[[Representation theory MOC]]
# Group representation theory MOC

The application of **representation theory** to [[Group theory MOC]]
studies groups via their [[Group representation|representations]] as linear maps.
The main category of interest here is  [[Category of group representations]].

[[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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