Representation Theory - Definitions and Concepts - Subrepresentations, Quotients, and Irreducible Representations

Subrepresentations, Quotients, and Irreducible Representations

If (W,ψ) is a representation of (say) a group G, and V is a linear subspace of W that is preserved by the action of G in the sense that g · v ∈ V for all v ∈ V (Serre calls these V stable under G), then V is called a subrepresentation: by defining φ(g) to be the restriction of ψ(g) to V, (V, φ) is a representation of G and the inclusion of V into W is an equivariant map. The quotient space W/V can also be made into a representation of G.

If W has exactly two subrepresentations, namely the trivial subspace {0} and W itself, then the representation is said to be irreducible; if W has a proper nontrivial subrepresentation, the representation is said to be reducible.

The definition of an irreducible representation implies Schur's lemma: an equivariant map α: V → W between irreducible representations is either the zero map or an isomorphism, since its kernel and image are subrepresentations. In particular, when V = W, this shows that the equivariant endomorphisms of V form an associative division algebra over the underlying field F. If F is algebraically closed, the only equivariant endomorphisms of an irreducible representation are the scalar multiples of the identity.

Irreducible representations are the building blocks of representation theory: if a representation W is not irreducible then it is built from a subrepresentation and a quotient that are both "simpler" in some sense; for instance, if W is finite dimensional, then both the subrepresentation and the quotient have smaller dimension.