Character table

Definition

A character table is a square matrix that provides information about the irreducible representations of a finite group, detailing how these representations act on group elements. Each row corresponds to an irreducible representation, while each column corresponds to a conjugacy class of the group. This structure reveals key properties of the group and is essential in understanding its representation theory and applications like the Peter-Weyl theorem.

5 Must Know Facts For Your Next Test

1. Character tables help determine the number of distinct irreducible representations for a finite group, which is crucial for understanding the group's structure.
2. The characters in the table are complex-valued functions that provide insight into how representations behave under group operations.
3. Character tables can be used to check if two groups are isomorphic, as isomorphic groups have identical character tables.
4. The orthogonality relations among rows and columns of the character table provide powerful tools for computations in representation theory.
5. In the context of compact Lie groups, character tables relate closely to harmonic analysis on these groups and their associated function spaces.

Review Questions

• How does a character table facilitate understanding the structure of a finite group?
  • A character table summarizes all the essential information about the irreducible representations of a finite group, allowing one to see how these representations relate to the group's structure. By examining the character table, one can easily identify properties such as the number of irreducible representations and their dimensions. Additionally, since each column corresponds to a conjugacy class, it highlights symmetries within the group, thus providing a deeper insight into its algebraic structure.
• Discuss the significance of orthogonality relations in character tables and their implications for representation theory.
  • Orthogonality relations in character tables state that different rows (representations) are orthogonal with respect to a specific inner product. This means that if two characters are from different irreducible representations, their inner product is zero. This property simplifies computations related to decomposing representations and helps in determining whether representations are distinct. Moreover, these relations are pivotal in proving various results in representation theory, making them fundamental tools for mathematicians.
• Evaluate how character tables connect with the Peter-Weyl theorem and contribute to harmonic analysis on groups.
  • Character tables serve as a bridge between representation theory and the Peter-Weyl theorem by providing detailed information about irreducible representations of compact groups. The Peter-Weyl theorem asserts that every continuous representation can be decomposed into these irreducible components, which are directly characterized by the data from character tables. As such, this connection allows for a comprehensive understanding of harmonic analysis on these groups, where one can analyze functions on compact groups using the framework established by their character tables.

Related terms

Irreducible Representation: A representation of a group that cannot be decomposed into a direct sum of smaller representations.

Conjugacy Class: A set of elements in a group that can be transformed into one another by conjugation, which is an important concept when analyzing group structure.

Peter-Weyl Theorem: A foundational result in representation theory that states every continuous representation of a compact group can be decomposed into finite-dimensional irreducible representations.