Character Table

Definition

A character table is a square matrix that summarizes the characters of the irreducible representations of a finite group over a given field. Each entry in the table represents the trace of a group element in a specific representation, providing important insights into the group's structure and its representations. The character table serves as a powerful tool for analyzing symmetries and decomposing representations into their irreducible components.

5 Must Know Facts For Your Next Test

1. The character table includes rows for each irreducible representation and columns for each conjugacy class of the group.
2. The values in a character table are complex numbers, specifically the traces of group elements in their respective representations.
3. The orthogonality relations among rows and columns of the character table provide significant insights into the relationships between different representations.
4. Character tables can be used to determine if two finite groups are isomorphic by comparing their respective character tables.
5. The size of the character table relates directly to the number of conjugacy classes and irreducible representations of the group.

Review Questions

• How do character tables help in understanding the relationships between different irreducible representations of a group?
  • Character tables provide an organized way to see how different irreducible representations relate to each other through their characters. By examining the entries in a character table, one can apply orthogonality relations that reveal connections between representations. This makes it easier to analyze how these representations decompose or combine within the larger context of the group's structure.
• Discuss the significance of orthogonality relations in character tables and what information they convey about a group's representations.
  • Orthogonality relations indicate that different irreducible representations are independent from one another when analyzed via their characters. The row orthogonality states that the inner product of characters from different irreducible representations is zero, while column orthogonality relates to conjugacy classes. This relationship provides critical information regarding how representations interact and helps determine the dimensions and structure of these representations within the group.
• Evaluate how character tables can be utilized to compare finite groups for potential isomorphism and provide an example illustrating this process.
  • Character tables serve as an effective tool for determining whether two finite groups are isomorphic by comparing their character tables. If two groups have different numbers of conjugacy classes or differing dimensions of irreducible representations, they cannot be isomorphic. For instance, if Group A has a character table with three conjugacy classes and Group B has four, we can conclude they are not isomorphic. However, if both groups share identical character tables, further analysis would be needed to confirm their isomorphism.

Related terms

Irreducible Representation:

A representation of a group that cannot be decomposed into smaller representations, meaning it has no proper invariant subspaces.

Trace: The trace of a linear transformation or matrix is the sum of its diagonal elements, which has important implications for the study of characters in representations.

Group Homomorphism: A structure-preserving map between two groups that respects the group operations, essential for understanding how representations relate to group properties.