12.1 Linear Representations and their Properties

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 dimension, character, and faithfulness of representations. We'll learn how to construct and classify representations, and explore their properties using tools like Schur's lemma and character theory.

Linear Representations of Finite Groups

Definition and Key Properties

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• Linear representation of a finite group 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 general linear group 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)
• Kernel of representation ρ contains G elements mapped to identity matrix in GL(V)
• Faithful representations preserve all group structure information with trivial kernels
• Image of representation ρ(G) forms GL(V) subgroup isomorphic to G/ker(ρ)
• Schur's lemma states for irreducible representations ρ and ρ' of G, intertwining operator T: V → V' must be zero or isomorphism

Advanced Concepts

• Center of group acts by scalar multiplication 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
• Tensor product representations combine existing representations
  • Example: Tensor product of two 2-dimensional representations of a dihedral group

Constructing Linear Representations

Basic Representations

• Trivial representation maps all group elements to 1×1 identity matrix
  • Example: Trivial representation of cyclic group C4
• Regular representation acts on group algebra F[G] by left multiplication
  • Dimension equals group order
  • Example: Regular representation of S3 on C[S3]
• Cyclic groups have one-dimensional irreducible representations using complex roots of unity
  • Example: Irreducible representations of C6 using 6th roots of unity

Group-Specific Representations

• Sign representation of symmetric group Sn maps even permutations to 1, odd to -1
  • Example: Sign representation of S4
• Dihedral groups Dn use 2-dimensional representations with rotation and reflection matrices
  • Example: 2-dimensional representation of D4 using 90-degree rotation and reflection matrices
• Decompose regular representation into irreducible representations using character theory
  • Example: Decomposition of regular representation of A4 into irreducibles
• Construct induced representations from subgroup representations
  • Example: Inducing a representation of C3 to S3

Classifying Linear Representations

Reducibility and Irreducibility

• Reducible representations decompose into non-trivial subrepresentations
• Irreducible representations cannot be further decomposed
• Maschke's theorem ensures complete reducibility for finite group representations over fields with characteristic not dividing group order
• Number of irreducible representations equals number of group conjugacy classes
  • Example: S4 has 5 conjugacy classes and 5 irreducible representations

Classification Techniques

• Apply Schur's orthogonality relations to determine irreducibility and decompose reducible representations
  • Example: Using orthogonality relations to decompose the permutation representation of S4
• Use character tables to classify representations
  • Example: Character table of D4 to identify its irreducible representations
• Employ Frobenius-Schur indicator to distinguish real, complex, and quaternionic representations
  • Example: Calculating Frobenius-Schur indicator for representations of A5

Equivalence of Linear Representations

Determining Equivalence

• Equivalent representations ρ and ρ' have invertible linear transformation T: V → V' satisfying T ∘ ρ(g) = ρ'(g) ∘ T for all g in G
• Characters remain invariant under equivalence, useful for determining representation equivalence
• Calculate inner product of characters to check representation equivalence
  • Example: Comparing characters of two 3-dimensional representations of S4

Advanced Equivalence Concepts

• Construct intertwining operator between representations to prove equivalence or inequivalence
  • Example: Constructing intertwining operator for two 2-dimensional representations of D3
• Irreducible representations of different dimensions are not equivalent (Schur's lemma)
• Analyze equivalence classes of induced representations
  • Example: Examining equivalence classes of representations induced from subgroups of A4
• Use representation theory techniques like character tables to classify and distinguish equivalence classes
  • Example: Using character table of Q8 (quaternion group) to identify equivalence classes of its representations