Character of a representation

Definition

The character of a representation is a function that assigns to each group element the trace of the corresponding linear transformation in a representation of that group. It is an important tool in understanding how representations behave, especially when analyzing subgroups and the relationships between different representations through concepts like induction and restriction.

5 Must Know Facts For Your Next Test

1. The character is constant on conjugacy classes, meaning it takes the same value for all elements that are conjugate to each other.
2. Characters are particularly useful for determining if two representations are equivalent; they can be compared by checking if their characters are identical.
3. In finite groups, characters can be used to derive important results like the orthogonality relations, which help in understanding the structure of representations.
4. The character of a trivial representation (where every group element acts as the identity) is always equal to the order of the group.
5. Characters can provide insight into subgroup structure through their behavior under induction and restriction, particularly in understanding how representations decompose.

Review Questions

• How does the character of a representation provide insights into conjugacy classes within a group?
  • The character of a representation is constant on conjugacy classes, meaning it assigns the same value to all elements that belong to the same conjugacy class. This property highlights how characters can be used to study the group's structure, as it allows us to reduce complex problems to simpler ones involving these classes. By examining how characters behave over conjugacy classes, we can infer information about the symmetry and relations within the group.
• Discuss how characters can be used to determine the equivalence of two representations and what implications this has for understanding group representations.
  • Characters serve as a powerful tool for determining whether two representations are equivalent. If two representations have identical characters for every group element, they are equivalent. This equivalence indicates that they provide the same information about the group and its structure. This relationship simplifies the study of representations, allowing mathematicians to classify and compare them based solely on their characters rather than their full structural details.
• Evaluate how characters play a role in understanding subgroup structures and their representations, particularly through induction and restriction processes.
  • Characters are instrumental in analyzing subgroup structures through induction and restriction processes. When we restrict a representation to a subgroup, the character can indicate how much of the original representation 'survives' in that subgroup. Conversely, inducing a character from a subgroup allows us to understand how it contributes to larger representations. The interactions between these characters under induction and restriction provide deep insights into how different parts of a group relate to each other and contribute to the overall representation theory.

Related terms

Representation:

A representation is a homomorphism from a group to the general linear group of a vector space, allowing the group to be expressed in terms of linear transformations.

Trace: The trace of a linear transformation is the sum of the diagonal elements of its matrix representation, providing a scalar value that gives insight into the transformation's properties.

Induced Representation: An induced representation is created from a smaller subgroup representation, allowing us to build larger representations while maintaining relationships between groups.