@pschwahn@AxelBoldt - Ugh! I should have stuck with rational representations, which is what people usually talk about when studying representations of linear algebraic groups.
I'm pretty sure that every finite-dimensional rational representation of GL(n,ℝ) is completely reducible, and every irreducible rational representation comes from one described by a by Young diagram, possibly tensored by a representation 𝑔↦det(𝑔)ⁿ where n is some integer.
(There is some overlap here since the nth exterior power of the tautologous representation, described by a Young diagram, is also the representation 𝑔↦det(𝑔).)
It's annoying that the basic facts about finite-dimensional representations of GL(n,ℝ) aren't on Wikipedia! Someday I'll have to put them on there... once I get enough references to make sure I'm not screwing up!