definition 5.147 Invariant subspace

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:6311 · p. 170

ground object -- no derivation owed

Rests on

depends_on definition 5.7 Vector subspace ¶
- depends_on definition 5.5 Linear combination ¶
  - 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 5.141 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 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶
- depends_on definition 4.48 Symmetric group ¶
  - depends_on definition 3.47 Bijective map ¶

Supports

depends_on definition 5.148 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 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 5.158 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 corollary A.377 The adjoint representation is irreducible ¶
      - depends_on example A.358 Rank two: $\mathfrak{su}(3)$ ¶
      - depends_on theorem 14.57 $\vect{3}\otimes\bar{\vect{3}} = \vect{1}\oplus\vect{8}$ ¶
      - depends_on theorem 102.12 Which quark combinations can be colour singlets ¶
    - 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 theorem A.407 Weyl's complete reducibility theorem ¶
    - depends_on lemma A.409 A Casimir invertible on a nontrivial irreducible module ¶
      - depends_on theorem A.410 Whitehead's first and second lemmas ¶
  - 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 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 5.149 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 ¶ ↺

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`	→	Vector subspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6321
`depends_on`	→	Representation of a group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6321
`depends_on`	←	Irreducible representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6340
`depends_on`	←	Totally reducible representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6347
`depends_on`	←	Invariance of the orthogonal complement	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6382
`depends_on`	←	Schur's first lemma	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7001
`depends_on`	←	Schur's second lemma	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7068