Representation theory of finite groups

TheoremMaschke theorem#

Maschke's theorem says that a representation of a finite group \(G\) over a field \(k\) whose characteristic does not divide the order of \(G\) splits along every invariant subspace: each subrepresentation has an invariant complement. It is the starting point of the representation theory of finite groups. It reduces the study of finite-dimensional representations to that of irreducible representations, and so makes the theory of characters and the structure of the group algebra accessible.

Statement

Let \(G\) be a finite group of order \(|G|\) and \(k\) a field in which \(|G| \neq 0\), that is, whose characteristic is \(0\) or a prime not dividing \(|G|\). Let \(V\) be a vector space over \(k\) on which \(G\) acts by linear maps, and let \(W \subseteq V\) be a subspace with \(gW \subseteq W\) for all \(g \in G\). Then there is a subspace \(U \subseteq V\), also stable under every \(g \in G\), such that

\[V = W \oplus U.\]

In the language of modules: a representation of \(G\) over \(k\) is the same thing as a module over the group algebra \(k[G]\), the vector space with basis \(G\) and multiplication extending that of \(G\), and invariant subspaces are its submodules. Under the same hypotheses on \(G\) and \(k\), the theorem says that every submodule of a \(k[G]\)-module has a complementary submodule: a submodule meeting it only in \(0\) and spanning the whole module together with it. A module with this property is called semisimple.

Proof

Choose any complementary subspace \(U_0\) of \(W\), ignoring the group action; this is possible because every subspace of a vector space has a complement. Let \(\pi \colon V \to W\) be the projection onto \(W\) along \(U_0\). It is linear and fixes \(W\) pointwise, but need not commute with the action of \(G\). Average it over the group:

\[\pi'(v) = \frac{1}{|G|} \sum_{g \in G} g\,\pi(g^{-1} v).\]

The division is allowed because \(|G| \neq 0\) in \(k\). Reindexing the sum shows \(\pi'(hv) = h\,\pi'(v)\) for every \(h \in G\). Since \(W\) is invariant, \(\pi'\) takes values in \(W\), and each term of the sum fixes every \(w \in W\), so \(\pi'(w) = w\). Thus \(\pi'\) is a projection onto \(W\) that commutes with \(G\), and its kernel \(U = \ker \pi'\) is an invariant complement of \(W\).

The same computation works over any commutative ring \(k\) in which \(|G|\) is invertible: if a \(k[G]\)-linear map has a \(k\)-linear left inverse, then averaging that left inverse over \(G\) gives a \(k[G]\)-linear left inverse. Over a field, where \(k\)-linear left inverses of injective maps always exist, it follows that every injective homomorphism of \(k[G]\)-modules has a left inverse that is itself a homomorphism of \(k[G]\)-modules.

Examples
  • Swapping two coordinates. Let the group of order \(2\) act on \(k^2\) by exchanging the coordinates. The line spanned by \((1, 1)\) is invariant. If the characteristic of \(k\) is not \(2\), the line spanned by \((1, -1)\) is an invariant complement. In characteristic \(2\) these two lines coincide, the line spanned by \((1, 1)\) is the only invariant line, and it has no invariant complement.
  • A modular example. Over a field of prime characteristic \(p\), the cyclic group of order \(p\) acts on \(k^2\) with a generator acting by the matrix with rows \((1, 1)\) and \((0, 1)\). The line spanned by the first basis vector is the only invariant line, so it has no invariant complement. The hypothesis on the characteristic cannot be dropped.
  • Infinite groups. The same matrix defines an action of the integers on \(\mathbb{C}^2\), with the same unique invariant line. Finiteness of the group cannot be dropped either, even in characteristic \(0\).
Consequences

Together with Schur's lemma, Maschke's theorem is the basis of the character theory of finite groups: over the complex numbers every finite-dimensional representation of a finite group decomposes into irreducible ones, whose multiplicities can be read off from its character. For finite-dimensional complex representations, an alternative proof averages an inner product instead of a projection, which makes every element of \(G\) act unitarily, and then takes the orthogonal complement of \(W\). The same averaging idea, with a sum replaced by an integral against an invariant measure, proves complete reducibility for finite-dimensional continuous complex representations of compact groups.

In Mathlib

Mathlib states the theorem for an arbitrary module over the group algebra, with no finite-dimensionality assumption and no separately given vector space structure. The group algebra is the monoid algebra of finitely supported functions from \(G\) to \(k\). The hypothesis on the characteristic is phrased as the image of the natural number \(|G|\) in \(k\) being nonzero, which for a field is the same as the characteristic not dividing \(|G|\), and a complement is expressed in the lattice of submodules: the two submodules have zero intersection and their sum is everything. The averaging step is proved over any commutative ring in which \(|G|\) is invertible. The consequences that every \(k[G]\)-module is semisimple, that \(k[G]\) is a semisimple ring, and that every representation of \(G\) over \(k\) is semisimple are recorded as instances derived from the theorem.

Written for Trough, under CC BY-SA 4.0.

Sources
MathlibMonoidAlgebra.Submodule.exists_isCompl
Uses 8
Statement usesGroup · Field · Characteristic of a ring · Representation · Complementary subspaces · Module
Proof usesLinear map · Kernel of a linear map

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.