Linear representations map finite groups to linear transformations, preserving group structure. They're a powerful tool for studying groups, allowing us to use linear algebra techniques to analyze abstract group properties.
This section introduces key concepts like , , and faithfulness of representations. We'll learn how to construct and classify representations, and explore their properties using tools like and character theory.
Linear Representations of Finite Groups
Definition and Key Properties
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of a G maps G to GL(V) as a homomorphism ρ: G → GL(V) (V denotes finite-dimensional vector space over field F, GL(V) represents of V)
Dimension of representation equals dimension of vector space V
Character of representation ρ (χρ: G → F) calculates as χρ(g) = Tr(ρ(g)) (Tr denotes matrix trace)
of representation ρ contains G elements mapped to identity matrix in GL(V)
Faithful representations preserve all group structure information with trivial kernels
of representation ρ(G) forms GL(V) subgroup isomorphic to G/ker(ρ)
Schur's lemma states for ρ and ρ' of G, T: V → V' must be zero or isomorphism
Advanced Concepts
Center of group acts by in irreducible representations (Schur's lemma)
Unitary representations preserve inner product on vector space V
Projective representations map G to PGL(V) (projective general linear group)
Example: Representation of SO(3) acting on spin-1/2 particles