🧩Representation Theory Unit 6 – Induced Representations

Key Concepts and Definitions

• Induced representations construct a representation of a group $G$ from a representation of a subgroup $H$
• Let $H$ be a subgroup of $G$ and $(\rho, V)$ a representation of $H$
• The induced representation $(\text{Ind}_H^G \rho, \text{Ind}_H^G V)$ is a representation of $G$
• Induction is a functor from the category of $H$-modules to the category of $G$-modules
• Frobenius reciprocity relates the induction functor to the restriction functor
  • Restriction functor takes a $G$-module and views it as an $H$-module
• Transitive $G$-sets correspond to cosets of subgroups (orbits under the action of $G$)
• Induction preserves irreducibility in certain cases (Mackey's irreducibility criterion)

Subgroups and Quotient Groups

• A subgroup $H$ of a group $G$ is a subset that is closed under the group operation and inverses
• The left cosets of $H$ in $G$ are the sets $gH = \{gh : h \in H\}$ for each $g \in G$
  • Right cosets are defined similarly as $Hg = \{hg : h \in H\}$
• The set of left cosets forms a partition of $G$ and is denoted $G/H$
• If $H$ is a normal subgroup, the set of cosets $G/H$ has a natural group structure (quotient group)
• The index of $H$ in $G$, denoted $[G:H]$, is the number of distinct left (or right) cosets
• Lagrange's theorem states that for a finite group $G$, the order of any subgroup $H$ divides the order of $G$
  • The order of $G$ is equal to the product of the order of $H$ and the index $[G:H]$

Construction of Induced Representations

• Let $H$ be a subgroup of $G$ and $(\rho, V)$ a representation of $H$
• The induced representation $\text{Ind}_H^G V$ is the vector space of functions $f: G \to V$ satisfying $f(hg) = \rho(h)f(g)$ for all $h \in H, g \in G$
  • The group $G$ acts on this space by left translation: $(gf)(x) = f(g^{-1}x)$
• Choose a set of representatives $\{g_i\}$ for the left cosets of $H$ in $G$
• For each $g_i$, define a function $f_i: G \to V$ by $f_i(g_j) = \delta_{ij}v$ for some fixed $v \in V$
  • The set $\{f_i\}$ forms a basis for $\text{Ind}_H^G V$
• The dimension of $\text{Ind}_H^G V$ is equal to $[G:H] \cdot \dim V$

Properties of Induced Representations

• Induction is transitive: if $K \leq H \leq G$, then $\text{Ind}_K^G \cong \text{Ind}_H^G \circ \text{Ind}_K^H$
• Induction preserves direct sums: $\text{Ind}_H^G(V \oplus W) \cong \text{Ind}_H^G V \oplus \text{Ind}_H^G W$
• Induction is left adjoint to restriction: $\text{Hom}_G(\text{Ind}_H^G V, W) \cong \text{Hom}_H(V, \text{Res}_H^G W)$ (Frobenius reciprocity)
• Mackey's formula relates induced representations from different subgroups
  • For subgroups $H, K \leq G$, $\text{Res}_K^G \text{Ind}_H^G \cong \bigoplus_{g \in K \backslash G / H} \text{Ind}_{K \cap gHg^{-1}}^K \text{Res}_{K \cap gHg^{-1}}^{gHg^{-1}} {}^g\rho$
• Induced representations can be used to construct irreducible representations (Mackey's irreducibility criterion)
  • If $\text{Res}_H^G \rho$ is irreducible and $\text{Res}_{gHg^{-1}}^G \rho \ncong \text{Res}_{gHg^{-1}}^G {}^g\rho$ for all $g \notin H$, then $\text{Ind}_H^G \rho$ is irreducible

Frobenius Reciprocity

• Frobenius reciprocity is a fundamental result relating induction and restriction of representations
• It states that for a subgroup $H \leq G$ and representations $(\rho, V)$ of $H$ and $(\pi, W)$ of $G$, there is a natural isomorphism:
  • $\text{Hom}_G(\text{Ind}_H^G V, W) \cong \text{Hom}_H(V, \text{Res}_H^G W)$
• This isomorphism is given by the map $\phi \mapsto \phi_e$, where $\phi_e(v) = \phi(f_v)(e)$ and $f_v(g) = \rho(g^{-1})v$
• Frobenius reciprocity allows for the computation of intertwining numbers between induced and restricted representations
• It is a key tool in the study of the representation theory of finite groups and Lie algebras
• Frobenius reciprocity can be generalized to the context of modules over algebras (adjunction between induction and restriction functors)

Applications in Physics and Chemistry

• Induced representations play a crucial role in the study of symmetry in quantum mechanics
• The symmetry group of a quantum system determines its allowed energy levels and transitions
  • Irreducible representations correspond to energy eigenspaces
• Crystal field theory uses induced representations to describe the splitting of atomic orbitals in transition metal complexes
  • The symmetry of the ligand environment induces a representation of the point group on the d-orbitals
• Vibrational modes of molecules can be classified using induced representations of the molecular point group
• The Poincaré group, the symmetry group of special relativity, has physically relevant induced representations (e.g., spin representations)
• Induced representations of the Lorentz group are used in the classification of elementary particles
• In quantum field theory, the construction of interacting theories often involves induced representations of the gauge group

Computational Techniques

• Computation of induced representations can be done using the formula $\text{Ind}_H^G \rho(g) = \sum_i \rho(h_i) \otimes e_{g_i}$, where $h_i = g_i^{-1}gg_i$
• The matrix elements of the induced representation can be computed using a set of coset representatives
• Character theory provides a way to decompose induced representations into irreducible components
  • The character of an induced representation is given by the induced character formula: $\chi_{\text{Ind}_H^G \rho}(g) = \frac{1}{|H|} \sum_{h \in H} \chi_\rho(hgh^{-1})$
• Mackey's formula can be used to compute the restriction of an induced representation to a subgroup
• Computation of intertwining numbers and Frobenius reciprocity can be done using character inner products
• The Littlewood-Richardson rule gives a combinatorial method for decomposing tensor products of induced representations of symmetric groups
• Computational algebra systems (e.g., GAP, Magma) have built-in functions for working with induced representations

Examples and Practice Problems

• Let $G = S_3$ and $H = \{e, (1 2)\}$. Compute the induced representation of the trivial representation of $H$.
• Show that the regular representation of a finite group $G$ is isomorphic to the induced representation of the trivial representation of the trivial subgroup.
• Let $G = D_4$ (dihedral group of order 8) and $H = \{e, r^2\}$. Compute the character table of $G$ using induced representations.
• Prove that if $H$ is a subgroup of index 2 in $G$, then $\text{Ind}_H^G \mathbb{C} \cong \mathbb{C} \oplus \mathbb{C}^-$, where $\mathbb{C}^-$ is the sign representation of $G$.
• Let $G = S_4$ and $H = S_3$ (as a subgroup fixing one element). Decompose the permutation representation of $G$ on $\mathbb{C}^4$ into irreducible representations using Frobenius reciprocity.
• Compute the induced representation of the standard representation of $S_2$ to $S_3$.
  • Decompose this representation into irreducible components.
• Let $G = GL(2, \mathbb{C})$ and $B$ be the subgroup of upper triangular matrices. Compute the character of the induced representation of the determinant character of $B$.