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Groups That Act Linearly

A representation is just a homomorphism from a group into invertible matrices. We unpack that one sentence: why it lets you do linear algebra on a group, what subrepresentations and irreducibility mean, and how the same object reads as a module over the group algebra.

The definition, slowly

You already know a group action is a map G × X → X. A representation is the special case where X is a vector space V over a field k and every group element acts as a linear map. Concretely, a representation of G is a homomorphism ρ: G → GL(V), where GL(V) is the group of invertible linear maps on V. Choosing a basis turns GL(V) into GL(n,k), so a representation is, very concretely, a rule that assigns to each g a matrix ρ(g) so that ρ(gh) = ρ(g)ρ(h).

The dimension of V is the degree of the representation. The simplest one is the trivial representation: every g acts as the identity on k. If ρ is injective the representation is faithful — it loses no information about G. The whole point is leverage: groups are slippery, but matrices we can compute eigenvalues, traces, and ranks of. A representation is a microscope that turns group structure into linear algebra.

Two representations of the cyclic group Z/3Z = {0,1,2}, generator g.

Trivial (degree 1):   rho(g) = [1]

A faithful degree-2 real representation = rotation by 120 degrees:
  rho(g) = [cos120, -sin120; sin120, cos120]
         = [-1/2, -sqrt(3)/2; sqrt(3)/2, -1/2]
Check the relation g^3 = e:
  rho(g)^3 = rotation by 360 deg = [1,0; 0,1]  (identity)  OK

Over C this same group also has the two 1-dim reps
  rho_j(g) = omega^j   where omega = e^{2 pi i/3},  j = 1, 2.
The same group, three different representations — one trivial, one real and faithful, two complex of degree 1.

Subrepresentations and irreducibility

Inside V some subspaces are special: W ⊆ V is a subrepresentation if it is G-invariant, meaning ρ(g)W ⊆ W for every g. Then ρ restricts to a representation on W. A nonzero representation is irreducible if its only G-invariant subspaces are 0 and V — it cannot be broken into a smaller block. Irreducibles are the atoms of the theory; the whole game is to find them and assemble everything else from them.

Watch the analogy with ring theory: a subrepresentation is to a representation what an ideal is to a ring, and irreducibility is the representation-theoretic cousin of simplicity. We will make this precise next: there is a single ring whose modules are exactly the representations of G.

Representations are modules

Here is the unifying idea. Form the group algebra k[G]: a vector space with basis the elements of G, where multiplication extends the group law linearly. Then a representation of G is exactly a module over k[G]. The action of a basis element g on a vector is just ρ(g), and a formal sum Σ c_g g acts by Σ c_g ρ(g). A subrepresentation becomes a submodule, an irreducible representation becomes a simple module.