proposition 5.151 prop:rep-not-totally-reducible

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:6410 · p. 171

Rests on

depends_on definition 5.148 Irreducible representation ¶
- depends_on definition 5.147 Invariant subspace ¶
  - depends_on definition 5.7 Vector subspace ¶
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶
  - depends_on definition 5.141 Representation of a group ¶
    - depends_on definition 4.21 Group ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
    - 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 4.33 Vector space ¶ ↺
  - 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-64 ¶

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type	direction	node	provenance	where
`depends_on`	→	Irreducible representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6413
`depends_on`	→	Totally reducible representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6413
`depends_on`	→	Invariance of the orthogonal complement	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6413
`proves`	←	ch:03-linear-algebra-representations@proof-64	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6416