Group Representations as Modules over the Group Algebra

Q: If representations and modules are the same thing, why keep both languages?

Because they make different things easy. Matrices let you compute and exhibit; modules let you form quotients, extensions and Hom-sets without choosing a basis, and give access to Schur's Lemma, Wedderburn–Artin and Krull–Schmidt. Character theory, a third language, is best for counting. Fluency in all three is the working skill.

Q: Does the correspondence need G to be finite?

No. The universal property and the dictionary hold for any group G and any commutative ring k. What needs finiteness is everything *else*: _k kG = |G| < ∈fty is what makes kG artinian, gives finitely many simple modules, and makes the radical nilpotent. For infinite G the dictionary survives and the structure theory does not.

Q: Why is char k dividing |G| such a sharp dividing line?

Because it is exactly when 1|G|_g g fails to exist in kG. That element is the averaging projector used to split every submodule off; without it, kG acquires a nonzero Jacobson radical and modules stop decomposing. Maschke's theorem shows the condition is not merely sufficient but necessary.

Q: Is a simple kG-module always finite-dimensional over k?

For G finite, yes: every simple module is a quotient of _kGkG, which has dimension |G|. For infinite G this fails — kG can have simple modules of infinite k-dimension.

Q: What is the difference between irreducible and absolutely irreducible?

M is irreducible if it is simple over kG; it is absolutely irreducible if it stays simple after extending scalars to every field extension of k, equivalently if End(_kGM) = k. The rational 2-dimensional representation of the cyclic group of order 3 over Q is irreducible but not absolutely irreducible: over Q() it breaks into two 1-dimensional pieces.

Q: How do direct sums and tensor products look on the module side?

Direct sums correspond directly. Tensor products are subtler: V ⊗_k W carries a G-action by g(v ⊗ w) = gv ⊗ gw, which is *not* the kG-module tensor product but comes from the coalgebra structure on kG — the diagonal map g g ⊗ g. That extra structure makes kG a Hopf algebra, and it is what makes the representation ring a ring.

Executive Summary

Representation theory asks how a group can act linearly. Ring theory asks how modules over an algebra decompose. For a finite group $G$ and a field $k$ these are literally the same question: the assignment $ρ \mapsto (module structure on V)$ is a bijection between $k$ -representations of $G$ and left $k G$ -modules, and it preserves every structural notion on both sides.

Everything else in this chapter is a consequence. Because $k G$ is finite-dimensional over $k$ when $G$ is finite, it is left and right artinian, so the Wedderburn–Artin theory and the Jacobson radical apply without further hypotheses. The single invariant that decides the character of the theory is whether $char k$ divides $| G |$ .

$\dim_{k} k G = | G |$ Finite-dimensional

ArtinianBoth sides

$char k ∣ | G |$ Modular case

1929Noether's reformulation

Overview

Let $k$ be a field and $G$ a group. The group algebra $k G$ is the $k$ -vector space with basis the set $G$ , with multiplication defined on basis elements by the group law and extended bilinearly. Its identity is $1_{G}$ , and $k \cdot 1_{G}$ sits in the centre, so $k G$ is a $k$ -algebra. When $G$ is finite, $\dim_{k} k G = | G |$ . The construction and its first properties are treated on Group Rings and Semigroup Rings: Construction and First Properties.

(\sum_{g \in G} a_{g} g) (\sum_{h \in G} b_{h} h) = \sum_{x \in G} (\sum_{g h = x} a_{g} b_{h}) x

(8.0)

Multiplication in $k G$ : a convolution over $G$ . All sums are finite.

A ** $k$ -representation** of $G$ is a homomorphism $ρ : G \to G L (V)$ for some $k$ -vector space $V$ . The point of the group algebra is that $ρ$ is exactly the data needed to make $V$ a $k G$ -module, and nothing is lost or added in the translation.

This is not a mere change of vocabulary. It imports Schur's Lemma, the Jacobson radical, Wedderburn–Artin and the Krull–Schmidt theorem into representation theory at zero cost, and it explains why the two hard cases of the subject — non-split fields and characteristic dividing $| G |$ — are the two hard cases of finite-dimensional algebra theory: division rings other than $k$ appearing as endomorphism rings, and a nonzero radical.

Learning Objectives

Construct $k G$ and state its universal property with respect to $k$ -algebras.
Prove that $k$ -representations of $G$ correspond bijectively to left $k G$ -modules.
Translate subrepresentation, irreducibility, equivalence and direct sum into module language.
Explain why $\dim_{k} k G = | G | < \infty$ makes $k G$ artinian and puts the whole of §3 and §4 at your disposal.
State Maschke's dichotomy and identify the ordinary and modular regimes.
Decompose $k C_{n}$ for $k = ℂ$ , $ℚ$ and $ℝ$ and read off the irreducible representations.

Definitions

Definition—Representation and its module

A ** $k$ -representation** of a group $G$ is a pair $(V, ρ)$ with $V$ a $k$ -vector space and $ρ : G \to G L (V)$ a group homomorphism. Its degree is $\dim_{k} V$ . Two representations $(V, ρ)$ and $(W, σ)$ are equivalent if there is a $k$ -isomorphism $f : V \to W$ with $f \circ ρ (g) = σ (g) \circ f$ for all $g \in G$ .

A subspace $U \subseteq V$ is a subrepresentation if $ρ (g) U \subseteq U$ for all $g$ . The representation is irreducible if $V \neq 0$ and its only subrepresentations are $0$ and $V$ , and completely reducible if $V$ is a sum of irreducible subrepresentations.

$k G$: The group algebra: free $k$ -module on $G$ with convolution product. Written $k [G]$ by many authors.
Augmentation $ε$: The $k$ -algebra map $k G \to k$ , $\sum a_{g} g \mapsto \sum a_{g}$ . Its kernel is the augmentation ideal, and it corresponds to the trivial representation.
Regular representation: $V = k G$ with $G$ acting by left translation; as a module this is $_{k G} k G$ , of degree $| G |$ .
Intertwining operator: A $k$ -linear $f : V \to W$ with $f ρ (g) = σ (g) f$ for all $g$ ; equivalently an element of ${Hom}_{k G} (V, W)$ .
Ordinary vs modular: Ordinary representation theory is the case $char k ∤ | G |$ (including characteristic $0$ ); modular is the case $char k = p$ with $p ∣ | G |$ .
$D_{i}$ and $n_{i}$: For a simple left $k G$ -module $M_{i}$ : $D_{i} = End (_{k G} M_{i})$ , a division ring by Schur's Lemma, and $n_{i} = \dim_{D_{i}} M_{i}$ .

Standing conventions: G is finite unless stated otherwise, all representations are finite-dimensional over k, modules are unital left modules, and char k is arbitrary.

Core Concepts

From a homomorphism to a module and back

Given $ρ : G \to G L (V)$ , define an action of $k G$ on $V$ by $(\sum a_{g} g) \cdot v = \sum a_{g} ρ (g) (v)$ . Bilinearity of the group-algebra product is exactly what makes this associative, and $ρ (1) = id$ makes it unital. Conversely, if $V$ is a left $k G$ -module then each $g \in G$ acts as a $k$ -linear map that is invertible with inverse the action of $g^{- 1}$ , so $g \mapsto (v \mapsto g v)$ is a homomorphism $G \to G L (V)$ .

{Hom}_{k -alg} (k G, A) ⟷ {Hom}_{Grp} (G, U (A))

(8.0a)

The universal property of $k G$ , for any $k$ -algebra $A$ . Taking $A = {End}_{k} (V)$ gives the representation correspondence, since $U ({End}_{k} V) = G L (V)$ .

The correspondence is not merely a bijection on objects. A $k$ -linear map between two representations intertwines the $G$ -actions precisely when it commutes with the action of every basis element $g$ of $k G$ , hence precisely when it is $k G$ -linear. So morphisms match as well, and the two categories are isomorphic.

What the dictionary translates

ρ : G \to G L (V)

algebra map

k G \to {End}_{k} V

left

k G

-module

V

Under the dictionary, the image of $k G$ in ${End}_{k} (V)$ is the enveloping algebra of the representation — the $k$ -span of $ρ (G)$ . The representation is faithful as a $G$ -representation iff $ρ$ is injective on $G$ ; it is faithful as a module iff the annihilator ideal is zero, which is a strictly stronger condition.

Why finiteness is doing work

For $G$ finite, $\dim_{k} k G = | G | < \infty$ . A finite-dimensional algebra is left and right artinian, so: $rad k G$ is nilpotent; $k G / rad k G$ is semisimple; there are only finitely many isomorphism classes of simple left $k G$ -modules; and every finitely generated module has a composition series, so Jordan–Hölder and Krull–Schmidt apply. None of this survives for infinite $G$ , which is why the infinite theory is a separate subject.

Key Results

Theorem(8.A)Representations are modules

Let $k$ be a commutative ring and $G$ a group. For every $k$ -algebra $A$ , restriction along $G ↪ k G$ is a bijection

{Hom}_{k -alg} (k G, A) \tilde{⟶} {Hom}_{Grp} (G, U (A)) .

Consequently the category of $k$ -representations of $G$ (with intertwining operators as morphisms) is isomorphic to the category of left $k G$ -modules (with $k G$ -homomorphisms), by an isomorphism that is the identity on underlying $k$ -modules and $k$ -linear maps.

Proof

The bijection. $k G$ is free as a $k$ -module with basis $G$ , so a $k$ -linear map $ϕ : k G \to A$ is uniquely determined by an arbitrary function $G \to A$ . Such a $ϕ$ is a $k$ -algebra homomorphism iff $ϕ (g h) = ϕ (g) ϕ (h)$ for all $g, h \in G$ and $ϕ (1_{G}) = 1_{A}$ — multiplicativity on a basis implies it everywhere, by bilinearity. A monoid homomorphism $G \to A$ automatically lands in $U (A)$ , because $ϕ (g) ϕ (g^{- 1}) = ϕ (1) = 1 = ϕ (g^{- 1}) ϕ (g)$ . Conversely every group homomorphism $G \to U (A)$ extends $k$ -linearly, and the extension is multiplicative by bilinearity. The two constructions are mutually inverse.

Objects. Take $A = {End}_{k} (V)$ for a $k$ -module $V$ . A left $k G$ -module structure on $V$ extending its $k$ -structure is precisely a $k$ -algebra map $k G \to {End}_{k} (V)$ , and $U ({End}_{k} V) = G L (V)$ . So such structures correspond to homomorphisms $G \to G L (V)$ .

Morphisms. Let $V, W$ carry $k G$ -structures with associated $ρ, σ$ , and let $f : V \to W$ be $k$ -linear. If $f$ is $k G$ -linear then $f (g v) = g f (v)$ , i.e. $f ρ (g) = σ (g) f$ . Conversely if $f$ intertwines all $ρ (g)$ then $f (\sum a_{g} g v) = \sum a_{g} f (ρ (g) v) = \sum a_{g} σ (g) f (v) = (\sum a_{g} g) f (v)$ by $k$ -linearity. Hence ${Hom}_{k G} (V, W)$ is exactly the set of intertwining operators, and composition and identities agree on both sides.

Corollary(8.B)Dictionary of structural notions

Under the correspondence of $(8 . A)$ : subrepresentations correspond to $k G$ -submodules; irreducible representations to simple modules; equivalence of representations to module isomorphism; direct sums to direct sums; complete reducibility to semisimplicity; and the regular representation of $G$ to the left regular module $_{k G} k G$ .

Proof

Each item is immediate once objects and morphisms match. For instance a subspace $U \subseteq V$ satisfies $ρ (g) U \subseteq U$ for all $g \in G$ iff $(\sum a_{g} g) U \subseteq U$ for all elements of $k G$ , since $U$ is closed under $k$ -linear combinations; so $G$ -stable subspaces and $k G$ -submodules are the same subsets.

Theorem(6.1)Maschke

Let $G$ be a finite group and $k$ a field. Then $k G$ is semisimple if and only if $char k ∤ | G |$ (in particular whenever $char k = 0$ ). Equivalently, every $k$ -representation of $G$ is completely reducible exactly in that case.

The proof and the converse direction are given on Maschke's Theorem and Semisimplicity of Group Rings; the averaging idempotent $\frac{1}{| G |} \sum_{g \in G} g$ is what fails to exist when $char k$ divides $| G |$ .

Remark(7.1)The four invariants

Fix a complete set $M_{1}, \dots, M_{r}$ of simple left $k G$ -modules, put $D_{i} = End (_{k G} M_{i})$ and $n_{i} = \dim_{D_{i}} M_{i}$ . These four data — $r$ , the $M_{i}$ , the $D_{i}$ , the $n_{i}$ — determine $k G / rad k G$ completely. That is the content of The Structure of kG modulo Its Radical, and every later result in this stream is a computation of one of them.

Proof Techniques and Method

How these arguments work, and which move is worth reusing.

Move 1

Check on a basis

$k G$ is free on $G$ . Any $k$ -linear statement about $k G$ reduces to a statement about group elements, and any multiplicative statement reduces to products of group elements. Almost every proof about $k G$ begins this way.

Move 2

Use the universal property

Rather than construct maps out of $k G$ by hand, specify them on $G$ and invoke $(8 . A)$ . This is how induced representations, tensor products and the augmentation are all defined cleanly.

Move 3

Transport a ring theorem

Once a representation-theoretic question is phrased in modules, look for the ring-theoretic theorem that answers it: Schur for endomorphisms, Wedderburn–Artin for decomposition, Krull–Schmidt for uniqueness, Nakayama for lifting.

The reusable idea is that a presentation of an algebra by generators from a group converts group data into ring data without loss. The same move produces skew group rings and crossed products when the action of $G$ on the coefficients is nontrivial.

Worked Example

Cyclic groups over $ℂ$ , $ℚ$ and $ℝ$

Let $G = ⟨ g ⟩$ be cyclic of order $n$ and $R = k G$ . Sending $g \mapsto x$ gives $k G ≅ k [x] / (x^{n} - 1)$ , so the representation theory of $G$ is the factorisation theory of $x^{n} - 1$ over $k$ .

Over $k = ℂ$

Here $x^{n} - 1 = \prod_{i = 0}^{n - 1} (x - ζ^{i})$ with $ζ$ a primitive $n$ th root of unity, and the factors are pairwise coprime, so

ℂ G ≅ \prod_{i = 0}^{n - 1} ℂ [x] / (x - ζ^{i}) ≅ ℂ^{n} .

(E.1)

There are $n$ simple modules $M_{i} = ℂ$ , with $g$ acting as multiplication by $ζ^{i}$ . All are $1$ -dimensional, so $ℂ$ is a splitting field for $G$ .

Over $k = ℚ$

Now $x^{n} - 1 = \prod_{d ∣ n} Φ_{d} (x)$ , a product of distinct irreducible cyclotomic polynomials, so

ℚ G ≅ \prod_{d ∣ n} ℚ [x] / (Φ_{d} (x)) ≅ \prod_{d ∣ n} ℚ (ζ_{d}) .

(E.2)

The simple module $N_{d} = ℚ (ζ_{d})$ has $g$ acting as multiplication by $ζ_{d}$ , and $\dim_{ℚ} N_{d} = ϕ (d)$ .

Here $D_{d} = End (N_{d}) = ℚ (ζ_{d})$ is generally larger than $ℚ$ , so $ℚ$ is not a splitting field unless $n \leq 2$ . The dimension count $| G | = \sum_{i} n_{i}^{2} \dim_{k} D_{i}$ reads $n = \sum_{d ∣ n} ϕ (d)$ — the classical identity, recovered as a statement about $ℚ G$ .

Over $k = ℝ$

Real irreducible factors of $x^{n} - 1$ are $x - 1$ , possibly $x + 1$ , and the quadratics $(x - ζ^{j}) (x - ζ^{- j})$ . For $n = 2 m + 1$ odd,

ℝ G ≅ ℝ \times \prod_{j = 1}^{m} ℂ,

(E.3)

one trivial $1$ -dimensional module and $m$ two-dimensional simple modules $V_{j}$ , on which $g$ acts as rotation by $2 π j / n$ .

For $n = 2 m$ even the answer is $ℝ \times ℝ \times \prod_{j = 1}^{m - 1} ℂ$ : two $1$ -dimensional modules ( $g \mapsto 1$ and $g \mapsto - 1$ ) and $m - 1$ rotation modules. Dimension check for $n$ even: $1 + 1 + 2 (m - 1) = 2 m = n$ . Extending scalars, $V_{j} \otimes_{ℝ} ℂ ≅ M_{j} \oplus M_{n - j}$ , so a real $2$ -dimensional irreducible splits into two complex $1$ -dimensional ones.

Process and Workflow

Fix the ground fieldRecord

char k

and whether it divides

| G |

; that decides whether

rad k G = 0

Form $k G$ and find its radicalIn the ordinary case it is zero. In the modular case start from the normal

p

-subgroups, which always contribute.

Decompose the semisimple quotientApply Wedderburn–Artin to

k G / rad k G

to obtain

r

, the

n_{i}

and the

D_{i}

Test for splittingIf some

D_{i} \neq k

, extend

k

to a splitting field; then

n_{i} = \dim_{k} M_{i}

and the arithmetic simplifies.

Translate backRead the simple modules as irreducible representations and, in the ordinary case, compute their characters.

Which framework should you work in?

MatricesExplicit

ρ (g)

as matrices — best for computation and for exhibiting a representation, worst for proofs, since a basis choice must be carried around.

ModulesBest for structure theory: submodules, quotients, extensions and Hom-sets are all available without choosing bases.

CharactersBest for counting and for orthogonality arguments, but only faithful invariants when

char k ∤ | G |

Comparison and Classification

The dictionary, term by term
Representation language	Module language	Comment
$ρ : G \to G L (V)$	left $k G$ -module $V$	bijective by $(8 . A)$
degree of $ρ$	$\dim_{k} V$	same number
$G$ -stable subspace	$k G$ -submodule	same subsets
irreducible	simple	nonzero, no proper nonzero submodule
equivalent	isomorphic	as $k G$ -modules
intertwining operator	element of ${Hom}_{k G} (V, W)$	Schur's Lemma applies
completely reducible	semisimple	sum of simples
regular representation	$_{k G} k G$	degree $\| G \|$
trivial representation	$k$ via the augmentation $ε$	kernel is the augmentation ideal
absolutely irreducible	simple with $End (_{k G} M) = k$	stays simple after any field extension

Which structural tools are available in which regime
	$char k = 0$	$char k ∤ \| G \|$	$char k ∣ \| G \|$	$G$ infinite
$k G$ artinian	yes	yes	yes	no
$k G$ semisimple	yes	yes	no	partial
$rad k G = 0$	yes	yes	no	partial
finitely many simple modules	yes	yes	yes	no
Krull–Schmidt for f.g. modules	yes	yes	yes	partial
characters determine the module	yes	partial	no	no

Which structural tools are available in which regime

Relationship Map

The correspondence is the hinge; everything downstream hangs on which side of Maschke's dichotomy you are.

$k$ -representations of finite $G$ — $=$ left $k G$ -modules
- $char k ∤ | G |$ — ordinary theory
  - $k G$ semisimple; every module is a direct sum of simples
  - character theory and orthogonality relations available
  - $k G ≅ \prod_{i} M_{n_{i}} (D_{i})$ with no radical
- $char k = p ∣ | G |$ — modular theory
  - $rad k G \neq 0$ ; indecomposable $\neq$ simple
  - normal $p$ -subgroups act trivially on simple modules
  - the count of simples drops to the $p$ -regular class count
- $k$ not a splitting field
  - some $D_{i} ⊋ k$ ; simple modules may split after extension
  - pass to a finite extension and use $(8.3)$

G

k G

k G / rad k G

\prod_{i} M_{n_{i}} (D_{i})

simple modules

M_{i}

Applications and Industry Use

Applications here means where this structure is used — inside mathematics and in the engineering and computing disciplines that consume it.

Computational algebra

Module-based algorithms

GAP and Magma represent a group representation internally as a module over a matrix algebra, not as a list of matrices with a group law. The MeatAxe splits such a module into composition factors; the dictionary is what makes the algorithm applicable to groups at all.

Physics and chemistry

Symmetry-adapted bases

Molecular vibration analysis and crystal field theory decompose a configuration space into irreducible representations of a point group. The projection operators used are the central idempotents of $ℂ G$ .

Signal processing

Generalised Fourier transforms

The isomorphism $ℂ G ≅ \prod_{i} M_{n_{i}} (ℂ)$ is exactly the Fourier transform on $G$ ; fast algorithms for it generalise the FFT from cyclic groups to arbitrary finite groups.

Coding theory

Group codes

A group code is an ideal of $k G$ . Cyclic codes are the case $G$ cyclic, where $k G ≅ k [x] / (x^{n} - 1)$ — the classical polynomial description is literally the group-algebra description.

The honest summary is that the dictionary is infrastructure: it is rarely quoted in a theorem statement, but nearly every theorem in the chapter is proved on the module side and read off on the representation side.

Design Considerations

Design considerations here means the choices made when modelling a problem with these algebraic structures.

Left or right? Left $k G$ -modules match the usual convention $ρ (g h) = ρ (g) ρ (h)$ with operators acting on the left. Right modules give an anti-homomorphism unless you also pass to $(k G)^{op}$ . The map $g \mapsto g^{- 1}$ gives $k G ≅ (k G)^{op}$ , so nothing is lost — but be explicit about which convention is in force.
Field or ring of coefficients? Over a field the theory is that of a finite-dimensional algebra. Over $ℤ$ or a complete discrete valuation ring you get integral representation theory, where modules need not be free and Krull–Schmidt can fail.
Which side do the scalars act on? Following Lam, elements of $D_{i} = End (_{k G} M_{i})$ are composed as right operators on $M_{i}$ , so that $End (M_{i})_{D_{i}} ≅ M_{n_{i}} (D_{i})$ with no opposite ring appearing. Choosing the other convention forces $D_{i}^{op}$ into every statement.
Finite-dimensional or not? Restricting to finite-dimensional representations is a genuine restriction even for finite $G$ only when $k$ is replaced by a general commutative ring; over a field, every simple $k G$ -module for finite $G$ is automatically finite-dimensional, being a quotient of $k G$ .

Standards and Notation

Standards here covers notation, symbol and markup standards, and reference implementations, rather than material or design codes.

Group algebra

k G

(Lam, Passman);

k [G]

(Curtis–Reiner, Serre)

Representation

ρ

σ

π

; degree written

\deg ρ

\dim_{k} V

Character

χ_{M}

χ_{ρ}

; the set of ordinary irreducible characters is

Irr (G)

Simple modules

M_{1}, \dots, M_{r}

(Lam);

S_{1}, \dots, S_{r}

L (λ)

elsewhere

MarkupPresentation MathML per ISO/IEC 40314; symbol conventions per ISO 80000-2

GAPGroupRing(k, G), GModuleByMats, MTX.CompositionFactors

MagmaGroupAlgebra, GModule, IrreducibleModules

SageGroupAlgebra(G, k), G.regular_representation()

Computational Notes

Computational notes cover algorithms, cost and library behaviour rather than manufacturing process.

Working with $k G$ computationally means working with a $| G |$ -dimensional algebra given by structure constants that are entirely $0$ or $1$ — a sparse and very well-behaved input.

Multiplication in $k G$ costs $O (| G |^{2})$ field operations by convolution; for abelian $G$ over a splitting field it drops to $O (| G | \log | G |)$ via the Fourier transform.
Constructing an irreducible representation is usually done not inside $k G$ but by the MeatAxe: take a module given by matrices, pick a random element, compute its characteristic polynomial, and use a nontrivial kernel to split. It is a Las Vegas algorithm, with each attempt costing $O (d^{3})$ over $𝔽_{q}$ for degree $d$ .
Over $ℚ$ , coefficient growth rather than the operation count dominates; implementations work modulo a prime and lift, or work over a cyclotomic field with a fixed conductor.
For $| G |$ beyond a few thousand the regular module is too large to handle directly. Practical work uses permutation modules, induction from subgroups, and condensation rather than $k G$ itself.

Failure Modes and Common Mistakes

Do not carry $char k = 0$ intuition into characteristic $p$ : complete reducibility, the orthogonality relations and the equality *number of irreducibles $=$ number of classes* all fail when $p ∣ | G |$ .
Do not conflate the group $G$ with the unit group $U (k G)$ . $G$ sits inside $U (k G)$ , usually as a very small subgroup, and describing the rest is a hard open problem.
Do not assume every $k G$ -module comes from a representation of $G$ on a finite-dimensional space when $G$ is infinite; for infinite $G$ the dictionary still holds but every finiteness statement on this page fails.

Historical Notes and Lessons Learned

1893Molien's hypercomplex systemsMolien studies the structure of finite-dimensional complex algebras and, in effect, decomposes $ℂ G$ into matrix algebras — the algebra-theoretic view before it had a name.
1896Frobenius introduces charactersFrobenius defines group characters via the factorisation of the group determinant, founding representation theory of finite groups without using modules at all.
1899Maschke's theoremMaschke proves complete reducibility in characteristic $0$ by averaging, isolating the role of $| G |$ being invertible.
1902–07Dickson opens the modular caseDickson studies representations over fields whose characteristic divides $| G |$ , where averaging is unavailable; these become known as modular representations.
1929Noether's reformulationEmmy Noether recasts representation theory as the module theory of the group algebra, making the results of Molien, Frobenius and Wedderburn instances of one structure theory. This is the viewpoint used throughout this page.
1935–47Brauer's modular theoryBrauer builds the systematic theory of modular representations, counting irreducibles by $p$ -regular classes and establishing splitting fields from the exponent of the group.

The methodological lesson is worth stating plainly: Frobenius's characters are invariants of an object he could not name, and Noether's contribution was to name the object. Once the representation is a module, the questions that were hard for special reasons become instances of questions that were already answered for general reasons.

Quick Reference

Definition

k G = ⨁_{g \in G} k g

with convolution product

Correspondence

ρ : G \to G L (V)

\leftrightarrow

left

k G

-module

V

Universal property

{Hom}_{k -alg} (k G, A) ≅ {Hom}_{Grp} (G, U (A))

Dimension

\dim_{k} k G = | G |

for finite

G

Maschke

k G

semisimple

\Leftrightarrow

char k ∤ | G |

Schur

D_{i} = End (_{k G} M_{i})

is a division ring

Character

χ_{M} (a) = tr (a ∣ M)

, constant on conjugacy classes

Trivial module

k

via the augmentation

ε (\sum a_{g} g) = \sum a_{g}

Reference points for the cyclic group of order n
$k$	$k C_{n}$	Number of simples	Splitting?
$ℂ$	$ℂ^{n}$	$n$	yes
$ℚ$	$\prod_{d ∣ n} ℚ (ζ_{d})$	number of divisors of $n$	only for $n \leq 2$
$ℝ$ , $n$ odd	$ℝ \times ℂ^{(n - 1) / 2}$	$(n + 1) / 2$	only for $n = 1$
$𝔽_{p}$ , $p ∤ n$	$\prod_{j} 𝔽_{p^{d_{j}}}$	number of $p$ -cyclotomic cosets mod $n$	iff all $d_{j} = 1$
$𝔽_{p}$ , $n = p$	$𝔽_{p} [t] / (t^{p})$	$1$	yes

Frequently Asked Questions

If representations and modules are the same thing, why keep both languages?

Because they make different things easy. Matrices let you compute and exhibit; modules let you form quotients, extensions and Hom-sets without choosing a basis, and give access to Schur's Lemma, Wedderburn–Artin and Krull–Schmidt. Character theory, a third language, is best for counting. Fluency in all three is the working skill.

Does the correspondence need

G

to be finite?

No. The universal property and the dictionary hold for any group $G$ and any commutative ring $k$ . What needs finiteness is everything else: $\dim_{k} k G = | G | < \infty$ is what makes $k G$ artinian, gives finitely many simple modules, and makes the radical nilpotent. For infinite $G$ the dictionary survives and the structure theory does not.

Why is

char k

dividing

| G |

such a sharp dividing line?

Because it is exactly when $\frac{1}{| G |} \sum_{g} g$ fails to exist in $k G$ . That element is the averaging projector used to split every submodule off; without it, $k G$ acquires a nonzero Jacobson radical and modules stop decomposing. Maschke's theorem shows the condition is not merely sufficient but necessary.

Is a simple

k G

-module always finite-dimensional over

k

For $G$ finite, yes: every simple module is a quotient of $_{k G} k G$ , which has dimension $| G |$ . For infinite $G$ this fails — $k G$ can have simple modules of infinite $k$ -dimension.

What is the difference between irreducible and absolutely irreducible?

$M$ is irreducible if it is simple over $k G$ ; it is absolutely irreducible if it stays simple after extending scalars to every field extension of $k$ , equivalently if $End (_{k G} M) = k$ . The rational $2$ -dimensional representation of the cyclic group of order $3$ over $ℚ$ is irreducible but not absolutely irreducible: over $ℚ (ω)$ it breaks into two $1$ -dimensional pieces.

How do direct sums and tensor products look on the module side?

Direct sums correspond directly. Tensor products are subtler: $V \otimes_{k} W$ carries a $G$ -action by $g (v \otimes w) = g v \otimes g w$ , which is not the $k G$ -module tensor product but comes from the coalgebra structure on $k G$ — the diagonal map $g \mapsto g \otimes g$ . That extra structure makes $k G$ a Hopf algebra, and it is what makes the representation ring a ring.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §8 (pp. 124–148); §7 for the algebra-theoretic background.
E. Noether, “Hyperkomplexe Größen und Darstellungstheorie”, Mathematische Zeitschrift 30 (1929).
C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley-Interscience, 1962.
J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer-Verlag, 1977.
D. S. Passman, The Algebraic Structure of Group Rings, Wiley-Interscience, 1977.
P. Webb, A Course in Finite Group Representation Theory, Cambridge Studies in Advanced Mathematics 161, Cambridge University Press, 2016.

AI Suggested Questions

Give the explicit central idempotents of $ℂ G$ for $G = S_{3}$ and verify they are orthogonal and sum to $1$ .
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Work out the $k G$ -module structure of the permutation module on the cosets of a subgroup $H$ , and relate it to induction from $H$ .
For which infinite groups $G$ and fields $k$ is $k G$ known to be semiprimitive?
Explain how the fast Fourier transform for finite groups is derived from $ℂ G ≅ \prod_{i} M_{n_{i}} (ℂ)$ .
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Representations and Modules

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

From a homomorphism to a module and back

What the dictionary translates

Why finiteness is doing work

Key Results

Proof Techniques and Method

Check on a basis

Use the universal property

Transport a ring theorem

Worked Example

Cyclic groups over $ℂ$ , $ℚ$ and $ℝ$

Over $k = ℂ$

Over $k = ℚ$

Over $k = ℝ$

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

Module-based algorithms

Symmetry-adapted bases

Generalised Fourier transforms

Group codes

Design Considerations

Standards and Notation

Computational Notes

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

References

AI Suggested Questions

Continue learning

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

From a homomorphism to a module and back

What the dictionary translates

Why finiteness is doing work

06Key Results

07Proof Techniques and Method

Check on a basis

Use the universal property

Transport a ring theorem

08Worked Example

Cyclic groups over ℂ, ℚ and ℝ

Over k=ℂ

Over k=ℚ

Over k=ℝ

09Process and Workflow

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Module-based algorithms

Symmetry-adapted bases

Generalised Fourier transforms

Group codes

13Design Considerations

14Standards and Notation

15Computational Notes

16Failure Modes and Common Mistakes

17Historical Notes and Lessons Learned

18Quick Reference

19Frequently Asked Questions

20Related KEVOS Topics

21References

22AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Key Results

Proof Techniques and Method

Worked Example

Cyclic groups over $ℂ$ , $ℚ$ and $ℝ$

Over $k = ℂ$

Over $k = ℚ$

Over $k = ℝ$

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

Design Considerations

Standards and Notation

Computational Notes

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions