Modular Representation Theory

Core Idea

Modular representation theory studies representations of finite groups over fields whose characteristic p divides the group order |G|. In this setting, Maschke's theorem fails — the group algebra is no longer semisimple, and representations need not decompose into irreducible summands. Indecomposable representations (which cannot be written as direct sums) need not be irreducible, and the Jordan-Hölder composition factors of a module become the primary objects of study. Brauer characters replace ordinary characters, and the representation theory organizes into blocks determined by p-local structure.

Explainer

In ordinary (characteristic 0) representation theory, Maschke's theorem guarantees complete reducibility: every representation splits into a direct sum of irreducibles. When the field has characteristic p dividing |G|, this fails catastrophically. The group algebra k[G] has a nonzero Jacobson radical J(k[G]) — a nilpotent ideal consisting of elements that act as zero on every simple module. The quotient k[G]/J(k[G]) is semisimple, but the radical introduces nontrivial extensions between simple modules, creating indecomposable modules that are not irreducible.

The simplest example is G = ℤ/pℤ over 𝔽_p. The group algebra 𝔽_p[ℤ/pℤ] ≅ 𝔽_p[x]/(x^p − 1) = 𝔽_p[x]/((x−1)^p) (since x^p − 1 = (x−1)^p in characteristic p). This is a local ring with unique maximal ideal (x−1). The only irreducible module is the trivial representation 𝔽_p, but there are p indecomposable modules of dimensions 1, 2, …, p, corresponding to Jordan blocks of size 1 through p for the element (x−1). The Krull-Schmidt theorem guarantees unique decomposition into indecomposables, which replaces the irreducible decomposition.

Brauer characters are the modular replacement for ordinary characters. For a p-regular element g (one whose order is coprime to p), the eigenvalues of ρ(g) are roots of unity of order coprime to p. These can be lifted uniquely to complex roots of unity via a fixed embedding of the multiplicative group of the algebraic closure into ℂ*. The Brauer character φ(g) is the sum of these lifted eigenvalues. Brauer characters satisfy orthogonality relations on p-regular classes, and the number of irreducible Brauer characters equals the number of p-regular conjugacy classes.

The representation theory organizes into blocks — indecomposable direct summands of the group algebra k[G] as a (k[G], k[G])-bimodule. Each block is controlled by a defect group, a p-subgroup of G that measures how far the block is from being semisimple. A block with trivial defect group is a full matrix algebra (semisimple), while a block with defect group of order p^d has p^d simple modules and a rich structure of indecomposable modules. Brauer's three main theorems relate the blocks of G to blocks of local subgroups (normalizers of p-subgroups), creating a deep connection between modular representation theory and the p-local structure of G.