Glossary

4.2 Derivation of Schur orthogonality relations

2 min read•july 25, 2024

are crucial in representation theory, linking of . 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

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  • Irreducible representations form building blocks of group theory unable to be broken down further
  • Matrix elements Dmnj(g)D^j_{mn}(g) represent specific entries in representation matrices
  • utilizes preserves volume under group operations
  • Integral formulation ∫dgDmnj(g)Dm′n′j′(g−1)\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 ∫dgDmnj(g)Dm′n′j′(g−1)=1djδjj′δmm′δnn′\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
      2. Using
      3. Simplifying with matrix element properties
  • Second relation ∫dgχj(g)χj′(g−1)=δjj′\int dg \chi^j(g) \chi^{j'}(g^{-1}) = \delta_{jj'}
    • Proof involves:
      1. Defining as representation trace
      2. Relating to first orthogonality relation
      3. Summing over indices

Interpretation and Extensions

Role of Kronecker delta function

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

Extension to unitary representations

  • satisfy D(g−1)=D(g)†D(g^{-1}) = D(g)^\dagger
  • Simplified integral formulation for unitary cases
  • First relation becomes ∫dgDmnj(g)Dm′n′j′(g)∗=1djδjj′δmm′δnn′\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 and (, )

Key Terms to Review (15)

Angular Momentum: Angular momentum is a physical quantity that represents the rotational inertia and angular velocity of a rotating body, quantifying the amount of rotational motion an object possesses. It plays a crucial role in understanding various phenomena in physics, particularly in systems governed by symmetries, such as conservation laws. Its importance is amplified in contexts where rotational motion interacts with symmetries, leading to essential results in representation theory.
Character: In representation theory, a character is a function that assigns to each group element the trace of its corresponding matrix representation. Characters provide deep insights into the structure of representations, revealing information about their irreducibility and symmetry properties.
Completeness relation: A completeness relation refers to an equation that expresses how a complete set of states or functions can reconstruct any element in a given space. In the context of representation theory, this concept plays a crucial role in understanding the relationships between different representations of a group, particularly how they can be combined to form new representations.
Crystal Structures: Crystal structures refer to the organized, repeating arrangement of atoms, ions, or molecules in a solid material, which defines its geometric shape and properties. These structures are crucial for understanding how materials interact and transform under different conditions, influencing characteristics like strength, conductivity, and optical behavior. The study of crystal structures plays a significant role in various areas such as solid-state physics, chemistry, and materials science.
Group integration: Group integration refers to the process by which elements of a group representation are combined to form a cohesive structure that reflects the symmetries and interactions of the group. This concept plays a significant role in various mathematical and physical contexts, allowing for the understanding of how different representations can be combined to analyze group properties effectively.
Haar measure: Haar measure is a type of measure defined on locally compact topological groups that allows for the integration of functions over the group in a way that is invariant under group translation. This means that the measure does not change when you shift the entire group, making it essential for understanding representation theory and harmonic analysis. Haar measure is crucial for establishing concepts like orthogonality and completeness of irreducible representations in the context of group representations.
Irreducible Representations: Irreducible representations are the simplest non-trivial representations of a group that cannot be decomposed into smaller representations. These representations form the building blocks of representation theory, and understanding them is essential for analyzing more complex structures within the field. They are closely tied to orthogonality relations, Schur's lemma, and applications such as Frobenius reciprocity and Clebsch-Gordan coefficients.
Kronecker Delta: The Kronecker delta is a function of two variables, typically denoted as \(\delta_{ij}\), that equals 1 when the indices are equal (i.e., \(i = j\)) and 0 when they are not (i.e., \(i \neq j\)). This function is essential in various mathematical contexts, particularly in linear algebra and representation theory, as it is used to express orthogonality relations among basis elements.
Linear Independence: Linear independence refers to a set of vectors in a vector space that cannot be expressed as a linear combination of each other. If a set of vectors is linearly independent, it means that none of the vectors can be written as a sum of multiples of the others, which is crucial for understanding the structure of vector spaces and their dimensions.
Matrix elements: Matrix elements are the components of matrices that represent linear transformations in vector spaces, often corresponding to the action of operators in representation theory. They provide a way to understand how different basis states interact and can be used to analyze symmetries and invariant properties of systems. In the context of Schur orthogonality relations, matrix elements help characterize representations of groups through their inner products.
Peter-Weyl Theorem: The Peter-Weyl Theorem is a fundamental result in representation theory that describes the decomposition of representations of compact topological groups into a direct sum of irreducible representations. This theorem connects the study of representations to harmonic analysis, allowing for a structured way to analyze how complex representations can be broken down into simpler, irreducible components, which is crucial in understanding the structure of the representation space.
Quantum Mechanics: Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. This theory introduces concepts such as wave-particle duality, uncertainty principle, and quantization, which are pivotal in understanding the behavior of particles and their interactions. Its principles have far-reaching implications across various fields, including representation theory, where it intersects with topics like matrix representations and orthogonality relations.
Schur Orthogonality Relations: Schur orthogonality relations are mathematical principles that establish the orthogonality of characters of finite groups, specifically in the context of representation theory. They provide a powerful framework for analyzing how different representations interact and are crucial for understanding the inner product of characters, leading to significant results in group theory and its applications.
Symmetry analysis: Symmetry analysis is a method used to identify and characterize the symmetrical properties of mathematical objects or physical systems. It plays a critical role in representation theory, helping to simplify complex problems by exploiting symmetrical structures, which can lead to powerful results such as the Schur orthogonality relations.
Unitary representations: Unitary representations are homomorphisms from a group to the group of unitary operators on a Hilbert space, preserving the group structure and ensuring that the inner product is maintained. They play a crucial role in quantum mechanics and representation theory, as they allow groups to act on complex vector spaces while preserving their geometric properties.
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