Key Concepts of Character Tables to Know for Representation Theory

Character tables are essential tools in representation theory, summarizing the characters of irreducible representations of finite groups. They reveal important relationships between group structure, conjugacy classes, and representation dimensions, making them invaluable for understanding group actions and symmetries.

1. Definition of character tables

   • A character table is a square matrix that summarizes the characters of all irreducible representations of a finite group.
   • Rows correspond to irreducible representations, while columns correspond to conjugacy classes of the group.
   • The entries are complex numbers representing the trace of the representation matrices for each class.

2. Properties of character tables

   • The sum of the squares of the dimensions of the irreducible representations equals the order of the group.
   • Each character is a class function, meaning it is constant on conjugacy classes.
   • The character table is symmetric with respect to the main diagonal for groups with certain properties (e.g., abelian groups).

3. Orthogonality relations

   • The rows of the character table are orthogonal with respect to a specific inner product defined on the space of class functions.
   • The inner product of two different rows is zero, while the inner product of a row with itself equals the order of the group.
   • This orthogonality provides a powerful tool for determining the irreducible representations of a group.

4. Character table construction

   • To construct a character table, first identify the conjugacy classes of the group.
   • Next, determine the dimensions of the irreducible representations, often using the group’s order and properties.
   • Finally, compute the characters for each representation and fill in the table accordingly.

5. Irreducible representations

   • An irreducible representation is a representation that cannot be decomposed into smaller representations.
   • The number of distinct irreducible representations equals the number of conjugacy classes in the group.
   • Each irreducible representation has a corresponding character that encapsulates its properties.

6. Class functions

   • Class functions are functions defined on the group that are constant on conjugacy classes.
   • Characters of representations are examples of class functions, which play a crucial role in representation theory.
   • The space of class functions has a natural inner product that facilitates the study of representations.

7. Great Orthogonality Theorem

   • This theorem states that the characters of irreducible representations are orthogonal with respect to the inner product defined on class functions.
   • It provides a comprehensive framework for understanding the relationships between different representations.
   • The theorem also implies that the character table can be used to extract significant information about the group structure.

8. Character table symmetries

   • Character tables exhibit symmetries, such as the symmetry about the main diagonal in certain groups.
   • The characters of complex conjugate representations are related, leading to specific patterns in the table.
   • Understanding these symmetries can simplify the analysis of the group’s representations.

9. Character table applications

   • Character tables are used to classify finite groups and understand their structure.
   • They facilitate the computation of group actions and can be applied in physics, particularly in quantum mechanics.
   • Character tables also aid in solving problems related to group homomorphisms and extensions.

10. Burnside's theorem

    • Burnside's theorem provides a method for counting the number of distinct orbits of a group action, which can be related to character tables.
    • It states that the number of orbits is equal to the average number of points fixed by the group elements.
    • This theorem connects group actions with representation theory, enhancing the understanding of character tables.