theorem 5.158 Schur's first lemma

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:6991 · p. 179

Rests on

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 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
  - depends_on definition 4.32 Field ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.18 Semigroup ¶ ↺
  - depends_on definition 4.31 Module ¶
    - depends_on definition 4.29 Action ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.28 Distributivity ¶ ↺
    - depends_on definition 4.21 Group ¶ ↺
    - depends_on definition 4.30 Ring ¶ ↺
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 4.43 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 5.148 Irreducible representation ¶
- depends_on definition 5.147 Invariant subspace ¶ ↺
proves proof ch:03-linear-algebra-representations@proof-67 ¶

Supports

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 proposition 102.6 Mesons are $q\bar{q}$, baryons are $qqq$ ¶
      - assumes experiment Deep Inelastic Scattering ¶
      - depends_on phenomenon 116.38 The quark spins do not account for the proton spin ¶
      - depends_on phenomenon 102.9 The $\Delta^{++}$ violates the spin-statistics theorem ¶
      - depends_on phenomenon 102.7 The nucleon magnetic moments come out right ¶
      - depends_on proposition 116.29 The Gross–Llewellyn Smith sum rule in the parton model ¶
  - 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 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 proposition A.408 The two groups vanish for the trivial module ¶
    - depends_on theorem A.410 Whitehead's first and second lemmas ¶
      - depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶
- 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 ¶ ↺

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`	→	Linear transformation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7001
`depends_on`	→	Invariant subspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7001
`depends_on`	→	Irreducible representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7001
`depends_on`	←	A Casimir acts as a number on an irreducible representation	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:580
`depends_on`	←	Invariant bilinear forms on a simple algebra	declared	appendices/A-long-proofs.tex:21002
`depends_on`	←	Schur, algebra form	declared	appendices/A-long-proofs.tex:19753
`depends_on`	←	Hadron masses obey an octet mass formula	declared	parts/11-qft-standard-model/04-qcd.tex:128
`depends_on`	←	Irreducible representations of an abelian group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7127
`depends_on`	←	Confinement at strong coupling	declared	parts/11-qft-standard-model/08-path-integral-quantization.tex:3384
`depends_on`	←	Which quark combinations can be colour singlets	declared	parts/11-qft-standard-model/04-qcd.tex:508
`proves`	←	ch:03-linear-algebra-representations@proof-67	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7004