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E-grāmata: Representation Theory of Symmetric Groups

(Universite Paris Sud, Orsay, France)
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Representation Theory of Symmetric GroupsPalielināt
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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 todays 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.

Recenzijas

"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

Preface xi
I Symmetric groups and symmetric functions
1(146)
1 Representations of finite groups and semisimple algebras
3(46)
1.1 Finite groups and their representations
3(10)
1.2 Characters and constructions on representations
13(5)
1.3 The non-commutative Fourier transform
18(9)
1.4 Semisimple algebras and modules
27(13)
1.5 The double commutant theory
40(9)
2 Symmetric functions and the Frobenius-Schur isomorphism
49(50)
2.1 Conjugacy classes of the symmetric groups
50(4)
2.2 The five bases of the algebra of symmetric functions
54(15)
2.3 The structure of graded self-adjoint Hopf algebra
69(9)
2.4 The Frobenius-Schur isomorphism
78(9)
2.5 The Schur-Weyl duality
87(12)
3 Combinatorics of partitions and tableaux
99(48)
3.1 Pieri rules and Murnaghan-Nakayama formula
99(9)
3.2 The Robinson-Schensted-Knuth algorithm
108(23)
3.3 Construction of the irreducible representations
131(9)
3.4 The hook-length formula
140(7)
II Hecke algebras and their representations
147(178)
4 Hecke algebras and the Brauer-Cartan theory
149(68)
4.1 Coxeter presentation of symmetric groups
151(10)
4.2 Representation theory of algebras
161(12)
4.3 Brauer-Cartan deformation theory
173(10)
4.4 Structure of generic and specialized Hecke algebras
183(24)
4.5 Polynomial construction of the q-Specht modules
207(10)
5 Characters and dualities for Hecke algebras
217(70)
5.1 Quantum groups and their Hopf algebra structure
218(12)
5.2 Representation theory of the quantum groups
230(22)
5.3 Jimbo-Schur-Weyl duality
252(11)
5.4 Iwahori-Hecke duality
263(9)
5.5 Hall-Littlewood polynomials and characters of Hecke algebras
272(15)
6 Representations of the Hecke algebras specialized at q = 0
287(38)
6.1 Non-commutative symmetric functions
289(10)
6.2 Quasi-symmetric functions
299(7)
6.3 The Hecke-Frobenius-Schur isomorphisms
306(19)
III Observables of partitions
325(174)
7 The Ivanov-Kerov algebra of observables
327(48)
7.1 The algebra of partial permutations
328(11)
7.2 Coordinates of Young diagrams and their moments
339(8)
7.3 Change of basis in the algebra of observables
347(7)
7.4 Observables and topology of Young diagrams
354(21)
8 The Jucys-Murphy elements
375(26)
8.1 The Gelfand-Tsetlin subalgebra of the symmetric group algebra
375(12)
8.2 Jucys-Murphy elements acting on the Gelfand-Tsetlin basis
387(9)
8.3 Observables as symmetric functions of the contents
396(5)
9 Symmetric groups and free probability
401(50)
9.1 Introduction to free probability
402(16)
9.2 Free cumulants of Young diagrams
418(8)
9.3 Transition measures and Jucys-Murphy elements
426(5)
9.4 The algebra of admissible set partitions
431(20)
10 The Stanley-Feray formula and Kerov polynomials
451(48)
10.1 New observables of Young diagrams
451(13)
10.2 The Stanley-Feray formula for characters of symmetric groups
464(15)
10.3 Combinatorics of the Kerov polynomials
479(20)
IV Models of random Young diagrams
499(130)
11 Representations of the infinite symmetric group
501(46)
11.1 Harmonic analysis on the Young graph and extremal characters
502(9)
11.2 The bi-infinite symmetric group and the Olshanski semigroup
511(16)
11.3 Classification of the admissible representations
527(11)
11.4 Spherical representations and the GNS construction
538(9)
12 Asymptotics of central measures
547(48)
12.1 Free quasi-symmetric functions
548(14)
12.2 Combinatorics of central measures
562(14)
12.3 Gaussian behavior of the observables
576(19)
13 Asymptotics of Plancherel and Schur-Weyl measures
595(34)
13.1 The Plancherel and Schur-Weyl models
596(6)
13.2 Limit shapes of large random Young diagrams
602(12)
13.3 Kerov's central limit theorem for characters
614(15)
Appendix
629(20)
Appendix A Representation theory of semisimple Lie algebras
631(18)
A.1 Nilpotent, solvable and semisimple algebras
631(4)
A.2 Root system of a semisimple complex algebra
635(6)
A.3 The highest weight theory
641(8)
References 649(12)
Index 661
Meliot, Pierre-Loic
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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