Induced representation

Definition

Induced representation is a way of constructing a representation of a group from a representation of one of its subgroups. This process is crucial in understanding how different representations can relate to each other, especially when dealing with larger groups built from smaller components. By inducing representations, we can analyze how characters behave under group actions and relate them to properties of the larger group through fundamental theorems.

5 Must Know Facts For Your Next Test

1. Induced representations are constructed using the notion of extending a representation from a subgroup to the entire group, usually denoted as ` ext{Ind}_H^G ho`, where `H` is a subgroup, `G` is the group, and ` ho` is the original representation.
2. The dimension of an induced representation can be computed using the index of the subgroup in the group and the dimension of the original representation.
3. Induced representations preserve certain properties like irreducibility under specific conditions, meaning that if the original representation is irreducible, so is its induced representation.
4. Characters associated with induced representations can be computed using the character table of both the subgroup and the group itself, making it easier to analyze their structure.
5. The process of inducing representations often involves summing over cosets, which plays a crucial role in how these representations interact with characters and other representations.

Review Questions

• How does the process of inducing a representation affect its properties compared to the original subgroup representation?
  • When inducing a representation, certain properties like irreducibility can be preserved under specific conditions. For instance, if the original representation from the subgroup is irreducible, it may still induce an irreducible representation in the larger group. However, this depends on the structure of both the subgroup and the larger group. Therefore, understanding these nuances helps in analyzing how representations evolve when transitioning from smaller subgroups to larger groups.
• Discuss how characters play a role in understanding induced representations and their significance in representation theory.
  • Characters are fundamental in understanding induced representations because they provide a compact way to analyze how representations behave under group actions. When you induce a representation, you can compute its character using characters from both the subgroup and the larger group. This relationship allows for deeper insights into the structure of groups and their representations, facilitating connections between different representations through their character tables.
• Evaluate Frobenius reciprocity and Mackey's theorem in relation to induced representations and their broader implications in representation theory.
  • Frobenius reciprocity establishes a critical connection between inducing and restricting representations, showing that these two processes are closely related. It states that there is an equivalence between morphisms of representations from `G` to `H` and morphisms from `H` to `G`. Mackey's theorem further refines this idea by providing conditions under which one can decompose induced representations into simpler components based on subgroup structures. Together, these concepts illustrate how intertwined different aspects of representation theory are and highlight their significance in analyzing group behavior.

Related terms

representation: A representation is a way to express group elements as matrices, allowing the study of group actions in a linear algebraic framework.

character: A character is a function that assigns a complex number to each group element, reflecting the trace of its corresponding matrix representation.

Frobenius reciprocity: Frobenius reciprocity relates the induction of representations from subgroups to the restriction of representations to subgroups, highlighting their interconnection.