You are previewing Representation Theory of the Symmetric Groups.
O'Reilly logo
Representation Theory of the Symmetric Groups

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.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. 1. Representation theory of finite groups
    1. 1.1 Basic facts
      1. 1.1.1 Representations
      2. 1.1.2 Examples
      3. 1.1.3 Intertwining operators
      4. 1.1.4 Direct sums and complete reducibility
      5. 1.1.5 The adjoint representation
      6. 1.1.6 Matrix coefficients
      7. 1.1.7 Tensor products
      8. 1.1.8 Cyclic and invariant vectors
    2. 1.2 Schur’s lemma and the commutant
      1. 1.2.1 Schur’s lemma
      2. 1.2.2 Multiplicities and isotypic components
      3. 1.2.3 Finite dimensional algebras
      4. 1.2.4 The structure of the commutant
      5. 1.2.5 Another description of HomG (W, V)
    3. 1.3 Characters and the projection formula
      1. 1.3.1 The trace
      2. 1.3.2 Central functions and characters
      3. 1.3.3 Central projection formulas
    4. 1.4 Permutation representations
      1. 1.4.1 Wielandt’s lemma
      2. 1.4.2 Symmetric actions and Gelfand’s lemma
      3. 1.4.3 Frobenius reciprocity for a permutation representation
      4. 1.4.4 The structure of the commutant of a permutation representation
    5. 1.5 The group algebra and the Fourier transform
      1. 1.5.1 L(G) and the convolution
      2. 1.5.2 The Fourier transform
      3. 1.5.3 Algebras of bi-K-invariant functions
    6. 1.6 Induced representations
      1. 1.6.1 Definitions and examples
      2. 1.6.2 First properties of induced representations
      3. 1.6.3 Frobenius reciprocity
      4. 1.6.4 Mackey’s lemma and the intertwining number theorem
  9. 2. The theory of Gelfand–Tsetlin bases
    1. 2.1 Algebras of conjugacy invariant functions
      1. 2.1.1 Conjugacy invariant functions
      2. 2.1.2 Multiplicity-free subgroups
      3. 2.1.3 Greenhalgebras
    2. 2.2 Gelfand–Tsetlin bases
      1. 2.2.1 Branching graphs and Gelfand–Tsetlin bases
      2. 2.2.2 Gelfand–Tsetlin algebras
      3. 2.2.3 Gelfand–Tsetlin bases for permutation representations
  10. 3. The Okounkov–Vershik approach
    1. 3.1 The Young poset
      1. 3.1.1 Partitions and conjugacy classes in Gn
      2. 3.1.2 Young frames
      3. 3.1.3 Young tableaux
      4. 3.1.4 Coxeter generators
      5. 3.1.5 The content of a tableau
      6. 3.1.6 The Young poset
    2. 3.2 The Young–Jucys–Murphy elements and a Gelfand–Tsetlin basis for Gn
      1. 3.2.1 The Young–Jucys–Murphy elements
      2. 3.2.2 Marked permutations
      3. 3.2.3 Olshanskii’s theorem
      4. 3.2.4 A characterization of the YJM elements
    3. 3.3 The spectrum of the Young–Jucys–Murphy elements and the branching graph of Gn
      1. 3.3.1 The weight of a Young basis vector
      2. 3.3.2 The spectrum of the YJM elements
      3. 3.3.3 Spec(n) = Cont(n)
    4. 3.4 The irreducible representations of Gn
      1. 3.4.1 Young’s seminormal form
      2. 3.4.2 Young’s orthogonal form
      3. 3.4.3 The Murnaghan–Nakayama rule for a cycle
      4. 3.4.4 The Young seminormal units
    5. 3.5 Skew representations and the Murnhagan–Nakayama rule
      1. 3.5.1 Skew shapes
      2. 3.5.2 Skew representations of the symmetric group
      3. 3.5.3 Basic properties of the skew representations and Pieri’s rule
      4. 3.5.4 Skew hooks
      5. 3.5.5 The Murnaghan–Nakayama rule
    6. 3.6 The Frobenius–Young correspondence
      1. 3.6.1 The dominance and the lexicographic orders for partitions
      2. 3.6.2 The Young modules
      3. 3.6.3 The Frobenius–Young correspondence
      4. 3.6.4 Radon transforms between Young’s modules
    7. 3.7 The Young rule
      1. 3.7.1 Semistandard Young tableaux
      2. 3.7.2 The reduced Young poset
      3. 3.7.3 The Young rule
      4. 3.7.4 A Greenhalgebra with the symmetric group
  11. 4. Symmetric functions
    1. 4.1 Symmetric polynomials
      1. 4.1.1 More notation and results on partitions
      2. 4.1.2 Monomial symmetric polynomials
      3. 4.1.3 Elementary, complete and power sums symmetric polynomials
      4. 4.1.4 The fundamental theorem on symmetric polynomials
      5. 4.1.5 An involutive map
      6. 4.1.6 Antisymmetric polynomials
      7. 4.1.7 The algebra of symmetric functions
    2. 4.2 The Frobenius character formula
      1. 4.2.1 On the characters of the Young modules
      2. 4.2.2 Cauchy’s formula
      3. 4.2.3 Frobenius character formula
      4. 4.2.4 Applications of Frobenius character formula
    3. 4.3 Schur polynomials
      1. 4.3.1 Definition of Schur polynomials
      2. 4.3.2 A scalar product
      3. 4.3.3 The characteristic map
      4. 4.3.4 Determinantal identities
    4. 4.4 The Theorem of Jucys and Murphy
      1. 4.4.1 Minimal decompositions of permutations as products of transpositions
      2. 4.4.2 The Theorem of Jucys and Murphy
      3. 4.4.3 Bernoulli and Stirling numbers
      4. 4.4.4 Garsia’s expression for χλ
  12. 5. Content evaluation and character theory of the symmetric group
    1. 5.1 Binomial coefficients
      1. 5.1.1 Ordinary binomial coefficients: basic identities
      2. 5.1.2 Binomial coefficients: some technical results
      3. 5.1.3 Lassalle’s coefficients
      4. 5.1.4 Binomial coefficients associated with partitions
      5. 5.1.5 Lassalle’s symmetric function
    2. 5.2 Taylor series for the Frobenius quotient
      1. 5.2.1 The Frobenius function
      2. 5.2.2 Lagrange interpolation formula
      3. 5.2.3 The Taylor series at infinity for the Frobenius quotient
      4. 5.2.4 Some explicit formulas for the coefficients cλr(m)
    3. 5.3 Lassalle’s explicit formulas for the characters of the symmetric group
      1. 5.3.1 Conjugacy classes with one nontrivial cycle
      2. 5.3.2 Conjugacy classes with two nontrivial cycles
      3. 5.3.3 The explicit formula for an arbitrary conjugacy class
    4. 5.4 Central characters and class symmetric functions
      1. 5.4.1 Central characters
      2. 5.4.2 Class symmetric functions
      3. 5.4.3 Kerov–Vershik asymptotics
  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
      1. 6.1.1 Words and lattice permutations
      2. 6.1.2 Pairs of partitions
      3. 6.1.3 James’ combinatorial theorem
      4. 6.1.4 Littlewood–Richardson tableaux
      5. 6.1.5 The Littlewood–Richardson rule
    2. 6.2 Randon transforms, Specht modules and orthogonal decompositions of Young modules
      1. 6.2.1 Generalized Specht modules
      2. 6.2.2 A family of Radon transforms
      3. 6.2.3 Decomposition theorems
      4. 6.2.4 The Gelfand–Tsetlin bases for Ma revisited
  14. 7. Finite dimensional *-algebras
    1. 7.1 Finite dimensional algebras of operators
      1. 7.1.1 Finite dimensional *-algebras
      2. 7.1.2 Burnside’s theorem
    2. 7.2 Schur’s lemma and the commutant
      1. 7.2.1 Schur’s lemma for a linear algebra
      2. 7.2.2 The commutant of a *-algebra
    3. 7.3 The double commutant theorem and the structure of a finite dimensional *-algebra
      1. 7.3.1 Tensor product of algebras
      2. 7.3.2 The double commutant theorem
      3. 7.3.3 Structure of finite dimensional *-algebras
      4. 7.3.4 Matrix units and central elements
    4. 7.4 Ideals and representation theory of a finite dimensional *-algebra
      1. 7.4.1 Representation theory of End(V)
      2. 7.4.2 Representation theory of finite dimensional *-algebras
      3. 7.4.3 The Fourier transform
      4. 7.4.4 Complete reducibility of finite dimensional *-algebras
      5. 7.4.5 The regular representation of a *-algebra
      6. 7.4.6 Representation theory of finite groups revisited
    5. 7.5 Subalgebras and reciprocity laws
      1. 7.5.1 Subalgebras and Bratteli diagrams
      2. 7.5.2 The centralizer of a subalgebra
      3. 7.5.3 A reciprocity law for restriction
      4. 7.5.4 A reciprocity law for induction
      5. 7.5.5 Iterated tensor product of permutation representations
  15. 8. Schur–Weyl dualities and the partition algebra
    1. 8.1 Symmetric and antisymmetric tensors
      1. 8.1.1 Iterated tensor product
      2. 8.1.2 The action of Gk on V⊗k
      3. 8.1.3 Symmetric tensors
      4. 8.1.4 Antisymmetric tensors
    2. 8.2 Classical Schur–Weyl duality
      1. 8.2.1 The general linear group GL(n, C)
      2. 8.2.2 Duality between GL(n, C) and Gk
      3. 8.2.3 Clebsch–Gordan decomposition and branching formulas
    3. 8.3 The partition algebra
      1. 8.3.1 The partition monoid
      2. 8.3.2 The partition algebra
      3. 8.3.3 Schur–Weyl duality for the partition algebra
  16. References
  17. Index