Representation Theory
Explore linear actions; characters and decomposition; construction and lie theory. Concepts and connections are available independently of video coverage.
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Characters
A finite-dimensional representation's character assigns the trace of each represented element.
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Group representations
A group representation is a homomorphism into invertible linear transformations.
Maschke's theorem
Finite-group representations are completely reducible when the field characteristic does not divide group order.
Schur's lemma
Nonzero intertwiners between irreducible representations are isomorphisms; scalar conclusions need field hypotheses.
Tensor products of representations
Actions on factor spaces induce a compatible action on their tensor product.
Complete reducibility
A representation is completely reducible when it is a direct sum of irreducible representations.
Knowledge connections
- Schur's lemma · Related · Map context
- Maschke's theorem · Related · Map context
- Group representations · Related · Map context
- Tensor products of representations · Related · Map context