Induced representations

Definition

Induced representations are a way of constructing a representation of a group from a representation of a subgroup. This process allows us to explore the relationship between groups and their subgroups, revealing how representations can be 'induced' to the larger group. This concept is crucial for understanding the interplay between different groups, especially in finite group theory and the applications of Frobenius reciprocity.

5 Must Know Facts For Your Next Test

1. Induced representations are created by taking a representation of a subgroup and extending it to the entire group, often using direct sums or integrals over cosets.
2. The process involves understanding how elements from the subgroup act on a vector space and then translating this action to the larger group setting.
3. Frobenius reciprocity plays an important role in establishing relationships between induced representations and those of subgroups, ensuring that there is consistency in how representations relate across different groups.
4. Inducing representations is particularly valuable in finite group theory, where it helps classify and understand the irreducible representations of groups by examining their subgroups.
5. The dimension of an induced representation can often be determined using character theory, which relates characters to irreducible representations and provides insights into the structure of the group.

Review Questions

• How does inducing a representation from a subgroup help us understand the structure of the larger group?
  • Inducing a representation from a subgroup allows us to extend the behavior of that subgroup's representation to the entire group. This helps reveal important structural properties and relationships between different parts of the group. By examining how elements from the subgroup act on vectors, we can uncover insights into how the larger group operates, as well as identify irreducible representations and their dimensions.
• What role does Frobenius reciprocity play in understanding induced representations, and how does it facilitate connections between groups?
  • Frobenius reciprocity establishes a fundamental link between the representations of a group and its subgroup by providing a way to compute homomorphisms between these representations. This principle ensures that we can analyze induced representations in two ways: by looking at them directly or through their relationship with subgroups. This dual perspective allows for deeper insights into both the group's structure and how representations can be effectively induced.
• Evaluate how inducing representations contributes to our classification of finite groups and their irreducible representations.
  • Inducing representations is crucial for classifying finite groups because it allows us to systematically build up larger representations from smaller ones based on subgroup behavior. This method helps identify all irreducible representations by leveraging character theory and understanding how dimensions relate across induced spaces. Furthermore, this process not only aids in representing complex groups but also highlights symmetries and underlying structures within finite groups that may not be apparent when considering them in isolation.

Related terms

Frobenius reciprocity: A principle that connects the representations of a group and its subgroup, stating that the space of homomorphisms between an induced representation and another representation can be calculated in two different ways.

representation: A homomorphism from a group to the general linear group of a vector space, allowing us to study group elements as linear transformations.

subgroup: A subset of a group that is itself a group under the operation defined on the original group, which plays a key role in the process of inducing representations.