Group representation theory MOC

Scope of group representation theory results

• All representations
  • Schur's Lemma
• All complex representations
  • Irreducible operators applied to an irreducible orthonormal basis transform in the product representation
  • Wigner-Eckart theorem
  • Idempotent of the complex group ring
  • Idempotent primitivity criterion
  • Equivalence of irreps on left ideals criterion
• All complex representations of compact groups
  • Every finite complex representation of a compact group is equivalent to a unitary representation
  • Orthonormality of irreps
  • Orthonormality of irreducible characters
  • The regular representation contains all irreducible representations
  • Irreducible orthonormal basis
  • Generalized projection operator of a representation
• All complex representations of finite groups groups
  • Irreducible character as function of an idempotent (might also work for compact groups)
• Complex representations of the symmetric group
  • Conjugacy classes of a symmetric group are determined by cycle structure
  • Characters of a finite symmetric group are real
  • Symmetrizer and antisymmetrizer elements are essentially idempotent and primitive
  • Trivial and alternating characters of a finite symmetric group in tensor product decomposition
  • Young operator

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