definition 5.141 Representation of a group

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:6181 · p. 169

ground object -- no derivation owed

Rests on

depends_on definition 4.21 Group ¶
- depends_on definition 4.11 Inverse element ¶
  - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
- depends_on definition 4.20 Monoid ¶
  - depends_on definition 4.9 Neutral element ¶ ↺
  - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.4 Internal binary operation; magma ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
      - depends_on definition 4.2 Binary operation ¶ ↺
depends_on definition 4.43 Group homomorphism ¶
- depends_on definition 4.21 Group ¶ ↺
- depends_on definition 3.43 Map ¶
  - depends_on definition 3.24 Quantifiers ¶
    - depends_on definition 3.23 Predicate ¶
      - depends_on definition 3.1 Logical proposition ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on definition 3.28 Set ¶
depends_on definition 4.48 Symmetric group ¶
- depends_on definition 3.47 Bijective map ¶

Supports

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 phenomenon 103.42 $CP$ violation in charm decays ¶
- 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 definition 25.42 Ordering rule ¶
    - 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 proposition 14.55 The fundamental, the antifundamental and the adjoint ¶
      - depends_on theorem A.354 Harish-Chandra, quoted: the labels separate ¶
    - depends_on lemma A.427 Invariant bilinear forms on a simple algebra ¶
      - depends_on proposition A.428 Uniqueness in degree zero ¶
    - depends_on lemma A.405 Schur, algebra form ¶
      - depends_on lemma A.406 Splitting a submodule of codimension one ¶
      - depends_on lemma A.409 A Casimir invertible on a nontrivial irreducible module ¶
    - 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 102.60 Confinement at strong coupling, and what it proves ¶
    - depends_on theorem 102.12 Which quark combinations can be colour singlets ¶
      - depends_on corollary 102.13 The selection rule of the eightfold way ¶
  - 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 ¶ ↺

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`	→	Group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6190
`depends_on`	→	Group homomorphism	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6190
`depends_on`	→	Symmetric group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6190
`depends_on`	←	Character	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6295
`depends_on`	←	Dimension of a representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6216
`depends_on`	←	Equivalent representations	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6287
`depends_on`	←	Faithful representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6223
`depends_on`	←	Invariant subspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6321
`depends_on`	←	Unitary representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6233