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El. knyga: Representation Theory of the Symmetric Groups: The Okounkov-Vershik Approach, Character Formulas, and Partition Algebras

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, (Universitą degli Studi di Roma 'La Sapienza', Italy), (Universitą degli Studi Roma Tre, Italy)
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Representation Theory of the Symmetric Groups: The Okounkov-Vershik Approach, Character Formulas, and Partition AlgebrasDidesnis paveikslėlis
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"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"--Provided by publisher.

A self-contained introduction to the representation theory of the symmetric groups, including an exhaustive exposition of the Okounkov-Vershik approach.

Recenzijos

"This beautifully written new book is a welcome addition... It is almost entirely self-contained, only assuming some basic group theory and linear algebra, yet it takes one to the forefront of recent advances in the area. It would be entirely suitable for a single semester or year-long graduate course, as it is replete with examples and exercises of varying difficulty. I suspect it will also find its way on to the shelf as a valuable reference work for researchers in the field, as it is an excellent complement to books of Kleshchev, Sagan, James, and James and Kerber." David John Hemmer, Mathematical Reviews

Daugiau informacijos

A self-contained introduction to the representation theory of the symmetric groups, including an exhaustive exposition of the OkounkovVershik approach.
Preface xiii
Representation theory of finite groups
1(58)
Basic facts
1(10)
Representations
1(1)
Examples
2(2)
Intertwining operators
4(1)
Direct sums and complete reducibility
5(1)
The adjoint representation
6(1)
Matrix coefficients
7(1)
Tensor products
8(2)
Cyclic and invariant vectors
10(1)
Schur's lemma and the commutant
11(8)
Schur's lemma
11(1)
Multiplicities and isotypic components
12(2)
Finite dimensional algebras
14(2)
The structure of the commutant
16(2)
Another description of HomG(W. V)
18(1)
Characters and the projection formula
19(8)
The trace
19(1)
Central functions and characters
20(2)
Central projection formulas
22(5)
Permutation representations
27(10)
Wielandt's lemma
27(3)
Symmetric actions and Gelfand's lemma
30(1)
Frobenius reciprocity for a permutation representation
30(5)
The structure of the commutant of a permutation representation
35(2)
The group algebra and the Fourier transform
37(14)
L(G) and the convolution
37(5)
The Fourier transform
42(4)
Algebras of bi-K-invariant functions
46(5)
Induced representations
51(8)
Definitions and examples
51(2)
First properties of induced representations
53(2)
Frobenius reciprocity
55(2)
Mackey's lemma and the intertwining number theorem
57(2)
The theory of Gelfand---Tsetlin bases
59(20)
Algebras of conjugacy invariant functions
59(10)
Conjugacy invariant functions
59(5)
Multiplicity-free subgroups
64(1)
Greenhalgebras
65(4)
Gelfand---Tsetlin bases
69(10)
Branching graphs and Gelfand---Tsetlin bases
69(2)
Gelfand---Tsetlin algebras
71(4)
Gelfand---Tsetlin bases for permutation representations
75(4)
The Okounkov-Vershik approach
79(77)
The Young poset
79(12)
Partitions and conjugacy classes in Gn
79(2)
Young frames
81(1)
Young tableaux
81(2)
Coxeter generators
83(2)
The content of a tableau
85(4)
The Young poset
89(2)
The Young-Jucys-Murphy elements and a Gelfand-Tsetlin basis for Gn
91(9)
The Young-Jucys-Murphy elements
92(1)
Marked permutations
92(3)
Olshanskii's theorem
95(3)
A characterization of the YJM elements
98(2)
The spectrum of the Young-Jucys-Murphy elements and the branching graph of Gn
100(10)
The weight of a Young basis vector
100(2)
The spectrum of the YJM elements
102(2)
Spec(n) = Cont(n)
104(6)
The irreducible representations of Gn
110(11)
Young's seminormal form
110(2)
Young's orthogonal form
112(4)
The Murnaghan---Nakayama rule for a cycle
116(2)
The Young seminormal units
118(3)
Skew representations and the Murnhagan---Nakayama rule
121(14)
Skew shapes
121(2)
Skew representations of the symmetric group
123(3)
Basic properties of the skew representations and Pieri's rule
126(4)
Skew hooks
130(2)
The Murnaghan-Nakayama rule
132(3)
The Frobenius---Young correspondence
135(10)
The dominance and the lexicographic orders for partitions
135(3)
The Young modules
138(2)
The Frobenius-Young correspondence
140(4)
Radon transforms between Young's modules
144(1)
The Young rule
145(11)
Semistandard Young tableaux
145(3)
The reduced Young poset
148(2)
The Young rule
150(3)
A Greenhalgebra with the symmetric group
153(3)
Symmetric functions
156(65)
Symmetric polynomials
156(15)
More notation and results on partitions
156(1)
Monomial symmetric polynomials
157(2)
Elementary, complete and power sums symmetric polynomials
159(6)
The fundamental theorem on symmetric polynomials
165(2)
An involutive map
167(1)
Antisymmetric polynomials
168(2)
The algebra of symmetric functions
170(1)
The Frobenius character formula
171(14)
On the characters of the Young modules
171(2)
Cauchy's formula
173(1)
Frobenius character formula
174(5)
Applications of Frobenius character formula
179(6)
Schur polynomials
185(14)
Definition of Schur polynomials
185(3)
A scalar product
188(1)
The characteristic map
189(4)
Determinantal identities
193(6)
The Theorem of Jucys and Murphy
199(22)
Minimal decompositions of permutations as products of transpositions
199(5)
The Theorem of Jucys and Murphy
204(4)
Bernoulli and Stirling numbers
208(5)
Garsia's expression for Χλ
213(8)
Content evaluation and character theory of the symmetric group
221(52)
Binomial coefficients
221(17)
Ordinary binomial coefficients: basic identities
221(3)
Binomial coefficients: some technical results
224(4)
Lassalle's coefficients
228(5)
Binomial coefficients associated with partitions
233(2)
Lassalle's symmetric function
235(3)
Taylor series for the Frobenius quotient
238(14)
The Frobenius function
238(4)
Lagrange interpolation formula
242(3)
The Taylor series at infinity for the Frobenius quotient
245(5)
Some explicit formulas for the coefficients cλ(m)
250(2)
Lassalle's explicit formulas for the characters of the symmetric group
252(11)
Conjugacy classes with one nontrivial cycle
252(2)
Conjugacy classes with two nontrivial cycles
254(4)
The explicit formula for an arbitrary conjugacy class
258(5)
Central characters and class symmetric functions
263(10)
Central characters
264(3)
Class symmetric functions
267(4)
Kerov-Vershik asymptotics
271(2)
Radon transforms, Specht modules and the Littlewood-Richardson rule
273(41)
The combinatorics of pairs of partitions and the Littlewood-Richardson rule
274(19)
Words and lattice permutations
274(3)
Pairs of partitions
277(4)
James' combinatorial theorem
281(3)
Littlewood-Richardson tableaux
284(6)
The Littlewood-Richardson rule
290(3)
Randon transforms, Specht modules and orthogonal decompositions of Young modules
293(21)
Generalized Specht modules
293(5)
A family of Radon transforms
298(5)
Decomposition theorems
303(4)
The Gelfand-Tsetlin bases for M revisited
307(7)
Finite dimensional -algebras
314(43)
Finite dimensional algebras of operators
314(4)
Finite dimensional -algebras
314(2)
Burnside's theorem
316(2)
Schur's lemma and the commutant
318(5)
Schur's lemma for a linear algebra
318(2)
The commutant of a -algebra
320(3)
The double commutant theorem and the structure of a finite dimensional -algebra
323(9)
Tensor product of algebras
323(2)
The double commutant theorem
325(2)
Structure of finite dimensional -algebras
327(4)
Matrix units and central elements
331(1)
Ideals and representation theory of a finite dimensional -algebra
332(9)
Representation theory of End(V)
332(2)
Representation theory of finite dimensional -algebras
334(2)
The Fourier transform
336(1)
Complete reducibility of finite dimensional -algebras
336(2)
The regular representation of a -algebra
338(1)
Representation theory of finite groups revisited
339(2)
Subalgebras and reciprocity laws
341(16)
Subalgebras and Bratteli diagrams
341(2)
The centralizer of a subalgebra
343(2)
A reciprocity law for restriction
345(2)
A reciprocity law for induction
347(4)
Iterated tensor product of permutation representations
351(6)
Schur-Weyl dualities and the partition algebra
357(45)
Symmetric and antisymmetric tensors
357(11)
Iterated tensor product
358(2)
The action of Gk on V∞k
360(1)
Symmetric tensors
361(4)
Antisymmetric tensors
365(3)
Classical Schur-Weyl duality
368(16)
The general linear group GL(n, C)
368(6)
Duality between GL(n, C) and Gk
374(4)
Clebsch-Gordan decomposition and branching formulas
378(6)
The partition algebra
384(18)
The partition monoid
385(6)
The partition algebra
391(2)
Schur-Weyl duality for the partition algebra
393(9)
References 402(7)
Index 409
Tullio Ceccherini-Silberstein is Professor in the Faculty of Engineering at the University of Sannio, Benevento. Fabio Scarabotti is Professor in the Faculty of Engineering at the University of Rome 'La Sapienza'. Filippo Tolli is Associate Professor in the Department of Mathematics at Roma Tre University.
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