theorem 5.160 Schur's second lemma

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:7061 · p. 180

Rests on

depends_on definition 5.53 Isomorphism ¶
- depends_on definition 5.37 Linear transformation ¶
  - 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 5.145 Equivalent representations ¶
- 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 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
  - depends_on definition 4.48 Symmetric group ¶
    - depends_on definition 3.47 Bijective map ¶
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 5.148 Irreducible representation ¶
- depends_on definition 5.147 Invariant subspace ¶ ↺
proves proof ch:03-linear-algebra-representations@proof-68 ¶

Supports

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 definition 25.44 Weyl and normal ordering ¶
      - depends_on definition 25.50 Star product and Moyal bracket ¶
      - depends_on definition 25.52 Wigner function ¶
      - depends_on example 25.45 Weyl ordering collapses ¶ ↺
      - depends_on proposition 25.46 Normal ordering disagrees already at degree two ¶
    - depends_on proposition 25.43 The size of the ambiguity ¶
      - depends_on proposition 25.48 Where the ambiguity does and does not bite ¶
- depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
- depends_on remark 24.55 What polarization costs ¶

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`	→	Isomorphism	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7068
`depends_on`	→	Equivalent representations	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7068
`depends_on`	→	Invariant subspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7068
`depends_on`	→	Irreducible representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7068
`depends_on`	←	The low-degree images are forced	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1381
`depends_on`	←	Groenewold–van Hove	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1356
`proves`	←	ch:03-linear-algebra-representations@proof-68	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7071