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Representation Theory
Explore linear actions; characters and decomposition; construction and lie theory. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Group homomorphisms
A homomorphism preserves group multiplication; its kernel measures failure of injectivity.
Video explanations to be addedVector spaces
A vector space specifies addition and scalar multiplication satisfying the vector-space axioms over a chosen field.
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Linear actions
Group representations
A group representation is a homomorphism into invertible linear transformations.
Video explanations to be addedInvariant subspaces
An invariant subspace is preserved by every transformation in a representation.
Video explanations to be addedIrreducible representations
An irreducible representation has no nonzero proper invariant subspace.
Video explanations to be addedIntertwining operators
An intertwiner is a linear map commuting with the represented group actions.
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Characters and decomposition
Characters
A finite-dimensional representation's character assigns the trace of each represented element.
Video explanations to be addedSchur's lemma
Nonzero intertwiners between irreducible representations are isomorphisms; scalar conclusions need field hypotheses.
Video explanations to be addedComplete reducibility
A representation is completely reducible when it is a direct sum of irreducible representations.
Video explanations to be addedMaschke's theorem
Finite-group representations are completely reducible when the field characteristic does not divide group order.
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Construction and Lie theory
Tensor products of representations
Actions on factor spaces induce a compatible action on their tensor product.
Video explanations to be addedInduced representations
A subgroup representation can be extended through an induction construction to the larger group.
Video explanations to be addedLie algebras
A Lie algebra has a bilinear alternating bracket satisfying the Jacobi identity.
Video explanations to be addedWeights and roots
Weights and roots organize representations and structure relative to suitable commuting subalgebras.
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