🧩Representation Theory Unit 7 – Frobenius Reciprocity & Mackey's Theorem

Key Concepts and Definitions

• Representation theory studies abstract algebraic structures by representing their elements as linear transformations of vector spaces
• Frobenius reciprocity relates induced representations and restricted representations in finite groups
• Mackey's theorem describes the composition of induced representations and restricted representations
• Induced representation constructs a representation of a group from a representation of a subgroup
  • Denoted as $Ind_H^G(\rho)$ where $\rho$ is a representation of subgroup $H$ and $G$ is the larger group
• Restricted representation takes a representation of a group and restricts it to a representation of a subgroup
  • Denoted as $Res_H^G(\pi)$ where $\pi$ is a representation of group $G$ and $H$ is the subgroup
• Character theory assigns to each representation a character, which is a class function on the group
• Intertwining number counts the multiplicity of one representation appearing in another representation

Historical Context and Development

• Representation theory has its roots in the study of permutation groups and matrix groups in the late 19th century
• Frobenius introduced the concept of induced representations in 1898 while studying the representation theory of finite groups
• Mackey developed his theorem in the 1950s as part of his work on the unitary representations of locally compact groups
• The development of representation theory was influenced by the needs of quantum mechanics and particle physics
  • Representations are used to describe the symmetries of physical systems
• Representation theory has grown into a vast and active area of mathematics with connections to many other fields
  • Including number theory, algebraic geometry, combinatorics, and mathematical physics
• The study of infinite-dimensional representations, such as those arising in the representation theory of Lie groups, has become increasingly important

Frobenius Reciprocity: Statement and Intuition

• Frobenius reciprocity states that for a finite group $G$, a subgroup $H$, and representations $\rho$ of $H$ and $\pi$ of $G$:
  • $Hom_G(Ind_H^G(\rho), \pi) \cong Hom_H(\rho, Res_H^G(\pi))$
• This isomorphism relates the intertwining operators between an induced representation and a representation of $G$ to the intertwining operators between the original representation of $H$ and the restricted representation
• Intuitively, Frobenius reciprocity says that the information contained in an induced representation is equivalent to the information contained in the original representation
• The isomorphism can be interpreted as a adjunction between the induction functor and the restriction functor
  • Induction is the left adjoint of restriction
• Frobenius reciprocity can be used to compute the character of an induced representation in terms of the character of the original representation
• The reciprocity formula has a natural interpretation in terms of the geometry of homogeneous spaces

Proof of Frobenius Reciprocity

• The proof of Frobenius reciprocity relies on the construction of explicit isomorphisms between the spaces of intertwining operators
• One direction of the isomorphism is given by the map that takes an intertwining operator $T: Ind_H^G(\rho) \to \pi$ to its restriction $T|_\rho: \rho \to Res_H^G(\pi)$
  • This map is well-defined because the induced representation is defined in terms of the original representation
• The other direction of the isomorphism is given by the map that takes an intertwining operator $S: \rho \to Res_H^G(\pi)$ and extends it to an intertwining operator $\tilde{S}: Ind_H^G(\rho) \to \pi$
  • This extension is possible because the induced representation is a universal object with respect to intertwining operators
• The proof that these maps are inverses of each other involves checking that the composition of the maps in either order gives the identity map
  • This verification relies on the explicit formulas for induction and restriction of representations
• The proof can be generalized to the case of infinite groups and continuous representations using the machinery of Hilbert spaces and bounded operators

Applications of Frobenius Reciprocity

• Frobenius reciprocity is a fundamental tool in the representation theory of finite groups and has numerous applications
• It can be used to compute the character table of a group from the character tables of its subgroups
  • This is particularly useful for groups that have a small number of conjugacy classes of subgroups
• Frobenius reciprocity is used in the classification of the irreducible representations of finite groups
  • It allows the construction of irreducible representations by inducing from representations of subgroups
• The reciprocity formula is used in the proof of the Mackey decomposition formula for induced representations
• Frobenius reciprocity has applications to the study of automorphic forms and the Langlands program
  • It relates automorphic representations of different groups
• In the representation theory of compact groups, Frobenius reciprocity is used to analyze the restriction of representations to subgroups
  • This is important in the theory of spherical functions and the Plancherel formula

Mackey's Theorem: Statement and Significance

• Mackey's theorem describes the composition of induction and restriction of representations
• For a finite group $G$, subgroups $H$ and $K$, and a representation $\rho$ of $H$, Mackey's theorem states:
  • $Res_K^G(Ind_H^G(\rho)) \cong \bigoplus_{x \in K \backslash G / H} Ind_{K \cap xHx^{-1}}^K(Res_{K \cap xHx^{-1}}^{xHx^{-1}}(\rho^x))$
  • Here, $\rho^x$ denotes the representation of $xHx^{-1}$ obtained by conjugating $\rho$ by $x$
• Mackey's theorem expresses the restriction of an induced representation as a direct sum of induced representations of certain subgroups
  • These subgroups are the intersections of $K$ with the conjugates of $H$
• The theorem is significant because it allows the computation of the character of a restricted induced representation
• Mackey's theorem is a generalization of Frobenius reciprocity
  • It reduces to Frobenius reciprocity in the case where $K = H$
• The theorem has important applications in the theory of automorphic forms and the representation theory of reductive groups over local fields

Proof of Mackey's Theorem

• The proof of Mackey's theorem involves analyzing the space of the induced representation $Ind_H^G(\rho)$ as a representation of the subgroup $K$
• The space of the induced representation consists of functions $f: G \to V$, where $V$ is the space of $\rho$, satisfying a certain transformation property under the action of $H$
• The action of $K$ on this space is given by $(k \cdot f)(g) = f(k^{-1}g)$
• The proof proceeds by decomposing the space of the induced representation into a direct sum of subspaces indexed by the double cosets $K \backslash G / H$
  • Each subspace consists of the functions supported on a particular double coset
• The action of $K$ on each subspace can be identified with the induced representation of the representation of $K \cap xHx^{-1}$ obtained by restricting and conjugating $\rho$
• The isomorphism in Mackey's theorem is then obtained by assembling these identifications
• The proof relies on the properties of the double coset decomposition and the Mackey decomposition formula for the restriction of an induced representation

Connections and Comparisons

• Frobenius reciprocity and Mackey's theorem are closely related to several other concepts in representation theory and beyond
• The reciprocity formula is an instance of a more general phenomenon known as Frobenius reciprocity for adjoint functors
  • This occurs in various settings, such as the representation theory of algebraic groups and the theory of Hecke algebras
• Mackey's theorem is related to the orbit method in the representation theory of Lie groups
  • The orbit method realizes irreducible representations as induced representations from stabilizers of orbits in the coadjoint representation
• The Mackey decomposition formula is analogous to the Peter-Weyl theorem in the representation theory of compact groups
  • Both theorems describe the decomposition of a representation into irreducible components
• Frobenius reciprocity and Mackey's theorem have categorical interpretations in terms of adjunctions and Mackey functors
  • These interpretations provide a unified perspective on various reciprocity and decomposition formulas
• The ideas behind Frobenius reciprocity and Mackey's theorem have been generalized to other settings, such as the representation theory of quantum groups and the theory of vertex operator algebras

Examples and Problem-Solving Strategies

• Concrete examples are essential for understanding Frobenius reciprocity and Mackey's theorem
• A basic example is the representation theory of the symmetric group $S_3$
  • The induced representation from the trivial representation of the subgroup $S_2$ decomposes as the sum of the trivial and the 2-dimensional irreducible representations of $S_3$
• Another example is the representation theory of the dihedral groups
  • Frobenius reciprocity can be used to compute the character table of these groups from the character tables of their cyclic subgroups
• When solving problems involving Frobenius reciprocity or Mackey's theorem, it is important to identify the relevant groups, subgroups, and representations
• The problem may require computing induced representations or restricted representations
  • This involves understanding the definition of these constructions and their properties
• In some cases, the problem may be solved by applying the reciprocity or decomposition formulas directly
  • In other cases, it may be necessary to use the formulas in combination with other techniques, such as character theory or the orbit method
• It is often helpful to work out small examples by hand to develop intuition for the general case
  • For example, one can compute the character table of a small group using Frobenius reciprocity and compare it with the character table obtained by other methods

Further Implications in Representation Theory

• Frobenius reciprocity and Mackey's theorem have far-reaching implications in representation theory and related fields
• The reciprocity formula is a key ingredient in the proof of the Artin induction theorem
  • This theorem states that the characters of a finite group are integer combinations of characters induced from linear characters of cyclic subgroups
• Mackey's theorem is used in the construction of the Weil representation of the metaplectic group
  • This representation plays a central role in the theory of theta functions and automorphic forms
• The ideas behind Frobenius reciprocity and Mackey's theorem have been generalized to the representation theory of infinite groups, such as Lie groups and p-adic groups
  • In this setting, the reciprocity and decomposition formulas involve integrals and distributions instead of sums
• Frobenius reciprocity and Mackey's theorem have applications to the representation theory of finite-dimensional algebras and quivers
  • They are used in the construction of the Auslander-Reiten quiver and the study of the module category
• The categorical perspective on Frobenius reciprocity and Mackey's theorem has led to the development of new algebraic structures, such as Frobenius algebras and Mackey functors
  • These structures have applications in topological field theory and algebraic topology
• The ideas behind Frobenius reciprocity and Mackey's theorem continue to inspire new developments in representation theory and related fields, such as the geometric Langlands program and the theory of categorification.