Character Table

Definition

A character table is a mathematical tool used in representation theory that summarizes the characters of a group for each of its irreducible representations. It provides crucial information about the symmetries and structure of a group by listing characters corresponding to each group element and representation, helping to analyze the group's representations and their properties.

5 Must Know Facts For Your Next Test

1. Character tables are square matrices where rows correspond to irreducible representations and columns correspond to conjugacy classes of a group.
2. The entries in a character table are complex numbers representing the trace of the corresponding group elements under each irreducible representation.
3. Each character in a character table satisfies certain orthogonality relations, which provide powerful tools for computing characters and understanding their properties.
4. The size of the character table is determined by the number of conjugacy classes, which corresponds to the number of distinct irreducible representations of the group.
5. Character tables can be used to easily derive important properties of groups, such as their order and abelian nature, making them essential in many applications of representation theory.

Review Questions

• How does the structure of a character table relate to the irreducible representations of a group?
  • A character table captures the essence of a group's irreducible representations by organizing them in a way that highlights their relationships with conjugacy classes. Each row corresponds to an irreducible representation, while each column represents a conjugacy class. This structure allows us to easily analyze how different representations behave with respect to specific group elements, revealing deep insights into the group's symmetry properties.
• Discuss the significance of orthogonality relations in understanding character tables and their implications for representation theory.
  • Orthogonality relations are fundamental to character tables as they govern how characters interact with one another. These relations state that characters from different irreducible representations are orthogonal when paired with each other over a certain inner product. This property allows mathematicians to compute unknown characters and verify existing ones, providing a structured way to explore the complexity of group representations and ensuring that the character table remains a useful tool in representation theory.
• Evaluate how character tables can be applied to determine properties like the abelian nature or order of a group, and what this reveals about its structure.
  • Character tables serve as powerful tools for deducing key properties of groups by analyzing their characters. For instance, if all characters are either 1 or -1, it indicates that the group is abelian. Additionally, by counting the dimensions of irreducible representations as given in the character table, one can derive the order of the group using formulas derived from Burnside's theorem. This application reveals not only structural aspects but also connects different areas within algebra through representation theory.

Related terms

Irreducible Representation: A representation that cannot be decomposed into smaller representations; it is the building block of all representations of a group.

Group Homomorphism: A function between two groups that preserves the group operation, often used to analyze relationships between different groups.

Orthogonality Relations: Mathematical relations that describe the orthogonality of characters from different irreducible representations, crucial for understanding the structure of character tables.