Modular Representation Theory | Semisimplicity, Indecomposables and Extensions

The dividing line: characteristic versus group order

Suppose G is finite and F is a field. Maschke’s theorem says that the group algebra F[G] is semisimple if charF does not divide |G|.

In that semisimple case, every representation can be decomposed into irreducible pieces. In modular representation theory, the characteristic p divides |G|. Then F[G] is generally not semisimple.

Non-modular case: decompose into irreducibles. Modular case: understand how irreducibles can be glued.

This shift—from pieces to extensions—is the defining conceptual change.

Why Maschke’s averaging proof fails

Suppose W is a G-invariant subspace of V and P:V→W is any projection. In the semisimple setting, we average over the group:

P_G=(1/|G|)Σ_{g∈G} ρ(g)Pρ(g)⁻¹.

The averaged projection commutes with G, so its kernel is an invariant complement to W.

But if charF=p and p divides |G|, then |G|=0 inside F. Division by |G| is impossible. The symmetrisation step cannot be performed in this way.

This is one of those rare moments where a single missing scalar changes an entire subject.

The smallest example: C_p over characteristic p

Let G=C_p=⟨g | g^p=e⟩ and let F have characteristic p. In the group algebra,

g^p−1=(g−1)^p

because the intermediate binomial coefficients vanish modulo p.

This means the polynomial x^p−1 has only one root, x=1, with multiplicity p. Instead of p distinct eigenvalues separating the action into one-dimensional modes, we can have nontrivial Jordan blocks with eigenvalue 1.

For example, over F_p the matrix

J = [1 1]
    [0 1]

satisfies suitable p-power unipotence conditions, and in characteristic 2 one has J²=I. The line spanned by the first basis vector is invariant, but there need not be an invariant complementary line.

The representation is reducible but indecomposable.

Reducible is not the same as decomposable

This distinction is central in modular theory.

• Reducible: V has a nonzero proper invariant subspace.
• Decomposable: V can be written as a direct sum V=V₁⊕V₂ of two nonzero invariant subspaces.
• Indecomposable: V cannot be expressed as such a direct sum.

Every irreducible representation is indecomposable, but an indecomposable representation need not be irreducible.

Irreducible means no proper invariant piece. Indecomposable means no direct-sum splitting.

A glued two-dimensional module

Consider a two-dimensional representation with basis e₁,e₂ where g acts by

g·e₁ = e₁
g·e₂ = e₂ + e₁.

The line Fe₁ is invariant and carries the trivial representation. The quotient V/Fe₁ is also trivial. So the composition factors are two trivial modules.

Yet V need not equal trivial⊕trivial as a G-module. The second copy is attached to the first through the off-diagonal term.

This is the simplest mental model of a non-split extension:

same irreducible pieces, different way of gluing them.

Short exact sequences

Extensions are encoded by short exact sequences

0 → A → E → B → 0.

The module E contains a submodule isomorphic to A, and the quotient is B.

If E≅A⊕B, the sequence splits. If not, E is a nontrivial extension of B by A.

Semisimple theory largely erases this distinction because every such sequence splits. Modular theory must retain it.

Composition series

Even when a module does not split into irreducibles, it can often be filtered by submodules

0=V₀⊂V₁⊂···⊂V_n=V

such that each quotient V_i/V_{i−1} is simple. These simple quotients are the composition factors.

The Jordan–Hölder theorem says the multiset of composition factors is independent of the chosen composition series.

But composition factors alone do not determine the module. Two modules may have the same factors arranged in different extension structures.

Why character theory must change

Ordinary complex characters rely heavily on semisimplicity. In characteristic p, ordinary trace functions do not by themselves recover the full modular structure. Jordan blocks with the same eigenvalues can have the same traces while representing different extensions.

Modular representation theory therefore introduces refined tools, including Brauer characters, which are defined on p-regular elements—elements whose order is not divisible by p.

Brauer characters recover an important analogue of ordinary character theory for simple modular representations, but extension and projective information still requires additional structure.

The group algebra viewpoint

A representation of G over F is the same as a left module over the group algebra F[G]. This viewpoint is especially powerful modularly because the failure of semisimplicity becomes a property of the algebra itself.

Instead of asking only which matrices represent G, we ask:

• What are the simple F[G]-modules?
• What are the indecomposable modules?
• Which modules are projective?
• What is the Jacobson radical of F[G]?
• How does F[G] decompose into blocks?
• Which extensions exist between simple modules?

The Jacobson radical

For a finite-dimensional algebra A, the Jacobson radical J(A) measures part of the obstruction to semisimplicity. The quotient A/J(A) is semisimple.

In the modular group algebra F[G], the radical can be nonzero. Nilpotent and extension behaviour lives there.

For G=C_p over F_p, the group algebra is isomorphic to

F_p[t]/((t−1)^p).

The element t−1 is nilpotent. That nilpotence is the algebraic source of the unipotent Jordan blocks seen in representations.

Simple modules versus indecomposable modules

In semisimple theory, once simple modules are classified, every finite-dimensional module is a direct sum of them. In modular theory, classifying simples is only the first layer.

We also need indecomposable modules. A module may have many simple composition factors yet refuse to split. The Krull–Schmidt theorem gives an important replacement principle: under standard finite-length conditions, a module decomposes as a direct sum of indecomposable modules, and the indecomposable summands are unique up to isomorphism and order.

Thus modular theory changes the atomic unit:

semisimple atoms = irreducibles; modular direct-sum atoms = indecomposables.

Projective modules

A module P is projective if maps from P lift across surjections. Equivalently, short exact sequences ending in P split.

Projective modules are important because they behave like modules with no extension obstruction on one side. Every simple module has a projective cover in the finite-dimensional group-algebra setting.

Projective indecomposable modules provide a second structural basis for the theory, especially in block decomposition and modular character theory.

Projective covers

A projective cover P→S of a simple module S is a projective module mapping onto S in a minimal essential way. The top P/radP is isomorphic to S.

The internal layers of P record how S is connected by extensions to other simple modules. Loewy series organise those radical layers.

Socle, radical and Loewy layers

The socle of a module is the sum of its simple submodules. The radical is the intersection of maximal submodules. Repeated radicals produce a filtration

V ⊃ radV ⊃ rad²V ⊃ ···.

The successive quotients are semisimple and are called Loewy layers.

A Loewy diagram is a compact way to show how simple factors are stacked and connected inside an indecomposable module.

Ext groups

Equivalence classes of extensions

0→A→E→B→0

are organised by Ext¹(B,A). A zero Ext¹ group means every such extension splits. A nonzero Ext¹ group means genuinely glued modules exist.

Higher Ext groups measure deeper homological relationships. This moves modular representation theory toward homological algebra.

Blocks of a group algebra

The group algebra decomposes into a direct product of indecomposable two-sided ideals called blocks. Simple modules belong to blocks, and extensions between simples in different blocks vanish.

Blocks partition the modular representation problem into smaller independent sectors.

group algebra → blocks → simple modules + projectives + extensions inside each block.

This is another example of representation theory searching for the correct decomposition even after complete reducibility fails at the module level.

Defect and p-local structure

Blocks carry defect groups that measure how strongly p-local subgroup structure influences the block. The deeper theory connects representations to Sylow p-subgroups, fusion and local group structure.

This is a distinctive feature of modular representation theory: the prime p is not merely a field parameter. It selects a local geometry inside the group.

Brauer characters

Ordinary character values are traces over characteristic zero. Brauer characters provide an analogue for representations in characteristic p by evaluating on p-regular elements and lifting suitable eigenvalues to characteristic zero roots of unity.

Irreducible Brauer characters correspond to simple modular representations over a sufficiently large splitting field. They allow decomposition-number calculations comparing characteristic-zero and characteristic-p theories.

Decomposition matrices

Suppose an ordinary irreducible representation is realised over an appropriate integral lattice and reduced modulo p. It may cease to be irreducible. The multiplicities of modular simple factors are recorded by decomposition numbers.

Arranged in a matrix, they connect ordinary irreducible characters with irreducible Brauer characters.

This is a controlled way to compare two representation worlds separated by characteristic.

Example: why C_p has only one simple module over F_p

Let G=C_p. Over F_p, the group algebra is local: F_p[G]≅F_p[u]/(u^p), where u=g−1.

The quotient by the radical (u) is F_p. Therefore there is only one simple module, the one-dimensional trivial module.

Yet there are indecomposable modules of dimensions 1 through p, represented by Jordan blocks with eigenvalue 1. So one simple module generates an entire family of nonisomorphic indecomposables through different nilpotent extension lengths.

This single example captures the modular shift perfectly: classification of simples is easy, classification of indecomposables is richer.

Jordan blocks as extension length

For C_p in characteristic p, let N be nilpotent and let g act by I+N. The relation g^p=I is compatible with N^p=0 because

(I+N)^p=I+N^p

in characteristic p.

A Jordan block of size r gives an r-dimensional indecomposable module. Its chain of invariant subspaces forms a composition series with r trivial composition factors.

The matrix’s nilpotent superdiagonal records how the trivial factors are glued together.

Restriction and induction in the modular setting

Restriction and induction still exist. Frobenius reciprocity still has a module-theoretic form. But semisimplicity can no longer be assumed when decomposing the results.

Induced modules may be indecomposable, projective or have nontrivial Loewy structure. Restriction may expose extensions that ordinary characters cannot see.

The previous Induced Representations, Restriction and Frobenius Reciprocity guide therefore remains structurally relevant, but the decomposition step must be upgraded.

Tensor products become more subtle

Tensor products still produce representations, but decomposition into irreducibles may be replaced by decomposition into indecomposables. Tensoring can interact nontrivially with projectivity and block structure.

This means the modular representation ring loses some of the clean semisimple intuition. The Grothendieck group can still record composition factors, but it forgets extension data. Two nonisomorphic modules can have the same class in the Grothendieck group.

Grothendieck groups: useful but lossy

The Grothendieck group K₀ records additive relations from short exact sequences:

[E]=[A]+[B] whenever 0→A→E→B→0.

Therefore split and non-split extensions have the same K₀ class. This is useful when counting composition factors but insufficient when the way they are glued matters.

Composition data is not extension data.

Stable module categories

For modular group algebras, projective modules can be treated as structurally trivial in a stable category, where maps factoring through projectives are identified with zero.

This viewpoint exposes periodicity, syzygies and cohomological phenomena that are difficult to see in the ordinary module category.

Group cohomology connection

Group cohomology is closely related to extension theory. For suitable modules, H¹ and higher cohomology groups can be interpreted through derivations and Ext groups over the group algebra.

This is one reason modular representation theory connects naturally to topology, homological algebra and arithmetic: once splitting fails, extension data becomes cohomological data.

Why p-groups are especially modular

If G is a finite p-group and F has characteristic p, then the group algebra F[G] is highly non-semisimple. Over an algebraically closed field of characteristic p, the trivial representation is the only simple module for a p-group.

Yet the category of finite-dimensional modules can be very rich. This makes p-groups natural laboratories for extension structure, projectives and cohomology.

Finite versus tame versus wild behaviour

For some algebras, indecomposable modules can be classified in finitely many families. For others, classification is tame. In wild representation type, the classification problem is at least as difficult as classifying pairs of matrices up to simultaneous similarity.

Modular group algebras can exhibit these difficult behaviours. Therefore there is no universal finite list of indecomposables analogous to the clean irreducible list in semisimple finite-group representation theory.

This is a structural limit, not a failure of technique.

What survives from ordinary representation theory

• Representations are still group homomorphisms into GL(V).
• Invariant subspaces and quotient modules still matter.
• Restriction, induction and tensor products still exist.
• Dual modules still exist.
• Group algebras still encode representations as modules.
• Characters still have analogues, especially Brauer characters.

What changes is the decomposition logic. Direct sums of irreducibles are no longer enough to classify modules.

What must be added

• indecomposable modules;
• composition series;
• radical and socle filtrations;
• projective covers;
• extensions and Ext groups;
• block theory;
• p-local subgroup structure;
• Brauer characters and decomposition numbers;
• homological and stable-category methods.

A modular diagnosis workflow

• Identify charF=p and test whether p divides |G|.
• If not, use ordinary semisimple theory.
• If yes, do not assume invariant complements exist.
• Find simple composition factors, but do not stop there.
• Check whether the module is indecomposable.
• Inspect radical, socle and Loewy layers.
• Use projective covers and block membership where appropriate.
• Use Ext groups to detect possible non-split extensions.
• Use Brauer characters for p-regular character information.
• Keep extension data separate from composition-factor data.

Common mistakes

• Using Maschke’s theorem without checking characteristic: the divisibility condition is essential.
• Equating reducible with decomposable: modular modules can be reducible but indecomposable.
• Assuming traces classify representations: trace can miss nilpotent extension structure.
• Listing composition factors as though they determine the module: they do not determine how factors are glued.
• Ignoring projectives: projective covers are central structural objects.
• Treating all primes alike: modular structure is specifically p-local.
• Expecting a complete finite classification of indecomposables: many cases are too complex for that.

Practice set

1. Maschke check

G has order 12 and F has characteristic 5. Is the representation theory semisimple by Maschke’s theorem?

Answer: yes, because 5 does not divide 12.

2. Modular check

G has order 12 and F has characteristic 3. Is this the modular case?

Answer: yes, because 3 divides 12.

3. Indecomposable distinction

Can a reducible representation be indecomposable?

Answer: yes. It may have invariant subspaces but no invariant direct-sum complement.

4. Composition factors

Do identical composition factors imply two modules are isomorphic?

Answer: no. The extension structure may differ.

5. Ext interpretation

What does a nonzero Ext¹(B,A) indicate?

Answer: non-split extensions of B by A can exist.

6. C_p in characteristic p

How many simple modules does C_p have over an algebraically closed field of characteristic p?

Answer: one, the trivial simple module.

Why modular representation theory matters

The subject is not merely ordinary representation theory with inconvenient arithmetic. It reveals information that semisimple theory has no need to record.

When every exact sequence splits, extension data disappears. When every module is a direct sum of simples, indecomposables add nothing new. When characteristic divides the group order, those simplifications vanish and the hidden architecture becomes visible.

That architecture connects representation theory to cohomology, local group theory, finite groups, algebraic topology, arithmetic and modern homological methods.

Semisimple theory asks which pieces occur. Modular theory must also ask how the pieces are attached.

Representation Mathematics — Batch 02

• Induced Representations, Restriction and Frobenius Reciprocity
• Representation Theory: Tensor Products, Duality, Symmetric and Exterior Powers
• Lie Group and Lie Algebra Representations | Highest Weights and Continuous Symmetry
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