetric groups and free probability --
9.1. Introduction to free probability --
9.2. Free cumulants of Young diagrams --
9.3. Transition measures and Jucys-Murphy elements --
9.4. The algebra of admissible set partitions --
10. The Stanley-Feray formula and Kerov polynomials --
10.1. New observables of Young diagrams --
10.2. The Stanley-Feray formula for characters of symmetric groups --
10.3. Combinatorics of the Kerov polynomials --
IV. Models of random Young diagrams --
11. Representations of the infinite symmetric group --
11.1. Harmonic analysis on the Young graph and extremal characters --
11.2. The bi-infinite symmetric group and the Olshanski semigroup --
11.3. Classification of the admissible representations --
11.4. Spherical representations and the GNS construction --
12. Asymptotics of central measures --
12.1. Free quasi-symmetric functions --
12.2. Combinatorics of central measures --
12.3. Gaussian behavior of the observables --
13. Asymptotics of Plancherel and Schur-Weyl measures --
13.1. The Plancherel and Schur-Weyl models --
13.2. Limit shapes of large random Young diagrams --
13.3. Kerov's central limit theorem for characters --
Appendix --
Appendix A Representation theory of semisimple Lie algebras --
A.1. Nilpotent, solvable and semisimple algebras --
A.2. Root system of a semisimple complex algebra --
A.3. The highest weight theory.
Series Title: CRC Press series on discrete mathematics and its applications.
Responsibility: Pierre-Loïc Méliot.

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