Linear character

Definition

A linear character is a homomorphism from a group to the multiplicative group of a field, often the complex numbers, that reflects the structure of the group in a one-dimensional way. This type of character can be thought of as a representation of the group that preserves the group operation and has a particularly simple form, making it easier to study properties such as irreducibility and induction in character theory.

5 Must Know Facts For Your Next Test

1. Linear characters are often associated with abelian groups since every linear character of an abelian group can be viewed as a group homomorphism.
2. In terms of representation theory, linear characters provide one-dimensional representations, making them easier to analyze than higher-dimensional representations.
3. The set of all linear characters of a group forms an abelian group under pointwise multiplication, which can provide important information about the original group's structure.
4. Every finite group has at least one linear character, specifically the trivial character that maps every group element to 1.
5. Linear characters are particularly useful in the study of the group's representation theory because they can help decompose more complex representations into simpler ones.

Review Questions

• How do linear characters differ from general characters in the context of group representations?
  • Linear characters are a specific subset of characters that map group elements to one-dimensional representations in the multiplicative group of a field. While general characters can have higher dimensions, linear characters simplify analysis by providing straightforward mappings. This distinction is important when considering how these characters can reveal structural properties of the group, especially for abelian groups.
• In what ways can linear characters contribute to understanding irreducible representations of finite groups?
  • Linear characters serve as one-dimensional irreducible representations, which means they cannot be further decomposed into smaller representations. By studying these linear characters, one can gain insights into the structure and behavior of more complex irreducible representations. Furthermore, because every irreducible representation over an algebraically closed field has a corresponding linear character, they play a crucial role in classification and analysis of finite group representations.
• Evaluate the implications of having multiple linear characters for an abelian group and how this affects its overall representation theory.
  • The presence of multiple linear characters for an abelian group indicates rich structural properties and allows for detailed exploration within its representation theory. Each linear character corresponds to a unique homomorphism, creating distinct one-dimensional representations that help form an abelian group under pointwise multiplication. This multiplicity leads to powerful tools for decomposing more complex representations into simpler components, facilitating understanding of interactions between different representations and their corresponding characters within the group's broader framework.

Related terms

Character: A character is a homomorphism from a group to the field of complex numbers, providing insight into the group's structure and representation.

Irreducible Representation: An irreducible representation is a representation that has no non-trivial invariant subspaces, meaning it cannot be decomposed into simpler representations.

Group Homomorphism: A group homomorphism is a function between two groups that preserves the group operation, mapping elements from one group to another in a way that reflects their structure.