12.3 Schur's Lemma and the Orthogonality Relations

Schur's Lemma is a key tool in representation theory, helping us understand how irreducible representations relate to each other. It shows that maps between different irreducible representations must be zero or isomorphisms, which is super useful for breaking down complex representations.

The orthogonality relations for characters are another big deal. They tell us how different irreducible representations are related and help us figure out how to split up more complicated representations into simpler pieces. This stuff is crucial for working with group representations in practice.

Schur's Lemma and its Implications

Statement and Proof of Schur's Lemma

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• Schur's Lemma states for irreducible representations ρ and σ of a group G, any G-homomorphism φ: V → W must be zero or an isomorphism
• Proof of Schur's Lemma considers kernel and image of φ as G-invariant subspaces and uses irreducibility of ρ and σ
• Only G-endomorphisms of an irreducible representation are scalar multiples of the identity
• Fundamental tool in representation theory for analyzing representation structure
• Crucial in establishing orthogonality relations for characters of irreducible representations

Applications and Extensions of Schur's Lemma

• Provides powerful tool for decomposing representations into irreducible components
• Extends to more general algebraic structures (algebras, Lie algebras)
• Helps analyze intertwining operators between representations
• Useful in proving Wedderburn's theorem on structure of semisimple algebras
• Establishes relationship between irreducible representations and minimal ideals in group algebras
• Aids in classifying finite-dimensional representations of certain groups (finite abelian groups)

Applying Schur's Lemma to Homomorphisms

Structure of Homomorphisms Between Representations

• Homomorphisms between non-isomorphic irreducible representations must be zero
• Space of G-homomorphisms between isomorphic irreducible representations is one-dimensional
• Endomorphism ring of an irreducible representation isomorphic to a division algebra
• Analyze structure of intertwining operators between representations using Schur's Lemma
• Prove Wedderburn's theorem on structure of semisimple algebras with Schur's Lemma
• Establish relationship between irreducible representations and minimal ideals in group algebras
• Classify all finite-dimensional representations of certain groups (finite abelian groups) using Schur's Lemma

Orthogonality Relations for Characters

Character Properties and Orthogonality

• Character of a representation defined as trace function with specific properties
• Characters of non-isomorphic irreducible representations orthogonal with respect to inner product on class functions
• First orthogonality relation inner product of characters of distinct irreducible representations equals zero
• Second orthogonality relation sum of squares of dimensions of irreducible representations equals group order
• Orthogonality relations interpreted in terms of group algebra structure and decomposition into simple components
• Lead to concept of character table for finite groups
• Imply linear independence of irreducible characters

Applications of Orthogonality Relations

• Compute multiplicity of irreducible representation in given representation using orthogonality relations
• Develop formula for projection operator onto irreducible component using characters and orthogonality relations
• Use character table and orthogonality relations to decompose regular representation of finite group
• Prove uniqueness of representation decomposition into irreducible components with orthogonality relations
• Construct explicit bases for isotypic components of representation using relations
• Determine decomposition of tensor products of representations with orthogonality relations
• Apply orthogonality relations in practical applications of representation theory (cryptography, quantum mechanics)

Decomposing Representations using Orthogonality

Techniques for Representation Decomposition

• Apply orthogonality relations to compute multiplicity of irreducible representation in given representation
• Develop projection operator formula onto irreducible component using characters and orthogonality relations
• Decompose regular representation of finite group using character table and orthogonality relations
• Prove uniqueness of representation decomposition into irreducible components
• Construct explicit bases for isotypic components of representation
• Determine decomposition of tensor products of representations
• Computational aspects of using orthogonality relations in practical applications (group theory software, physics simulations)

Examples of Representation Decomposition

• Decompose representation of symmetric group S3 into irreducible components
• Analyze decomposition of permutation representation of dihedral group D8
• Compute multiplicities of irreducible representations in tensor product of two representations
• Decompose regular representation of cyclic group Cn into irreducible components
• Determine structure of induced representation from subgroup to full group using orthogonality relations
• Analyze representation of quaternion group Q8 and its decomposition
• Decompose representation arising from action of rotational symmetry group on spherical harmonics