Glossary

7.3 Mackey's theorem and its consequences

2 min read•july 25, 2024

is a powerful tool for breaking down complex group representations into simpler parts. It's like a Swiss Army knife for mathematicians, helping them understand how representations of smaller groups fit into bigger ones.

The theorem's applications are wide-ranging, from analyzing symmetric groups to studying . It's a key player in representation theory, making tricky problems more manageable and revealing hidden patterns in group structures.

Mackey's Theorem and Its Applications

Mackey's theorem for induced representations

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  • Mackey's theorem describes utilizing
  • Key components involve groups HH, KK (subgroups of GG), representation ρ\rho of HH, and double cosets KgHKgH
  • Formula decomposes as IndHG(ρ)≅⨁g∈K\G/HIndK∩gHg−1K(ResK∩gHg−1gHg−1(gρ))\text{Ind}_H^G(\rho) \cong \bigoplus_{g \in K\backslash G/H} \text{Ind}_{K \cap gHg^{-1}}^K(\text{Res}_{K \cap gHg^{-1}}^{gHg^{-1}}(g\rho))
  • Interpretation sums over double coset representatives, applies and operations, and conjugates representation
  • Examples include decomposing representations of symmetric groups (S_n) and special linear groups (SL(n,R))

Proof of Mackey's theorem

  • Proof steps begin with definition, apply , use restriction and induction properties
  • Key properties used include transitivity of induction and compatibility with direct sums
  • Double coset decomposition partitions GG into KgHKgH double cosets
  • established through bijection between basis elements
  • GG-module structure verified by checking compatibility with group action
  • Example: proving theorem for ( D_8) decomposition

Decomposition of induced representations

  • Application steps identify groups GG, HH, KK, determine double cosets KgHKgH, compute intersections K∩gHg−1K \cap gHg^{-1}, calculate restrictions and conjugations, induce to KK
  • Examples include decomposing representations of symmetric groups (S_4) and orthogonal groups (O(3))
  • Simplification techniques use character theory and exploit group symmetries
  • Process illustrated with step-by-step decomposition of specific induced representation (representation of S_3 induced to S_4)

Applications of Mackey's theorem

  • Restriction of induced representations formula: ResKG(IndHG(ρ))≅⨁g∈K\G/HIndK∩gHg−1K(ResK∩gHg−1gHg−1(gρ))\text{Res}_K^G(\text{Ind}_H^G(\rho)) \cong \bigoplus_{g \in K\backslash G/H} \text{Ind}_{K \cap gHg^{-1}}^K(\text{Res}_{K \cap gHg^{-1}}^{gHg^{-1}}(g\rho))
  • Induction of restricted representations relates to original representation and analyzes multiplicities
  • Composition of induction and restriction compares with original representation and identifies new irreducible components
  • Subgroup analysis applications include examining normal subgroups and
  • Example: analyzing restriction of induced representation from cyclic subgroup to another subgroup in S_4

Consequences in representation theory

  • Theoretical implications deepen understanding of induced representations and connect subgroup structure to representations
  • Practical applications compute character tables and analyze representation stability
  • Extensions apply to on homogeneous spaces and
  • Classification problems use theorem to identify irreducible representations and construct representations from smaller groups
  • Related theorems include and
  • Computational aspects develop algorithms for decomposing induced representations and implement in computer algebra systems (GAP, SageMath)

Key Terms to Review (23)

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.
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.
Decomposition of induced representation: The decomposition of induced representation is the process of breaking down an induced representation into a direct sum of irreducible representations. This concept is essential for understanding how complex representations can be analyzed in simpler components, which can be particularly useful when applying Mackey's theorem to establish relationships between different groups and their representations. This breakdown not only aids in calculations but also deepens the understanding of the underlying structure of representations.
Dihedral group: The dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. It captures the essence of how geometric shapes can be manipulated while preserving their structure, making it a key example in understanding group actions and orbits, as well as representations of groups.
Double cosets: Double cosets are subsets formed in the context of group theory, where a group is partitioned by two subgroups. Specifically, given a group G and two subgroups H and K, the double coset of an element g in G with respect to H and K is the set of all elements of the form h*g*k for h in H and k in K. This concept plays a vital role in understanding the interactions between different groups through induction and restriction functors, as well as in the formulation and applications of Mackey's theorem.
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 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.
G-module structure: A g-module structure refers to a mathematical framework in which a module is equipped with the action of a group g, allowing for the study of representations of g within the context of modules. This structure is essential for analyzing how group elements act on vector spaces or other algebraic structures, providing insights into their properties and behavior. Understanding g-module structures is vital when applying Mackey's theorem, as it lays the groundwork for examining the decompositions and interactions between different representations under group actions.
Harmonic analysis: Harmonic analysis is a branch of mathematics that studies the representation of functions or signals as the superposition of basic waves, often using techniques from Fourier analysis. It connects various mathematical concepts such as group representations and symmetry, playing a crucial role in understanding how functions behave under transformations.
Induced representation: 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.
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 Representation: An irreducible representation is a linear representation of a group that cannot be decomposed into smaller, non-trivial representations. This concept is crucial in understanding how groups act on vector spaces, as irreducible representations form the building blocks from which all representations can be constructed, similar to prime numbers in arithmetic.
Irving Segal: Irving Segal was a prominent mathematician known for his significant contributions to representation theory, particularly in the context of induced representations and the study of unitary representations of groups. His work laid foundational aspects for Mackey's theorem, which deals with how representations can be constructed from smaller subgroups, illustrating the importance of the relationship between a group and its subgroups.
Isomorphism of Vector Spaces: An isomorphism of vector spaces is a bijective linear map between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that if two vector spaces are isomorphic, they are structurally the same, allowing for a one-to-one correspondence between their elements while maintaining the algebraic structure. In the context of representation theory, understanding isomorphisms can provide insights into the relationships between different representations and their associated vector spaces.
Mackey's Theorem: Mackey's Theorem provides a fundamental framework in representation theory, particularly dealing with the relationship between representations of a group and its subgroups. It characterizes how to decompose representations when passing from a group to its subgroups, establishing a crucial connection between induced representations and characters. The theorem highlights the way in which irreducible representations can be understood through their restrictions and how characters behave under induction.
Orthogonal Group: The orthogonal group, denoted as O(n), is the group of all n x n orthogonal matrices, which are square matrices whose columns and rows are orthonormal vectors. This group plays a crucial role in various mathematical fields, including representation theory, as it preserves inner products and thus geometric structures in Euclidean spaces. Orthogonal groups can be studied through their representations, where Mackey's theorem provides insights into the behavior of such representations under the action of subgroups.
P-adic group representation theory: p-adic group representation theory studies representations of p-adic groups, which are groups defined over p-adic numbers. These representations play a crucial role in number theory and algebraic geometry, providing insights into the structure and properties of these groups. The connections between representations and various mathematical concepts, including Mackey's theorem, reveal how they interact with different subgroups and lead to decompositions that help understand the overall structure of the group.
Paul Halmos: Paul Halmos was a prominent mathematician known for his contributions to various fields, including functional analysis, operator theory, and probability. He played a significant role in the development of representation theory, especially through his work that laid the groundwork for concepts like Mackey's theorem, which connects group representations with measure theory and harmonic analysis.
Quantum Mechanics: Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. This theory introduces concepts such as wave-particle duality, uncertainty principle, and quantization, which are pivotal in understanding the behavior of particles and their interactions. Its principles have far-reaching implications across various fields, including representation theory, where it intersects with topics like matrix representations and orthogonality relations.
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.
Special Linear Group: The special linear group, denoted as $$SL(n, ext{F})$$, is the group of n x n matrices with determinant equal to 1, where F is a field. This group plays a crucial role in various areas of mathematics, including representation theory, as it captures symmetries that preserve volume and orientation in vector spaces. The structure and properties of the special linear group help in understanding more complex groups and their representations, linking them to Mackey's theorem and related concepts.
Symmetric group: The symmetric group, denoted as $$S_n$$, is the group of all permutations of a finite set of $$n$$ elements, capturing the essence of rearranging objects. This group is fundamental in understanding how groups act on sets, with its elements representing all possible ways to rearrange the members of the set, leading to various applications in algebra and combinatorics.
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