07.01.07 · representation-theory / foundations

Induced representation

shipped3 tiersLean: partial

Anchor (Master): Frobenius 1898 *Über Relationen zwischen den Charakteren einer Gruppe und denen ihrer Untergruppen*; Serre §7; Fulton-Harris §3.3

Intuition [Beginner]

The induced representation is a construction that takes a representation $W$ of a subgroup $H \leq G$ and produces a representation of the larger group $G$ . The procedure is to "lift" $W$ to $G$ by allowing it to be transported around by all the cosets of $H$ in $G$ .

The intuition: think of $W$ as living over the single coset $H$ in $G$ . The induced representation creates a copy of $W$ for each coset of $H$ , and lets $G$ permute these copies according to how it permutes cosets, mixed with the original $H$ -action on each individual copy.

The formula is concrete: $dim Ind_{H}^{G} W = [G : H] \cdot dim W$ , the index of $H$ in $G$ times the dimension of $W$ . This is one of the most powerful constructions in representation theory: it lets you build representations of larger groups from understood representations of subgroups, and it is adjoint (in a precise sense) to the restriction operation that takes a $G$ -representation and forgets down to an $H$ -representation. This adjoint relationship is Frobenius reciprocity (unit 07.01.08).

Frobenius introduced the construction in 1898 as part of his investigations into how characters of subgroups relate to characters of the full group. Induction is the foundation of Mackey theory (Mackey 1949–52), the theory of $GL_{n} (Q_{p})$ -representations, the local Langlands programme, and modern automorphic forms.

Visual [Beginner]

The induced representation: a copy of the subgroup-representation $W$ over each coset of $H$ in $G$ , with $G$ permuting the copies and acting within each according to $H$ .

Worked example [Beginner]

Take $G = S_{3}$ and $H = Z /3 Z = {e, (123), (132)}$ , the cyclic subgroup of order 3. The index is $[S_{3} : H] = 6/3 = 2$ .

Let $W = C_{ω}$ be the 1-dimensional representation of $H$ where the generator $(123)$ acts by $ω = e^{2 π i /3}$ . Then $dim Ind_{H}^{S_{3}} W = 2 \cdot 1 = 2$ .

The induced representation has basis indexed by coset representatives ${e, (12)}$ — two copies of $W$ , one for each coset. The action of $(123) \in S_{3}$ on the basis vector indexed by the identity coset multiplies by $ω$ (since $(123) \in H$ acts on $W$ by $ω$ ); the action of $(12)$ swaps the cosets.

What this tells us: starting from a 1-dimensional representation of a subgroup of order 3, we built a 2-dimensional representation of $S_{3}$ . By Frobenius reciprocity, this induced representation contains the standard 2-dimensional representation of $S_{3}$ — and indeed for $ω \neq = 1$ the induced representation is exactly the standard representation. For $ω = 1$ (the principal representation of $H$ ), the induced representation contains both the principal and the sign characters of $S_{3}$ .

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $G$ be a group, $H \leq G$ a subgroup, and $(W, ρ)$ a representation of $H$ over a field $k$ . The induced representation $Ind_{H}^{G} W$ is constructed as follows.

Tensor-product (left-module) construction. Form the $k H$ -bimodule $k G$ (with left $k G$ -action and right $k H$ -action), and define

Ind_{H}^{G} W := k G \otimes_{k H} W,

where $k G$ is viewed as a right $k H$ -module via right multiplication and $W$ as a left $k H$ -module via $ρ$ . The left $k G$ -action on the first factor descends to give the $G$ -action on the induced representation.

Cosets-and-functions construction. Choose coset representatives $g_{1}, \dots, g_{r}$ for $G / H$ (so $G = ⨆_{i} g_{i} H$ ). Then

Ind_{H}^{G} W ≅ i = 1 ⨁ r g_{i} \otimes W ≅ W^{\oplus r}, r = [G : H],

as $k$ -vector spaces. The action of $g \in G$ permutes the summands according to the action of $G$ on $G / H$ , with twists by $H$ : if $g \cdot g_{i} H = g_{j} H$ , then $g \cdot g_{i} = g_{j} h_{g, i}$ for some $h_{g, i} \in H$ , and $g$ sends $g_{i} \otimes w$ to $g_{j} \otimes ρ (h_{g, i}) w$ .

Equivalent definitions.

Functions transformation: $Ind_{H}^{G} W$ is isomorphic to the space of functions $f : G \to W$ satisfying $f (g h) = ρ (h)^{- 1} f (g)$ for all $g \in G$ , $h \in H$ , with $G$ acting by left translation $(g^{'} \cdot f) (g) := f (g^{' - 1} g)$ .
Adjoint construction: $Ind_{H}^{G}$ is the left adjoint to the restriction functor $Res_{H}^{G} : Rep (G) \to Rep (H)$ — see Frobenius reciprocity, unit 07.01.08.

Theorem (Character of induced representation, Frobenius 1898). For finite $G$ and $H \leq G$ , the character of $Ind_{H}^{G} W$ at $g \in G$ is

χ_{Ind_{H}^{G} W} (g) = \frac{1}{∣ H ∣} x \in G : x^{- 1} g x \in H \sum χ_{W} (x^{- 1} g x) .

Equivalently, summing over coset representatives,

χ_{Ind_{H}^{G} W} (g) = i : g_{i}^{- 1} g g_{i} \in H \sum χ_{W} (g_{i}^{- 1} g g_{i}),

where the sum runs over those coset representatives $g_{i}$ for which $g_{i}^{- 1} g g_{i} \in H$ ^{[Serre §7.2]}.

Key theorem with proof [Intermediate+]

Theorem (Induced character formula). Let $G$ be a finite group, $H \leq G$ , $W$ an $H$ -representation with character $ψ$ . Then

χ_{Ind_{H}^{G} W} (g) = \frac{1}{∣ H ∣} x \in G : x^{- 1} g x \in H \sum ψ (x^{- 1} g x) .

Proof. Choose coset representatives $g_{1}, \dots, g_{r}$ for $G / H$ , so $G = ⨆_{i} g_{i} H$ . As a $k$ -vector space, $Ind_{H}^{G} W$ has basis ${g_{i} \otimes w_{j}}$ where ${w_{j}}$ is a basis of $W$ . The $G$ -action on $g \cdot (g_{i} \otimes w_{j})$ : write $g g_{i} = g_{σ (i)} h_{i}$ for some permutation $σ$ of cosets and some $h_{i} \in H$ . Then

g \cdot (g_{i} \otimes w_{j}) = (g g_{i}) \otimes w_{j} = (g_{σ (i)} h_{i}) \otimes w_{j} = g_{σ (i)} \otimes ρ (h_{i}) w_{j} .

The matrix entry of $g$ on ${g_{i} \otimes w_{j}}$ takes $(g_{i} \otimes w_{j})$ to $(g_{σ (i)} \otimes ρ (h_{i}) w_{j})$ . The trace contributes only from indices $i$ where $σ (i) = i$ , i.e., $g g_{i} \in g_{i} H$ , equivalently $g_{i}^{- 1} g g_{i} \in H$ . For such $i$ , the contribution to the trace is $tr (ρ (h_{i})) ∣_{V} = ψ (h_{i}) = ψ (g_{i}^{- 1} g g_{i})$ .

Summing over fixed cosets:

χ_{Ind} (g) = i : g_{i}^{- 1} g g_{i} \in H \sum ψ (g_{i}^{- 1} g g_{i}) .

To convert to the symmetric formula, observe that as $x$ ranges over the coset $g_{i} H$ , the conjugate $x^{- 1} g x$ ranges over $H$ -conjugates of $g_{i}^{- 1} g g_{i}$ . The class function $ψ$ is invariant on $H$ -conjugacy classes, so

x \in g_{i} H \sum ψ (x^{- 1} g x) = ∣ H ∣ \cdot ψ (g_{i}^{- 1} g g_{i}),

provided $g_{i}^{- 1} g g_{i} \in H$ , and zero otherwise. Summing:

x \in G : x^{- 1} g x \in H \sum ψ (x^{- 1} g x) = i : g_{i}^{- 1} g g_{i} \in H \sum ∣ H ∣ ψ (g_{i}^{- 1} g g_{i}) = ∣ H ∣ \cdot χ_{Ind} (g) . □

This induced-character formula is one of the most useful tools in representation theory: it lets you compute characters of induced representations by summing the subgroup-character over conjugates that land back in the subgroup ^{[Frobenius 1898]}.

Bridge. The construction here builds toward later units of the strand, where the same pattern is taken up at higher structure. The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Exercise 3 (medium, numeric).

Compute the character of $Ind_{H}^{S_{3}} χ_{ω}$ where $H = Z /3 Z$ embedded as the alternating subgroup, and $χ_{ω}$ sends the generator $(123)$ to $ω = e^{2 π i /3}$ .

Hint

Apply the Frobenius induced-character formula at each conjugacy class of $S_{3}$ .

Answer

Coset representatives: ${e, (12)}$ . So we sum over $i = 1, 2$ where $g_{i}^{- 1} g g_{i} \in H$ .

At $g = e$ : $χ_{Ind} (e) = χ_{ω} (e) + χ_{ω} (e) = 1 + 1 = 2$ (both cosets contribute).

At $g = (12)$ (a transposition): $e^{- 1} (12) e = (12) \in / H$ . $(12)^{- 1} (12) (12) = (12) \in / H$ . No contribution; $χ_{Ind} ((12)) = 0$ .

At $g = (123)$ (a 3-cycle): $e^{- 1} (123) e = (123) \in H$ , contributing $χ_{ω} ((123)) = ω$ . $(12)^{- 1} (123) (12) = (12) (123) (12) = (132) \in H$ , contributing $χ_{ω} ((132)) = ω^{2}$ . Total: $χ_{Ind} ((123)) = ω + ω^{2} = - 1$ .

So $χ_{Ind} = (2, 0, - 1)$ on classes ${e}, {$ trans $}, {$ 3-cycle $}$ . This matches the standard representation $χ_{std}$ of $S_{3}$ . So $Ind_{H}^{S_{3}} χ_{ω} ≅ V_{std}$ .

Exercise 6 (hard, proof).

State and prove the Mackey decomposition formula for $Res_{K}^{G} \circ Ind_{H}^{G} W$ where $H, K \leq G$ .

Hint

Decompose $G$ into double cosets $K \ G / H$ .

Answer

Mackey's formula. For $H, K \leq G$ and a representation $W$ of $H$ ,

Res_{K}^{G} Ind_{H}^{G} W ≅ x \in K \ G / H ⨁ Ind_{K \cap x H x^{- 1}}^{K} Res_{K \cap x H x^{- 1}}^{x H x^{- 1}} (^{x} W),

where the sum runs over double-coset representatives $x \in K \ G / H$ , $^{x} W$ denotes the $x H x^{- 1}$ -representation obtained by twisting $W$ via $w \mapsto x w x^{- 1}$ .

Proof sketch. Write $G = ⨆_{x} K x H$ as a disjoint union of double cosets. For each double coset, the corresponding piece of $Ind_{H}^{G} W ≅ k G \otimes_{k H} W$ , restricted to $K$ , is itself a $K$ -induced representation (from the subgroup $K \cap x H x^{- 1}$ ). Combining gives the formula.

This decomposes the restriction of an induced representation as a sum of induced representations from intersection subgroups, indexed by double cosets. It is the key tool in Mackey theory (Mackey 1949–52) and underlies many irreducibility criteria.

Exercise 7 (hard, proof).

Use the induced-character formula to compute $⟨ χ_{Ind_{H}^{G} W}, χ_{V} ⟩_{G}$ in terms of $⟨ χ_{W}, χ_{Res_{H}^{G} V} ⟩_{H}$ (Frobenius reciprocity at the character level).

Hint

Substitute the induced-character formula and reorder the sum.

Answer

By definition,

⟨ χ_{Ind W}, χ_{V} ⟩_{G} = \frac{1}{∣ G ∣} g \in G \sum χ_{Ind W} (g) \overline{χ_{V} (g)} .

Substitute the induced-character formula:

= \frac{1}{∣ G ∣} g \sum \frac{1}{∣ H ∣} x : x^{- 1} g x \in H \sum χ_{W} (x^{- 1} g x) \overline{χ_{V} (g)} .

Change variables: let $h = x^{- 1} g x \in H$ (so $g = x h x^{- 1}$ ). As $g$ ranges over $G$ and $x$ ranges over $G$ with $x^{- 1} g x \in H$ , the pair $(g, x)$ ranges over pairs $(x h x^{- 1}, x)$ with $h \in H$ , $x \in G$ . So the double sum becomes

= \frac{1}{∣ G ∣∣ H ∣} x \in G \sum h \in H \sum χ_{W} (h) \overline{χ_{V} (x h x^{- 1})} .

Use $\overline{χ_{V} (x h x^{- 1})} = \overline{χ_{V} (h)}$ (class function) and the inner sum becomes

h \sum χ_{W} (h) \overline{χ_{V} (h)} = ∣ H ∣ \cdot ⟨ χ_{W}, χ_{Res V} ⟩_{H} .

The outer sum over $x$ is just $∣ G ∣$ copies, so the $∣ G ∣$ 's cancel:

⟨ χ_{Ind W}, χ_{V} ⟩_{G} = ⟨ χ_{W}, χ_{Res V} ⟩_{H} . □

This is Frobenius reciprocity at the character level. The full categorical adjunction $Hom_{G} (Ind_{H}^{G} W, V) ≅ Hom_{H} (W, Res_{H}^{G} V)$ is unit 07.01.08.

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has the basic Representation.ind construction (induced representation from a subgroup) in Mathlib.RepresentationTheory.Induced. Frobenius reciprocity is partially formalised; the induced character formula can be stated.

[object Promise]

Advanced results [Master]

Mackey's irreducibility criterion (1951). The induced representation $Ind_{H}^{G} W$ is irreducible iff for every $g \in G ∖ H$ , the conjugate representation $^{g} W$ on $g H g^{- 1}$ is disjoint (no common irreducible constituents) from $W$ on $H \cap g H g^{- 1}$ . This was the foundational result of Mackey theory (Mackey, On induced representations of groups, 1951).

Brauer's induction theorem (1951). Every irreducible character of a finite group $G$ over $C$ is a $Z$ -linear combination of characters $Ind_{E}^{G} ψ$ where $E$ ranges over elementary subgroups (subgroups of the form $C \times P$ with $C$ cyclic, $P$ a $p$ -group, $g cd (∣ C ∣, p) = 1$ ) and $ψ$ ranges over 1-dimensional characters of such $E$ . This deep theorem powers Artin's conjecture on Artin $L$ -functions and the rationality of character values.

Induced representations of Lie groups. For a Lie group $G$ with closed subgroup $H$ , the induced representation $Ind_{H}^{G} W$ on a finite-dimensional unitary representation $W$ of $H$ is constructed as smooth sections of the homogeneous vector bundle $G \times_{H} W \to G / H$ . For $G$ semisimple and $H$ a parabolic subgroup, the induced representations are parabolic induction, the building blocks of admissible representations of real reductive Lie groups (Harish-Chandra) and of $GL_{n} (Q_{p})$ representations (Bernstein-Zelevinsky).

Local Langlands correspondence. The local Langlands correspondence for $GL_{n} (F)$ (Harris-Taylor 2001, Henniart 2000) parameterises irreducible admissible representations of $GL_{n} (F)$ ( $F$ a non-archimedean local field) by $n$ -dimensional representations of the Weil-Deligne group of $F$ . The match between "induced representation of $GL_{n} (F)$ from a parabolic subgroup" and "Galois representation reducible as a sum" is one of the most striking features of the correspondence.

Frobenius's theorem on Frobenius groups. A finite group $G$ acting transitively on a set $X$ such that every non-identity element fixes at most one point is called a Frobenius group. Frobenius (1901) proved that the elements of $G$ fixing no point, together with the identity, form a normal subgroup. The proof crucially uses induced characters; no proof avoiding character theory has been found.

Induction and tensor product (Mackey's projection formula). The projection formula relates induction and tensor product:

Ind_{H}^{G} (W \otimes Res_{H}^{G} V) ≅ (Ind_{H}^{G} W) \otimes V,

a $G$ -equivariant isomorphism for any $H$ -representation $W$ and $G$ -representation $V$ . This is the non-commutative analogue of "factor out what you can": if part of the input is already a $G$ -representation, you can pull it out.

Synthesis. This construction generalises the pattern fixed in 07.01.01 (group representation), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

The induced-character formula is proved in the key-theorem section. Frobenius reciprocity at the character level is proved in Exercise 7 by direct manipulation of characters; the categorical version is in unit 07.01.08. Induction in stages and the dimension formula are proved in Exercises 2 and 5 respectively. Mackey's decomposition formula (Exercise 6) is the core of Mackey theory; its proof requires double-coset analysis and is in Serre §7.4 or Curtis-Reiner §10. Brauer's induction theorem is one of the deepest results in finite-group character theory; the proof occupies a chapter in Serre, Linear Representations of Finite Groups Part III (Chapters 9–10), using the integrality of character values via algebraic-integer arguments. The Lie-theoretic statements (parabolic induction, local Langlands) are stated without proof here; standard references are Knapp Representation Theory of Semisimple Groups, Bump Lie Groups Ch. 8, and Bushnell-Henniart The Local Langlands Conjecture for $GL (2)$ .

Connections [Master]

Group representation 07.01.01 — induction is one of the four basic operations on representations (alongside direct sum, tensor product, restriction).
Frobenius reciprocity 07.01.08 — induction is left adjoint to restriction; this adjunction is the structural backbone of Mackey theory.
Regular representation 07.01.05 — the regular representation is $Ind_{{e}}^{G} C$ , the induced representation from the identity subgroup.
Character of a representation 07.01.03 — the induced character formula computes characters explicitly.
Tensor product of representations 07.01.06 — the projection formula connects induction with tensor products via Mackey decomposition.
Group 01.02.01 — induction is a construction depending on the subgroup structure of $G$ .
Highest weight representation 07.03.01 — parabolic induction (the Lie-group analogue) is fundamental to the construction of admissible representations.
Cartan-Weyl classification 07.04.01 — induced representations from parabolic subgroups are central to the structure of complex/real reductive Lie group representations.

Historical & philosophical context [Master]

Ferdinand Frobenius introduced the induced representation in his 1898 paper Über Relationen zwischen den Charakteren einer Gruppe und denen ihrer Untergruppen (Sitzungsberichte der königlich preussischen Akademie der Wissenschaften zu Berlin), continuing his investigations of finite-group character theory begun in 1896. The motivation was directly stated in the title: relations between characters of a group $G$ and characters of its subgroups $H \leq G$ . Frobenius defined what is now called Frobenius reciprocity at the level of characters and gave the explicit induced-character formula proved in the key-theorem section.

The brilliance of Frobenius's construction was that it produced representations of $G$ from any starting point — even a 1-dimensional character of an arbitrarily small subgroup. By varying the subgroup and the input character, one could in principle construct enough representations to span the irreducible representations of $G$ . This intuition was made precise much later by Brauer's induction theorem (1951): every irreducible character of $G$ is a $Z$ -linear combination of characters induced from 1-dimensional characters of elementary subgroups. The depth of Brauer's theorem comes from the integrality of character values, which Frobenius could not have anticipated in 1898.

George Mackey systematised induced representations in the late 1940s and early 1950s. His three-paper series On induced representations of groups (Amer. J. Math. 1949), Induced representations of locally compact groups, I (Ann. Math. 1952), and II (Ann. Math. 1953) extended the theory from finite groups to locally compact topological groups, providing the framework for Harish-Chandra's classification of admissible representations of real reductive Lie groups (1950s–1970s) and Bernstein-Zelevinsky's classification of admissible representations of $GL_{n}$ over a $p$ -adic field (1976–77). The local Langlands correspondence — proved for $GL_{n}$ by Harris-Taylor (2001) and Henniart (2000), extended to general reductive groups by Fargues-Scholze (2021) — sits at the modern apex of induced-representation theory, parameterising representations of $p$ -adic groups by Galois-side data via parabolic induction and Bernstein blocks.

Bibliography [Master]

Frobenius, Über Relationen zwischen den Charakteren einer Gruppe und denen ihrer Untergruppen, Sitzungsberichte (1898) — original construction.
Mackey, On induced representations of groups, Amer. J. Math. 73 (1951) — Mackey theory foundations.
Mackey, Induced representations of locally compact groups, I, Ann. Math. 55 (1952), and II, Ann. Math. 58 (1953) — extension to topological groups.
Brauer, A characterization of the characters of groups of finite order, Ann. Math. 57 (1953) — Brauer's induction theorem.
Serre, Linear Representations of Finite Groups — §7, induced representations and Frobenius reciprocity, with Brauer's theorem in Part III.
Fulton & Harris, Representation Theory: A First Course — §3.3.
Bump, Lie Groups — Ch. 8, induced representations of Lie groups.
Knapp, Representation Theory of Semisimple Groups — parabolic induction.
Bushnell & Henniart, The Local Langlands Conjecture for $GL (2)$ — modern application.
Curtis, Pioneers of Representation Theory (1999) — historical account of Frobenius's work.

Prerequisites

07.01.01
01.02.01

Tier anchors

beginner: Strogatz analogous level for lifting a subgroup-representation to the full group
intermediate: Serre §7; Fulton-Harris §3.3
master: Frobenius 1898 *Über Relationen zwischen den Charakteren einer Gruppe und denen ihrer Untergruppen*; Serre §7; Fulton-Harris §3.3

References

TODO_REF
Frobenius — Über Relationen zwischen den Charakteren einer Gruppe und denen ihrer Untergruppen (1898) · Sitzungsberichte 1898, the original construction of induced representations
TODO_REF
Serre — Linear Representations of Finite Groups · §7, induced representations and characters
TODO_REF
Fulton-Harris — Representation Theory: A First Course · §3.3, induced and restricted representations
TODO_REF
Bump — Lie Groups · Ch. 8, induced representations of Lie groups
TODO_REF
Mackey — Induced representations of locally compact groups (1949-52) · Mackey theory of induced representations

Lean module

Codex.RepresentationTheory.Foundations.InducedRepresentation

Reviewer

TBD

Estimated time

beginner: 14m
intermediate: 42m
master: 65m