Modular Representation Theory

Executive summary

The interesting case is when the characteristic divides the order

Maschke's theorem makes k[G] semisimple when the characteristic of k does not divide |G|, so every module is projective and all higher cohomology vanishes. When the characteristic does divide the order, the group algebra has infinite global dimension, modules have infinite resolutions, and the cohomology ring becomes a genuine geometric object. Support varieties turn that ring into a space, and the representation theory is organised by geometry.

Learning objectives

State Maschke's theorem and its failure.
Explain why global dimension becomes infinite.
Define complexity and support varieties.
State the Quillen stratification and its consequence.

Section 01The two regimes

Ordinarychar k ∤ |G|

k[G] is semisimple by Maschke's theorem. Every module is projective and injective; Hⁿ(G, M) = 0 for n ≥ 1. Representation theory is character theory.

Modularchar k | |G|

k[G] is not semisimple. Global dimension is infinite, resolutions never terminate, and cohomology is non-zero in infinitely many degrees.

Why Maschke fails

The averaging idempotent (1/|G|)∑g requires |G| to be invertible. Without it there is no canonical projection onto a submodule, and short exact sequences need not split. The failure is exactly the failure of an averaging argument, as with the Casimir in the Lie algebra case.

The smallest example
Setting	Behaviour
k[ℤ/p], char k = p	k[x]/(x^p) — a local ring
Indecomposable modules	k[x]/(xⁱ) for 1 ≤ i ≤ p
Projectives	Only the free module k[G] itself
Cohomology H^*(ℤ/p, k)	Non-zero in every degree, periodic
Global dimension	Infinite

Section 02Complexity and support varieties

The complexity of a module is the polynomial rate of growth of the ranks in its minimal projective resolution. It is zero exactly for projective modules, and bounded above by the p-rank of the group.

Stage 01Cohomology ringH^*(G, k) is a finitely generated graded-commutative k-algebra — the Evens–Venkov theorem.
Stage 02Its spectrumThe maximal ideal spectrum is an affine variety, the support variety of the trivial module.
Stage 03Module supportEach module M has a support variety, the subvariety defined by the annihilator of Ext^*(M, M).
Stage 04Dimension = complexityThe dimension of the support variety equals the complexity of the module, tying geometry to resolution growth.

Quillen stratification

The variety is stratified by contributions from the elementary abelian p-subgroups, with the maps induced by restriction. So the geometry of the cohomology ring is assembled from the elementary abelian subgroups — the geometric counterpart of Sylow detection.

Section 03Structural tools

Tool

Relative projectivity and vertices

Every indecomposable module is relatively projective with respect to a p-subgroup, its vertex, unique up to conjugacy. This is relative homological algebra doing structural work.

Tool

Green correspondence

A bijection between indecomposable modules with a given vertex for G and for the normaliser of that vertex, reducing questions to smaller groups.

Tool

Blocks

The group algebra decomposes into blocks; each has a defect group measuring how far it is from semisimple. Blocks of defect zero behave like the ordinary case.

Tool

Stable module category

Quotient the module category by projectives. It is triangulated, and support varieties classify its thick subcategories.

Tool

Tate cohomology

Extends cohomology to negative degrees for finite groups, and is the natural home for the stable category.

Tool

Periodicity

Modules of complexity 1 are exactly those with periodic resolutions — the representation-theoretic face of periodic cohomology.

ReferenceFrequently asked questions

Why is the cohomology ring finitely generated?

By the Evens–Venkov theorem, proved using the norm map and an induction over subgroups. Finite generation is what makes the spectrum a genuine algebraic variety and the whole geometric approach possible.

Is modular representation theory relative homological algebra?

In large part. Vertices and sources are defined by relative projectivity with respect to subgroups, so the projective-class machinery of this collection is exactly the right framework.

What does complexity zero mean?

That the minimal resolution terminates, so the module is projective. Complexity 1 means periodic resolutions; higher complexity means polynomially growing ranks, and the exponent is the dimension of the support variety.

NavigateContinue in this stream

Curated next steps from this page. The site also surfaces algorithmically related reading below.

ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Modular Representation Theory. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Modular Representation Theory as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—representation, theory, support, section, modular—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Modular Representation Theory?