Representation theory MOC

Group representation theory MOC

The application of representation theory to Group theory MOC studies groups via their representations as linear maps. The main category of interest here is .

Scope of group representation theory results

Fundamentals

• Group representation, Equivalence of group representations
• Group character, Character table
• Reducibility of representations, Decomposition of a representation (Maschke's theorem)
• Group ring

Schur's orthogonalities

• Schur's lemma (general)
• Orthonormality of irreps (finite, compact)
• Orthonormality of irreducible characters (finite, compact)
• Irreducible orthonormal basis

Theorems

• Irrep dimension theorem
• Wigner-Eckart theorem

Irrep operations

• Direct sum of representations
• Tensor product of group representations

Linear objects

• Generalized projection operator of a representation
• Representation operator (i.e. irreducible operators)

Representation theory of particular groups

• Representation theory of finite symmetric groups

Misc

• McKay graph

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