REFERENCE SHEET

Representations & invariant subspaces

Replace abstract symmetries by matrices that preserve composition.

Before this Homomorphisms & quotients, Vector spaces & linear maps

ρ: G → GL(V), ρ(gh)=ρ(g)ρ(h)

Definitions

Representation

A group homomorphism into the invertible linear maps of a vector space V over a specified field. Its dimension is dim V.

Invariant subspace

W⊆V is invariant if ρ(g)W⊆W for every g. A nonzero representation is irreducible if its only invariant subspaces are 0 and V.

Intertwiner

A linear map T:V→W with Tρ(g)=σ(g)T for all g. An invertible intertwiner makes the representations equivalent.

Key theorems

Maschke theorem

Every finite-dimensional representation of a finite group over a field whose characteristic does not divide |G| is a direct sum of irreducible representations.

Schur lemma

An intertwiner between irreducible representations is zero or an isomorphism. For a finite-dimensional irreducible representation over an algebraically closed field, every endomorphism intertwiner is scalar.

Notable applications

Symmetry reduction

Decompose linear problems into invariant blocks before computing.

Fourier analysis

Characters of finite abelian groups provide elementary Fourier modes.

Work it out

Try before revealing

01 / Decompose

C₂ acts on ℝ² by swapping coordinates. Find invariant one-dimensional subspaces.

Hint

Use symmetric and antisymmetric vectors.

Solution

span{(1,1)} carries the trivial action; span{(1,−1)} carries the sign action. Their direct sum is ℝ².

02 / Explain

Is the trivial representation on ℂ² irreducible?