definition 4.48 Symmetric group

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:1636 · p. 77

ground object -- no derivation owed

Rests on

depends_on definition 3.47 Bijective map ¶

Supports

depends_on definition 4.49 Permutation group ¶
- depends_on proposition 4.50 Cardinal of the permutation group ¶
  - depends_on proposition 4.52 $S_3$ is a smallest non-abelian group ¶
- depends_on proposition 4.53 The subgroups of $S_3$ ¶
- depends_on theorem 4.56 Cayley's theorem ¶
depends_on definition 5.32 Levi–Civita symbol; cross product ¶
- depends_on definition 5.131 Isotropic Cartesian tensor ¶
  - depends_on definition A.672 The octahedral rotation group ¶
    - depends_on proposition A.673 Cubic tensors of rank four ¶
      - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
  - depends_on lemma 5.132 Parity constraint ¶
    - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
      - depends_on lemma A.742 Isotropic representation ¶
      - depends_on lemma A.671 The three products are independent ¶
      - depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
      - depends_on remark A.677 The lower ranks, and why an isotropic solid is not piezoelectric ¶
      - depends_on remark 30.27 Two results this chapter borrows from Part II ¶
      - depends_on remark 13.7 What the classification is used for ¶
- depends_on lemma 5.34 Contraction of two Levi–Civita symbols ¶
  - depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶
- depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
  - depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶ ↺
  - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶ ↺
depends_on definition 5.141 Representation of a group ¶
- depends_on definition 5.146 Character ¶
- depends_on definition 5.142 Dimension of a representation ¶
- depends_on definition 5.145 Equivalent representations ¶
  - depends_on proposition 5.155 Averaging trick ¶
    - depends_on theorem 101.62 The mixing matrix is unitary ¶
      - depends_on proposition 101.63 Parameter counting, and why three generations permit $CP$ violation ¶
      - depends_on theorem 103.11 GIM suppression at one loop ¶
  - depends_on theorem 5.160 Schur's second lemma ¶
    - depends_on lemma 25.39 The low-degree images are forced ¶
      - depends_on example 25.45 Weyl ordering collapses ¶
    - depends_on theorem 25.38 Groenewold–van Hove ¶
      - depends_on corollary 25.41 Quantization is not a functor ¶
      - depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
      - depends_on remark 24.55 What polarization costs ¶
- depends_on definition 5.143 Faithful representation ¶
- depends_on definition 5.147 Invariant subspace ¶
  - depends_on definition 5.148 Irreducible representation ¶
    - depends_on proposition 5.161 Irreducible representations of an abelian group ¶
    - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶
    - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶
      - depends_on theorem 101.62 The mixing matrix is unitary ¶ ↺
    - depends_on theorem 5.158 Schur's first lemma ¶
      - depends_on corollary 14.18 A Casimir acts as a number on an irreducible representation ¶
      - depends_on lemma A.427 Invariant bilinear forms on a simple algebra ¶
      - depends_on lemma A.405 Schur, algebra form ¶
      - depends_on phenomenon 102.1 Hadron masses obey an octet mass formula ¶
      - depends_on proposition 5.161 Irreducible representations of an abelian group ¶ ↺
      - depends_on theorem 106.72 Confinement at strong coupling ¶
      - depends_on theorem 102.12 Which quark combinations can be colour singlets ¶
    - depends_on theorem 5.160 Schur's second lemma ¶ ↺
  - depends_on definition 5.149 Totally reducible representation ¶
    - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
    - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
  - depends_on lemma 5.150 Invariance of the orthogonal complement ¶
    - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
    - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
  - depends_on theorem 5.158 Schur's first lemma ¶ ↺
  - depends_on theorem 5.160 Schur's second lemma ¶ ↺
- depends_on definition 5.144 Unitary representation ¶
  - depends_on lemma 5.150 Invariance of the orthogonal complement ¶ ↺
  - depends_on proposition 5.155 Averaging trick ¶ ↺
depends_on notation 4.51 Cycle notation ¶
depends_on proposition 4.72 The automorphism group ¶
- depends_on definition 4.73 External semidirect product ¶
  - depends_on example 4.78 The Euclidean group ¶
    - depends_on proposition 18.20 The Galilean group ¶
      - depends_on corollary 18.24 Galilean composition of velocities ¶
      - depends_on remark 18.22 Superseded, and in exactly what sense ¶
  - depends_on example 4.79 The Poincaré group ¶
  - depends_on proposition 4.74 The semidirect product is a group ¶
    - depends_on proposition 4.75 The two factors inside the semidirect product ¶
      - depends_on example 4.79 The Poincaré group ¶ ↺
      - depends_on proposition 4.77 When a semidirect product is direct ¶
  - depends_on proposition 18.20 The Galilean group ¶ ↺
  - depends_on theorem 4.76 Internal characterization of the semidirect product ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Bijective map	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1642
`depends_on`	←	Permutation group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1667
`depends_on`	←	Levi–Civita symbol; cross product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1489
`depends_on`	←	Representation of a group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6190
`depends_on`	←	Cycle notation	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1770
`depends_on`	←	The automorphism group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2656