Correlation Tables to Know for Representation Theory

Correlation tables are essential tools in Representation Theory, summarizing irreducible representations and their relationships. They help analyze symmetry in molecular systems, subgroups, and direct products, providing insights into molecular shapes, vibrational modes, and physical properties across various contexts.

1. Character tables

   • Provide a concise summary of the irreducible representations of a group.
   • Include characters, which are traces of the representation matrices.
   • Facilitate the determination of symmetry properties in molecular systems.

2. Correlation tables for subgroups

   • Show how irreducible representations of a group relate to those of its subgroups.
   • Help in understanding the behavior of representations under group restrictions.
   • Useful for analyzing the symmetry of smaller systems within larger groups.

3. Correlation tables for direct products

   • Illustrate how to construct representations of a direct product of groups.
   • Allow for the combination of representations from individual groups.
   • Essential for studying systems with multiple independent symmetries.

4. Correlation tables for symmetry groups

   • Detail the relationships between symmetry operations and their corresponding representations.
   • Aid in identifying how different symmetry elements interact within a system.
   • Important for classifying molecular and crystal symmetries.

5. Correlation tables for point groups

   • Focus on the symmetry operations of molecules that do not change their center of mass.
   • Provide insight into the molecular shape and its symmetry properties.
   • Crucial for predicting spectroscopic behavior and chemical reactivity.

6. Correlation tables for space groups

   • Extend point group analysis to three-dimensional periodic structures.
   • Include translations along with symmetry operations.
   • Important for crystallography and understanding crystal structures.

7. Correlation tables for irreducible representations

   • List the irreducible representations of a group and their dimensions.
   • Serve as a foundation for building more complex representations.
   • Key for simplifying the analysis of physical systems in quantum mechanics.

8. Correlation tables for reducible representations

   • Show how reducible representations can be decomposed into irreducible components.
   • Help in identifying the symmetry properties of complex systems.
   • Useful for understanding the behavior of systems under various symmetry operations.

9. Correlation tables for symmetry operations

   • Map out the effects of symmetry operations on molecular orbitals and vibrations.
   • Provide a framework for analyzing how symmetry influences physical properties.
   • Essential for predicting the outcomes of spectroscopic experiments.

10. Correlation tables for molecular vibrations

    • Relate vibrational modes of molecules to their symmetry properties.
    • Help in determining which vibrational modes are active in infrared or Raman spectroscopy.
    • Important for understanding molecular dynamics and stability.