Glossary

7.2 Applications of Frobenius reciprocity

2 min read•july 25, 2024

is a powerful tool in representation theory, connecting induced and restricted representations. It helps us understand how representations of a group relate to those of its subgroups, simplifying complex calculations.

This theorem is crucial for decomposing representations, computing , and analyzing group-subgroup relationships. It's the foundation for many important results in representation theory, from branching rules to theorems.

Understanding Frobenius Reciprocity

Decomposition of restricted representations

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  • states ⟨IndHG(V),W⟩G=⟨V,ResHG(W)⟩H\langle \text{Ind}_H^G(V), W \rangle_G = \langle V, \text{Res}_H^G(W) \rangle_H applies to finite-dimensional representations of finite groups (, symmetric groups)
  • Decomposition uses inner product of characters and applies Frobenius reciprocity to relate induced and restricted representations
  • Steps for decomposition:
    1. Identify group G and subgroup H (G = S4, H = S3)
    2. Determine of G
    3. Calculate restricted characters
    4. Use Frobenius reciprocity to find multiplicities
  • Branching rules describe representation decomposition when restricted to a subgroup derived using Frobenius reciprocity (SO(3) to SO(2))

Character values of induced representations

  • for χIndHG(V)(g)=1∣H∣∑x∈GχV(x−1gx)\chi_{\text{Ind}_H^G(V)}(g) = \frac{1}{|H|} \sum_{x \in G} \chi_V(x^{-1}gx) summed over elements where x−1gx∈Hx^{-1}gx \in H
  • Frobenius reciprocity for character computations relates inner products of characters of G to those of H
  • Steps for computing induced character values:
    1. Start with original representation character
    2. Apply character formula
    3. Use and to simplify calculations ()
  • generalizes character formula for induced representations useful for complex group structures (semidirect products)

Transitivity of induction and restriction

  • Transitivity of induction: IndKG(IndHK(V))≅IndHG(V)\text{Ind}_K^G(\text{Ind}_H^K(V)) \cong \text{Ind}_H^G(V) for subgroups H≤K≤GH \leq K \leq G
  • : ResHK(ResKG(W))≅ResHG(W)\text{Res}_H^K(\text{Res}_K^G(W)) \cong \text{Res}_H^G(W)
  • Frobenius reciprocity in transitive induction: ⟨IndHG(V),W⟩G=⟨V,ResHG(W)⟩H\langle \text{Ind}_H^G(V), W \rangle_G = \langle V, \text{Res}_H^G(W) \rangle_H simplifies calculations involving multiple inductions or restrictions
  • Problem-solving strategies:
    1. Identify subgroup chain (H < K < G)
    2. Apply transitivity properties to simplify expressions
    3. Use Frobenius reciprocity to switch between induction and
  • relates restriction of induced representation to induced representations of intersections (Mackey decomposition)

Representations of groups vs subgroups

  • describes relationship between irreducible representations of group and normal subgroups uses Frobenius reciprocity in proofs
  • Restriction and induction functors:
    • Restriction: from G representations to H representations
    • Induction: from H representations to G representations
    • Adjoint functors related by Frobenius reciprocity
  • states every character of is Z-linear combination of characters induced from elementary subgroups (p-groups, cyclic groups)
  • states every character is rational linear combination of characters induced from cyclic subgroups
  • Applications in , , and ()

Key Terms to Review (23)

Algebraic number theory: Algebraic number theory is a branch of mathematics that deals with the properties of numbers and the relationships between them, particularly focusing on algebraic integers and their extensions. This field studies how different number systems relate to each other through various algebraic structures, leading to insights about the solutions of polynomial equations and their implications in other areas like representation theory. Understanding these interactions can reveal essential features, such as orthogonality relations and their consequences, and explore applications like Frobenius reciprocity.
Artin's Induction Theorem: Artin's Induction Theorem is a key result in representation theory that provides a way to relate the representations of a subgroup to the representations of the whole group. It essentially states that if you have a representation of a subgroup, you can 'induce' a representation of the entire group from it, preserving certain properties. This theorem is vital for understanding how representations behave under the processes of induction and restriction, and it ties into fundamental concepts like Frobenius reciprocity and Mackey's theorem.
Brauer's Induction Theorem: Brauer's Induction Theorem is a fundamental result in representation theory that relates the representations of a group to those of its subgroup through induction and restriction processes. It establishes a method for understanding how representations can be extended from subgroups to the whole group, bridging the gap between smaller and larger symmetry structures. This theorem plays a crucial role in applications of Frobenius reciprocity, providing a framework for analyzing how representations behave under these transformations.
Character formula: The character formula is a mathematical expression that relates the characters of representations of groups to their decomposition into irreducible components. It helps in understanding how different representations correspond with each other and can be particularly useful in calculating characters in the context of Frobenius reciprocity, which connects representations of groups with those of their subgroups and quotient groups.
Character values: Character values are complex numbers associated with the irreducible representations of a group, reflecting how group elements act within these representations. These values provide insight into the structure of the group, allowing for analysis of various properties like symmetry and decomposition of representations. They play a key role in understanding orthogonality relations and Frobenius reciprocity in representation theory.
Class field theory: Class field theory is a fundamental branch of algebraic number theory that describes the relationship between abelian extensions of number fields and their ideal class groups. This theory provides a way to understand how Galois groups of extensions can be interpreted through the lens of arithmetic properties, linking them to important concepts like reciprocity laws and L-functions.
Class Functions: Class functions are functions defined on the elements of a group that only depend on the conjugacy classes of those elements. They play a significant role in representation theory, particularly in analyzing representations and their properties, such as irreducibility and character theory. Class functions are particularly useful for studying symmetries and can be applied to derive important results like Burnside's theorem, facilitate understanding irreducible representations, and leverage Frobenius reciprocity in the context of group actions.
Clifford Theory: Clifford Theory is a fundamental result in representation theory that connects representations of a group with those of its subgroups, particularly highlighting how induction and restriction functors behave in this context. It provides a framework for understanding how the representations of a group can be constructed from the representations of its subgroups, which plays a significant role in analyzing more complex structures like symmetric and alternating groups. By emphasizing the relationships between different groups, Clifford Theory aids in various applications including the analysis of character theory and decomposition of representations.
Conjugacy Classes: Conjugacy classes are subsets of a group formed by grouping elements that are related through conjugation, meaning if one element can be transformed into another via an inner automorphism. Each conjugacy class represents a distinct behavior of elements in a group and plays a crucial role in understanding the structure of groups, especially when constructing character tables, analyzing irreducible representations, applying Frobenius reciprocity, and utilizing Mackey's theorem.
Cyclic Groups: A cyclic group is a type of group that can be generated by a single element, meaning every element in the group can be expressed as a power of that generator. These groups play a crucial role in various areas, such as understanding irreducible representations, character theory, and the applications of Frobenius reciprocity. The structure of cyclic groups helps to simplify complex problems by leveraging their inherent properties.
Double coset formula: The double coset formula is a mathematical expression used in representation theory that relates the decomposition of representations of a group into representations of its subgroups. This formula helps to analyze how a group action can be broken down when considering two subgroups, providing a powerful tool for understanding the structure and relationships between different representations.
Finite group: A finite group is a set equipped with a binary operation that satisfies the group properties (closure, associativity, identity, and invertibility) and has a finite number of elements. This concept is crucial for understanding various topics in representation theory, as the structure and properties of finite groups significantly influence their representations and character theory.
Frobenius Reciprocity: Frobenius reciprocity is a fundamental concept in representation theory that describes a relationship between induced representations and restricted representations of groups. It states that there is a natural correspondence between homomorphisms from an induced representation to a representation and homomorphisms from the original representation to the restricted representation, facilitating the transition between different levels of group representations.
Frobenius Reciprocity Theorem: The Frobenius Reciprocity Theorem is a fundamental result in representation theory that relates the induction and restriction of representations of groups. It essentially states that the inner product of a representation induced from a subgroup and a representation restricted to that subgroup can be expressed in terms of the inner products of the representations on the original group, highlighting the deep connections between these two processes.
Induced representations: 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.
Induction: Induction is a method of reasoning that establishes the truth of a statement by proving it for a base case and then showing that if it holds for an arbitrary case, it also holds for the next case. This technique is especially useful in areas like representation theory, where it helps in constructing representations and understanding their properties, connecting foundational concepts with complex applications in group theory and algebra.
Irreducible Representations: Irreducible representations are the simplest non-trivial representations of a group that cannot be decomposed into smaller representations. These representations form the building blocks of representation theory, and understanding them is essential for analyzing more complex structures within the field. They are closely tied to orthogonality relations, Schur's lemma, and applications such as Frobenius reciprocity and Clebsch-Gordan coefficients.
Mackey's Formula: Mackey's Formula is a result in representation theory that relates the characters of representations of a group to those of its subgroups. It provides a way to compute the character of an irreducible representation of a group in terms of the characters of its subgroup representations and their respective actions. This concept is crucial for understanding how representations behave under induction and restriction, as well as in applications of Frobenius reciprocity and finite group theory.
Modular representation theory: Modular representation theory studies representations of groups over fields with characteristic dividing the order of the group. This theory is essential for understanding how groups can be represented through matrices and linear transformations in contexts where standard representation theory may not apply, particularly when working with finite groups and their representations in modular arithmetic settings.
Permutation groups: Permutation groups are mathematical structures that describe the symmetries of a set by representing the set's elements as permutations, or rearrangements. Each element in a permutation group corresponds to a bijective function that maps the set onto itself, showcasing how elements can be transformed while preserving their relationships. This concept plays a crucial role in understanding the actions of groups on sets and is essential for various applications, including combinatorics and the study of symmetry in algebra.
Representation Stability: Representation stability refers to a phenomenon in representation theory where the dimensions of the spaces of representations for a given group, as it varies with respect to some parameter (like degree or size), exhibit a predictable and stable pattern. This concept highlights that as the group grows or changes, the representations maintain certain structural similarities, often leading to insights about their reducibility, equivalence, and how they relate to other groups, especially in specific applications like Frobenius reciprocity.
Restriction: Restriction refers to the process of limiting a representation of a group to a smaller subgroup. This concept allows us to study how representations behave when we focus on just a part of the group, providing insight into the relationship between different representations and their induced counterparts.
Transitivity of Restriction: Transitivity of restriction is a concept in representation theory that describes how the restriction of representations can be understood across different groups in a chain. It shows that if you restrict a representation from a group to a subgroup, and then restrict that result further to another subgroup, you can directly relate it to the representation restricted from the original group to the second subgroup. This property highlights the interconnection between representations of various groups and their subgroups.
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