4.2 Derivation of Schur orthogonality relations

Schur orthogonality relations are crucial in representation theory, linking matrix elements of irreducible representations. They provide a powerful tool for analyzing group structures and their representations, forming the foundation for many applications in physics and mathematics.

These relations, expressed through integrals over group elements, reveal the orthogonality between different representations and within matrix elements. This property is key in decomposing representations and understanding the structure of group algebras.

Matrix Elements and Integral Formulation

Integral formulation of Schur orthogonality

Top images from around the web for Integral formulation of Schur orthogonality

• 

  Representation theory of finite groups - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

• 

  SU(3) [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Representation theory of finite groups - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

1 of 3

Top images from around the web for Integral formulation of Schur orthogonality

• 

  Representation theory of finite groups - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

• 

  SU(3) [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Representation theory of finite groups - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

1 of 3

• Irreducible representations form building blocks of group theory unable to be broken down further
• Matrix elements $D^j_{mn}(g)$ represent specific entries in representation matrices
• Group integration utilizes Haar measure preserves volume under group operations
• Integral formulation $\int dg D^j_{mn}(g) D^{j'}_{m'n'}(g^{-1})$ integrates over entire group
• Normalization accounts for group volume and representation dimensionality

Proof of Schur orthogonality relations

• First relation $\int dg D^j_{mn}(g) D^{j'}_{m'n'}(g^{-1}) = \frac{1}{d_j} \delta_{jj'} \delta_{mm'} \delta_{nn'}$
  • Proof involves:
    1. Applying completeness relation
    2. Using Peter-Weyl theorem
    3. Simplifying with matrix element properties
• Second relation $\int dg \chi^j(g) \chi^{j'}(g^{-1}) = \delta_{jj'}$
  • Proof involves:
    1. Defining character as representation trace
    2. Relating to first orthogonality relation
    3. Summing over indices

Interpretation and Extensions

Role of Kronecker delta function

• Kronecker delta $\delta_{ij} = 1$ if $i = j$, 0 otherwise appears in orthogonality relations
• $\delta_{jj'}$ indicates orthogonality between different representations
• $\delta_{mm'}$ and $\delta_{nn'}$ show orthonormality within a representation
• Ensures linear independence of matrix elements and completeness of irreducible representations (SU(2), SO(3))

Extension to unitary representations

• Unitary representations satisfy $D(g^{-1}) = D(g)^\dagger$
• Simplified integral formulation for unitary cases
• First relation becomes $\int dg D^j_{mn}(g) D^{j'}_{m'n'}(g)^* = \frac{1}{d_j} \delta_{jj'} \delta_{mm'} \delta_{nn'}$
• Second relation remains unchanged
• Applications in quantum mechanics and symmetry analysis (angular momentum, crystal structures)