definition 5.144 Unitary representation

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:6226 · p. 169

ground object -- no derivation owed

Rests on

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 4.3 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 5.141 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 ¶
depends_on equation 5.171 eq:rep-unitary-group ¶

Supports

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 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 proposition 5.155 Averaging trick ¶
- depends_on theorem 101.62 The mixing matrix is unitary ¶ ↺

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`	→	Inner product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6233
`depends_on`	→	Representation of a group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6233
`depends_on`	→	eq:rep-unitary-group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6233
`depends_on`	←	Invariance of the orthogonal complement	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6382
`depends_on`	←	Averaging trick	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6603