proposition 5.153 prop:rep-unitary-completely-reducible

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:6467 · p. 172

Rests on

depends_on corollary 5.29 Orthogonal decomposition ¶
- depends_on definition 5.18 Inner product ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 5.7 Vector subspace ¶
  - depends_on definition 5.5 Linear combination ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on proposition 5.28 Gram–Schmidt ¶
  - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.5 Linear combination ¶ ↺
  - depends_on definition 5.18 Inner product ¶ ↺
  - depends_on definition 5.27 Orthonormal basis ¶
    - depends_on definition 5.26 Orthogonal basis ¶
      - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.24 Orthogonal vectors ¶
    - depends_on definition 5.25 Unit vector ¶
      - depends_on definition 5.19 Norm ¶
  - depends_on equation 5.44 eq:lin-norm-assoc ¶
  - proves proof ch:03-linear-algebra-representations@proof-5 ¶
- proves proof ch:03-linear-algebra-representations@proof-6 ¶
depends_on definition 5.148 Irreducible representation ¶
- depends_on definition 5.147 Invariant subspace ¶
  - depends_on definition 5.7 Vector subspace ¶ ↺
  - depends_on definition 5.141 Representation of a group ¶
    - depends_on definition 4.21 Group ¶ ↺
    - depends_on definition 4.43 Group homomorphism ¶
      - depends_on definition 4.21 Group ¶ ↺
      - depends_on definition 3.43 Map ¶
    - depends_on definition 4.48 Symmetric group ¶
      - depends_on definition 3.47 Bijective map ¶
depends_on definition 5.149 Totally reducible representation ¶
- depends_on definition 5.147 Invariant subspace ¶ ↺
- depends_on equation 5.52 eq:lin-orthogonal-complement ¶
depends_on lemma 5.150 Invariance of the orthogonal complement ¶
- depends_on definition 5.147 Invariant subspace ¶ ↺
- depends_on definition 5.144 Unitary representation ¶
  - depends_on definition 5.18 Inner product ¶ ↺
  - depends_on definition 5.141 Representation of a group ¶ ↺
  - depends_on equation 5.171 eq:rep-unitary-group ¶
- depends_on equation 4.25 eq:alg-hom-inverse ¶
- depends_on equation 5.52 eq:lin-orthogonal-complement ¶ ↺
- depends_on equation 5.183 eq:rep-unitary-invariance ¶
- proves proof ch:03-linear-algebra-representations@proof-63 ¶
proves proof ch:03-linear-algebra-representations@proof-65 ¶

Supports

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 ¶

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`	→	Orthogonal decomposition	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6485
`depends_on`	→	Irreducible representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6485
`depends_on`	→	Totally reducible representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6485
`depends_on`	→	Invariance of the orthogonal complement	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6485
`depends_on`	←	The mixing matrix is unitary	declared	parts/11-qft-standard-model/03-weak-interactions.tex:3000
`proves`	←	ch:03-linear-algebra-representations@proof-65	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6488