07.02.01 · representation-theory / character

Maschke's theorem

shipped3 tiersLean: partial

Anchor (Master): Heinrich Maschke 1899 *Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen...*; Lang Ch. XVIII; Curtis-Reiner

Intuition [Beginner]

Maschke's theorem is the cornerstone result of finite-group representation theory: every representation of a finite group $G$ over a field of characteristic 0 (or any field whose characteristic does not divide $∣ G ∣$ ) decomposes as a direct sum of irreducible representations. In other words, the irreducibles are atoms and every representation is a finite disjoint union of these atoms.

The intuition: think of a representation as a vector space with a $G$ -action. Maschke's theorem says you can always find a basis adapted to the $G$ -action — a basis where the action is "block-diagonal" with each block being an irreducible representation. There's no genuine entanglement that prevents the splitting.

The proof uses a beautiful trick called averaging over $G$ : given any subrepresentation $W \subset V$ , find a complementary $G$ -invariant subspace $W^{'}$ by starting with any complement and averaging it under the $G$ -action. The averaging needs $\frac{1}{∣ G ∣}$ to make sense — and that fails precisely when $∣ G ∣$ is divisible by the field characteristic.

Heinrich Maschke proved this in 1899 in Mathematische Annalen. When the averaging trick fails (characteristic dividing $∣ G ∣$ ), one enters modular representation theory — Brauer's 1935 framework, where representations are no longer atomic and a richer structure (blocks, defect groups) takes over.

Visual [Beginner]

A finite-group representation decomposed into irreducible blocks: each block is invariant under $G$ , and the blocks together span the whole space.

Worked example [Beginner]

The cyclic group $Z /3 Z$ acting on $R^{3}$ by cyclic permutation of coordinates: $ρ (1)$ sends $(x_{1}, x_{2}, x_{3})$ to $(x_{3}, x_{1}, x_{2})$ .

This 3-dimensional real representation is reducible: the line $R (1, 1, 1)$ is a 1-dimensional invariant subspace (every cyclic permutation fixes the all-ones vector). So $V$ has a 1-dim subrepresentation; can we find a complementary 2-dim invariant subspace?

By Maschke (working over $R$ since $∣ G ∣ = 3 \neq = 0$ in $R$ ), yes: the orthogonal complement of $R (1, 1, 1)$ — the plane ${(x_{1}, x_{2}, x_{3}) : x_{1} + x_{2} + x_{3} = 0}$ — is invariant under cyclic permutation. So $R^{3} = R (1, 1, 1) \oplus {x_{1} + x_{2} + x_{3} = 0}$ as $G$ -representations.

What this tells us: the 3-dim permutation representation of $Z /3 Z$ on $R^{3}$ decomposes as a 1-dim principal piece plus a 2-dim "standard" piece. Over $C$ , the 2-dim piece further splits into two 1-dim pieces (eigenspaces of $ρ (1)$ for the eigenvalues $ω = e^{2 π i /3}$ and $ω^{2}$ ). So over $C$ , the regular representation of $Z /3 Z$ decomposes into three 1-dim irreducible pieces: $C^{3} = χ_{0} \oplus χ_{1} \oplus χ_{2}$ .

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $G$ be a group and $k$ a field. A representation $(V, ρ)$ of $G$ is completely reducible (or semisimple) if every $G$ -invariant subspace $W \subseteq V$ has a $G$ -invariant complement $W^{'}$ , i.e., $V = W \oplus W^{'}$ with both $W, W^{'}$ being $G$ -subrepresentations. Equivalently, $V$ decomposes as a direct sum of irreducible subrepresentations.

Theorem (Maschke 1899). Let $G$ be a finite group and $k$ a field whose characteristic does not divide $∣ G ∣$ (in particular, any field of characteristic 0). Then every finite-dimensional representation of $G$ over $k$ is completely reducible ^{[Serre §1.3]}.

Equivalent formulation. The group algebra $k G$ is semisimple as a $k$ -algebra, i.e., $k G ≅ \prod_{i} M_{n_{i}} (D_{i})$ for some division algebras $D_{i}$ over $k$ (Artin-Wedderburn).

Counter-example. Maschke fails in characteristic $p$ dividing $∣ G ∣$ . Take $G = Z / p Z$ and $V = F_{p}^{2}$ with the generator acting by $(1011)$ . The line $F_{p} \cdot (0 1)$ is invariant (the action sends $(0 1)$ to itself), but no transverse line is invariant: any vector $(1 a)$ is sent to $(1 a + 1)$ , so the line through $(1 a)$ is not preserved. Hence $V$ is reducible (it has the invariant line) but not completely reducible (no invariant complement to that line).

This failure launches modular representation theory — Brauer's 1935 framework, where representations of $G$ over fields of characteristic $p ∣ ∣ G ∣$ exhibit a richer structure (indecomposable but not irreducible modules, blocks, defect groups, decomposition matrices).

Generalisations.

Compact Lie groups. Every finite-dimensional continuous representation of a compact Lie group on a complex vector space is completely reducible. The proof uses Haar measure and the unitarian trick (Weyl 1925–26).
Reductive Lie algebras. Finite-dimensional representations of complex semisimple Lie algebras are completely reducible (Weyl's theorem on complete reducibility).
Reductive algebraic groups. Over algebraically closed characteristic-0 fields, finite-dimensional representations of reductive algebraic groups are completely reducible. In characteristic $p$ , the Mumford conjecture (proved by Haboush 1975) gives a partial replacement.

Key theorem with proof [Intermediate+]

Theorem (Maschke). Let $G$ be a finite group, $k$ a field with $∣ G ∣$ invertible in $k$ . Every finite-dimensional $k$ -representation of $G$ is completely reducible.

Proof (averaging-projection argument). Let $(V, ρ)$ be a finite-dimensional $G$ -representation over $k$ , and let $W \subseteq V$ be a $G$ -invariant subspace. We construct a $G$ -invariant complement $W^{'}$ .

Step 1 (start with any projection). Since $V$ is finite-dimensional, pick any $k$ -linear projection $π : V \to W$ , i.e., a linear map satisfying $π (v) \in W$ for all $v$ and $π ∣_{W} = id_{W}$ . (Such $π$ exists by linear algebra: extend a basis of $W$ to a basis of $V$ and project onto $W$ along the chosen complement.) The projection $π$ is in general not $G$ -equivariant.

Step 2 (averaging). Define the averaged projection

\tilde{π} := \frac{1}{∣ G ∣} g \in G \sum ρ (g) \circ π \circ ρ (g)^{- 1} : V \to V .

The factor $\frac{1}{∣ G ∣}$ is well-defined because $∣ G ∣$ is invertible in $k$ .

Step 3 (verification). Verify three properties of $\tilde{π}$ :

(a) Image lies in $W$ . For any $v \in V$ and $g \in G$ , $ρ (g)^{- 1} (v) \in V$ , $π (ρ (g)^{- 1} v) \in W$ , and since $W$ is $G$ -invariant, $ρ (g) (π (ρ (g)^{- 1} v)) \in W$ . So each summand lies in $W$ , hence so does the average.

(b) $\tilde{π} ∣_{W} = id_{W}$ . For $w \in W$ , $ρ (g)^{- 1} w \in W$ (since $W$ is $G$ -invariant), so $π (ρ (g)^{- 1} w) = ρ (g)^{- 1} w$ ( $π$ is identity on $W$ ). Then $ρ (g) π (ρ (g)^{- 1} w) = ρ (g) ρ (g)^{- 1} w = w$ . Each summand equals $w$ , so the average is $\frac{1}{∣ G ∣} \cdot ∣ G ∣ w = w$ .

ρ (h) \circ \tilde{π} = \frac{1}{∣ G ∣} g \in G \sum ρ (h) ρ (g) π ρ (g)^{- 1} = \frac{1}{∣ G ∣} g \sum ρ (h g) π ρ (g)^{- 1} = \frac{1}{∣ G ∣} g^{'} \sum ρ (g^{'}) π ρ (g^{'})^{- 1} ρ (h) = \tilde{π} \circ ρ (h),

where we substituted $g^{'} = h g$ in the sum (which is bijective on $G$ ).

Step 4 (define complement). Set $W^{'} := ker (\tilde{π})$ . Since $\tilde{π}$ is $G$ -equivariant, $W^{'}$ is a $G$ -invariant subspace.

Step 5 (decomposition). From (a) and (b), $\tilde{π}$ is a projection onto $W$ . By the standard projection identity, $V = im (\tilde{π}) \oplus ker (\tilde{π}) = W \oplus W^{'}$ .

Step 6 (induction). Apply the same argument to the subrepresentations $W$ and $W^{'}$ to further decompose. Since $dim V$ is finite, this process terminates with $V$ written as a direct sum of irreducible $G$ -subrepresentations. $□$

Remark. The proof gives an explicit construction of the complement $W^{'}$ via the averaged projection. The Hermitian-inner-product variant of the proof: choose any inner product on $V$ , average it over $G$ to make it $G$ -invariant, and take $W^{'}$ to be the orthogonal complement of $W$ with respect to the averaged inner product. Both proofs require $\frac{1}{∣ G ∣}$ and fail in characteristic $p ∣ ∣ G ∣$ .

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 1 (easy, proof).

Show that the averaging trick yields a $G$ -invariant inner product, given any starting inner product on $V$ .

Hint

Average $⟨ v, w ⟩$ over $ρ (g) v, ρ (g) w$ for $g \in G$ .

Answer

Let $⟨ \cdot, \cdot ⟩_{0}$ be any Hermitian inner product on $V$ . Define

⟨ v, w ⟩ := \frac{1}{∣ G ∣} g \in G \sum ⟨ ρ (g) v, ρ (g) w ⟩_{0} .

Hermiticity, positive-definiteness, and linearity follow from the corresponding properties of $⟨ \cdot, \cdot ⟩_{0}$ (sum of Hermitian PD forms is Hermitian PD, divided by a positive constant).

$G$ -invariance: for $h \in G$ ,

⟨ ρ (h) v, ρ (h) w ⟩ = \frac{1}{∣ G ∣} g \sum ⟨ ρ (g) ρ (h) v, ρ (g) ρ (h) w ⟩_{0} = \frac{1}{∣ G ∣} g \sum ⟨ ρ (g h) v, ρ (g h) w ⟩_{0} = \frac{1}{∣ G ∣} g^{'} \sum ⟨ ρ (g^{'}) v, ρ (g^{'}) w ⟩_{0} = ⟨ v, w ⟩,

substituting $g^{'} = g h$ . So the averaged form is $G$ -invariant. $□$

This gives an alternative proof of Maschke: any subrepresentation $W$ has its $G$ -invariant orthogonal complement $W^{⊥}$ in the averaged inner product, providing the splitting.

Exercise 3 (medium, proof).

Show by direct computation that the 2-dim representation of $Z / p Z$ over $F_{p}$ given by the generator acting via $(1011)$ is reducible but not completely reducible.

Hint

Find an invariant line, then check whether any complementary line is invariant.

Answer

Let $A = (1011)$ . The generator $1 \in Z / p$ acts by $A$ , and $A^{p} = I$ (since the matrix is unipotent, $(A - I)^{2} = 0$ and $A^{p} = (I + (A - I))^{p} = I + p (A - I) + (2 p) (A - I)^{2} = I$ in $F_{p}$ ).

Invariant line. The vector $(0 1)$ satisfies $A (0 1) = (0 1)$ , so $W = F_{p} \cdot (0 1)$ is $G$ -invariant.

No invariant complement. Suppose a 1-dim invariant complement $W^{'} = F_{p} \cdot (1 a)$ exists for some $a \in F_{p}$ . Then $A (1 a) = (1 a + 1) \in W^{'}$ , so there exists $λ \in F_{p}$ with $(1 a + 1) = λ (1 a)$ . The second coordinate gives $λ = 1$ , but then $a + 1 = a$ , contradiction.

So $V$ is reducible (has the invariant line $W$ ) but not completely reducible (no invariant complement). Maschke fails because $∣ G ∣ = p$ is not invertible in $F_{p}$ . $□$

Exercise 6 (hard, proof).

Prove the analogue of Maschke's theorem for compact Lie groups: every finite-dimensional continuous representation of a compact Lie group $G$ on a complex vector space is completely reducible.

Hint

Replace $\frac{1}{∣ G ∣} \sum_{g}$ with $\int_{G} \cdot d μ (g)$ for the Haar measure $μ$ .

Answer

For a compact Lie group $G$ with normalised Haar measure $μ$ (so $μ (G) = 1$ ), the averaging trick adapts. Given an inner product $⟨ \cdot, \cdot ⟩_{0}$ on $V$ , define

⟨ v, w ⟩ := \int_{G} ⟨ ρ (g) v, ρ (g) w ⟩_{0} d μ (g) .

This is well-defined (the integrand is continuous and $G$ is compact, so the integral converges absolutely) and is $G$ -invariant: for $h \in G$ ,

⟨ ρ (h) v, ρ (h) w ⟩ = \int_{G} ⟨ ρ (g h) v, ρ (g h) w ⟩_{0} d μ (g) = \int_{G} ⟨ ρ (g^{'}) v, ρ (g^{'}) w ⟩_{0} d μ (g^{'}) = ⟨ v, w ⟩,

by translation-invariance of Haar measure ( $d μ (g) = d μ (g h^{- 1}) = d μ (g^{'})$ ).

Now any $G$ -invariant subspace $W$ has $G$ -invariant orthogonal complement $W^{⊥}$ in this inner product, giving the splitting. By induction on $dim V$ , $V$ decomposes as a direct sum of irreducibles. $□$

This is Weyl's unitarian trick: the averaging-over-Haar-measure replacement of the discrete sum $\frac{1}{∣ G ∣} \sum_{g}$ extends Maschke from finite groups to compact groups. For non-compact groups, no such averaging exists, and the structure of representations is dramatically more complex.

Exercise 7 (hard, conceptual).

Explain how Maschke's failure in characteristic $p$ dividing $∣ G ∣$ launches modular representation theory.

Hint

In characteristic $p ∣ ∣ G ∣$ , representations are no longer atomic — describe what replaces complete reducibility.

Answer

In characteristic $p$ dividing $∣ G ∣$ , Maschke fails: there exist representations $V$ of $G$ over $F_{p}$ that are reducible but not completely reducible. The Jordan-Hölder series $0 \subset V_{1} \subset V_{2} \subset \dots \subset V$ has irreducible quotients but no direct-sum splitting in general. The category of $F_{p} G$ -modules is no longer semisimple — it has nontrivial extensions $Ext^{1}$ between simples.

Modular representation theory (Brauer 1935+) studies this situation. Key features:

Brauer characters. In place of ordinary characters (which are well-defined for irreducibles in $char 0$ ), one defines Brauer characters on $p$ -regular elements (those of order coprime to $p$ ), with values in characteristic 0 (lifted from roots of unity).
Decomposition matrices. The relation between irreducible representations in $char 0$ (ordinary characters) and irreducibles in $char p$ (Brauer characters) is encoded in the decomposition matrix: an integer matrix with rows indexed by ordinary irreducibles, columns by modular irreducibles, entries giving the multiplicity of the modular irreducible in the reduction-mod- $p$ of the ordinary one.
Blocks. The group algebra $F_{p} G$ decomposes as a direct product of indecomposable two-sided ideals — blocks. Each block has an associated defect group (a $p$ -subgroup of $G$ measuring "how badly" Maschke fails for that block). Blocks with defect 0 behave like ordinary semisimple pieces; blocks with positive defect have richer module theory.
Open conjectures. Many central conjectures of finite-group theory — Alperin's weight conjecture, Broué's abelian defect conjecture, the McKay conjecture (recently proven for unitary groups, Cabanes-Späth) — concern modular representations. The structure of modular representation theory remains an active research area, with deep connections to algebraic geometry (perverse sheaves on Springer fibres) and homological algebra (derived categories, equivalences between blocks).

So Maschke's failure is not a limitation but a doorway to one of the richest fields of modern algebra.

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has Maschke's theorem in Mathlib.RepresentationTheory.Maschke: the statement that every $C G$ -submodule of a finite-dimensional $C G$ -module has a complement is formalised, with the explicit averaging-projection construction.

[object Promise]

Advanced results [Master]

Artin-Wedderburn structure theorem. Maschke implies that $C G$ is a semisimple $C$ -algebra; by Artin-Wedderburn (1907 / 1927), every finite-dimensional semisimple $C$ -algebra is isomorphic to a finite product of matrix algebras:

C G ≅ V \prod End_{C} (V) ≅ V \prod M_{d i m V} (C),

with the product running over irreducible representations $V$ . This is the Wedderburn decomposition of the group algebra. Over $R$ , the analogous decomposition involves matrix algebras over $R, C, H$ — the real Frobenius-Schur classification of irreducible representations into real, complex, and quaternionic types.

Rickard equivalences and Broué's conjecture. In modular representation theory, blocks of $F_{p} G$ that share the same defect group are conjectured (Broué 1990) to be derived equivalent. The Rickard-Linckelmann theory of splendid Rickard tilting complexes provides a categorification of Maschke's complete-reducibility failure: blocks are "as semisimple as possible" relative to their defect group structure. This is the most active branch of modern modular representation theory.

Schur indices. For a finite group $G$ and an absolutely irreducible representation $V$ over $\overline{Q}$ , the Schur index $m_{V}$ measures the obstruction to defining $V$ over the smaller field $Q$ : $V$ is realisable over a number field $K$ iff $m_{V}$ divides the local Schur indices at completions of $K$ . The Brauer-Speiser theorem bounds Schur indices for symmetric and alternating groups.

Maschke for Hopf algebras. Maschke's theorem generalises to finite-dimensional Hopf algebras over a field: a finite-dimensional Hopf algebra $H$ is semisimple iff there exists an integral $Λ \in H$ with $ε (Λ) \neq = 0$ (Larson-Sweedler 1969). For $H = k G$ a group algebra, the integral is $Λ = \sum_{g} g$ and $ε (Λ) = ∣ G ∣$ , recovering Maschke. For quantum groups at roots of unity, the integral vanishes and a richer modular structure emerges.

Reductive algebraic groups. For a reductive algebraic group $G$ over an algebraically closed field $k$ of characteristic 0, finite-dimensional rational representations of $G$ are completely reducible (Weyl). In characteristic $p > 0$ , complete reducibility fails in general, replaced by the Mumford conjecture (Haboush 1975): a $G$ -stable line bundle on a $G$ -orbit closure has a $G$ -equivariant section vanishing only on the boundary. This is the foundation of geometric invariant theory in positive characteristic.

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 averaging-projection proof of Maschke's theorem is presented in the key-theorem section. The Hermitian-inner-product proof (Exercise 1) is the alternative using $G$ -invariant unitarisation. The compact-group analogue (Exercise 6) uses Haar measure as the integration replacement. Artin-Wedderburn is proved in any standard graduate algebra text (Lang Ch. XVIII, Curtis-Reiner Vol. I); the proof factors any semisimple algebra into matrix algebras over division rings via the Jacobson density theorem and primitive idempotents. The Lie-theoretic and modular generalisations (Weyl complete reducibility, Brauer's modular character theory, Broué's conjecture) are referenced rather than proved here; they occupy the bulk of Knapp's Lie Groups Beyond an Introduction and the entire monograph of Navarro Character Theory and the McKay Conjecture.

Connections [Master]

Group representation 07.01.01 — Maschke is the structural theorem organising all of finite-group representation theory.
Schur's lemma 07.01.02 — Maschke (semisimplicity) plus Schur (irreducible morphisms) together give the structure of finite-group representations.
Character of a representation 07.01.03 — characters are well-defined invariants because of complete reducibility — every representation has a character determined by its irreducible decomposition.
Character orthogonality 07.01.04 — orthogonality combined with Maschke gives the multiplicity formula for irreducibles in any representation.
Regular representation 07.01.05 — the regular representation $C G$ decomposes via Maschke into irreducible isotypic components, giving the dimension formula $\sum (dim V)^{2} = ∣ G ∣$ .
Cartan-Weyl classification 07.04.01 — the analogous compact-Lie-group statement is foundational to the classification of representations of compact semisimple Lie groups.
Highest weight representation 07.03.01 — Weyl's complete-reducibility theorem for semisimple Lie algebras is the Lie-theoretic analogue of Maschke.

Historical & philosophical context [Master]

Heinrich Maschke (1853–1908) was a German-American mathematician who proved his theorem in 1899, in a paper titled Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind (Mathematische Annalen 52). The verbose German title translates roughly as Proof of the theorem that those finite linear substitution groups in which some entirely vanishing coefficients occur are intransitive. The substantive content — every finite-group representation over a characteristic-0 field is completely reducible — is the foundation of the entire modern theory.

Maschke's proof used the averaging trick we present in the key-theorem section. The trick was already present implicitly in earlier work of Frobenius and Schur on group characters — the averaging operator $\frac{1}{∣ G ∣} \sum_{g} ρ (g) f (g) ρ (g^{- 1})$ appears throughout Frobenius's 1896–1898 papers — but Maschke gave the cleanest formulation and the cleanest proof of complete reducibility itself. His paper appeared in Mathematische Annalen, then under the editorship of David Hilbert and Felix Klein, and was widely read.

The 20th century extended Maschke's framework dramatically. Hermann Weyl's 1925–26 papers on compact semisimple Lie groups used the unitarian trick — averaging over Haar measure — to prove the analogous complete-reducibility theorem in the continuous setting. This required Alfréd Haar's 1933 construction of Haar measure on locally compact groups (proved for general compact groups by André Weil 1940), giving Weyl's unitarian trick a rigorous foundation. The Lie-algebra analogue — Weyl's theorem on complete reducibility for semisimple Lie algebras — was proved separately, with three different proofs (Weyl's analytic, Brauer's algebraic, Whitehead's cohomological).

Richard Brauer's 1935 paper Über die Darstellung von Gruppen in Galois'schen Feldern (Math. Ann. 110) launched modular representation theory: the systematic study of finite-group representations in characteristic $p$ dividing $∣ G ∣$ , where Maschke fails. Brauer introduced modular characters (defined on $p$ -regular elements), blocks (decomposition of the group algebra into indecomposable two-sided ideals), and defect groups (measuring how far each block is from being semisimple). Modern modular representation theory — through the work of Alperin, Broué, Lusztig, Rickard, and others — sits among the deepest and most active areas of finite-group theory, with conjectures (Alperin's weight conjecture, Broué's abelian defect conjecture, the McKay conjecture) connecting modular characters to ordinary characters via local-global principles. The McKay conjecture for the prime 2 was proven by Malle-Späth in 2016; the conjecture for unitary groups by Cabanes-Späth in 2024.

Bibliography [Master]

Maschke, Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind, Mathematische Annalen 52 (1899) — the original.
Brauer, Über die Darstellung von Gruppen in Galois'schen Feldern, Mathematische Annalen 110 (1935) — modular representation theory.
Weyl, Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen, I–IV, Mathematische Zeitschrift (1925–26) — unitarian trick for compact Lie groups.
Serre, Linear Representations of Finite Groups — §1, classical proof.
Fulton & Harris, Representation Theory: A First Course — §1.1, modern textbook proof.
Lang, Algebra — Ch. XVIII, Maschke and Artin-Wedderburn.
Curtis & Reiner, Representation Theory of Finite Groups and Associative Algebras — comprehensive reference for ordinary and modular representation theory.
Navarro, Character Theory and the McKay Conjecture (2018) — modern modular representation theory and global-local principles.
Linckelmann, Block Theory of Finite Group Algebras (2018) — modern derived-equivalence approach.
Curtis, Pioneers of Representation Theory (1999) — historical account including Maschke and Brauer.

Prerequisites

07.01.01

Tier anchors

beginner: Strogatz analogous level for the splitting of finite-group representations into atoms
intermediate: Serre §1; Fulton-Harris §1.1
master: Heinrich Maschke 1899 *Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen...*; Lang Ch. XVIII; Curtis-Reiner

References

TODO_REF
Maschke — Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind (1899) · Mathematische Annalen 52 (1899), the original complete-reducibility theorem
TODO_REF
Serre — Linear Representations of Finite Groups · §1, Maschke's theorem
TODO_REF
Fulton-Harris — Representation Theory: A First Course · §1.1, complete reducibility for finite groups
TODO_REF
Lang — Algebra · Ch. XVIII, semisimple algebras and Maschke
TODO_REF
Curtis-Reiner — Representation Theory of Finite Groups and Associative Algebras · Vol. I, semisimplicity of group algebras

Lean module

Codex.RepresentationTheory.Character.Maschke

Reviewer

TBD

Estimated time

beginner: 12m
intermediate: 38m
master: 60m