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Why is an invariant vector subspace sometimes called a representation? For example in Lie algebras, say su(3), the subspace characterized by the highest weight (1,0) is an irreducible representation of dimension 3 of su(3).

However, a representation of a Lie algebra is a Lie algebra homomorphism from the algebra to a subspace of the so called general linear algebra of some vector space. Or in other words, the representation is a map that assigns elements of the algebra to elements of the set of linear endomorphisms of some vector space.

In the previous example, the subspace (1,0) is a subspace in which the action of the endomorphisms maps its elements into themselves. So by the definition, the irreducible representation should be the mapping that associates the endomorphisms to the elements of the algebra and not the space in which they act.

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To add to what @Qmechanic says, note also that for a representation $\rho:\mathfrak g\to \mathfrak{gl}(V)$ of a Lie algebra $\mathfrak g$ acting on a vector space $V$, a vector subspace $W$ of $V$ is called an invariant subspace of the representation provided $\rho(X)w\in W$ for all $X\in\mathfrak g$ and $w\in W$. A representation $\rho$ is said to be irreducible provided it has no invariant subspaces except for $\{0\}$ and $V$. This is why one encounters physicists "identifying irreducible representations with invariant subspaces." Strictly speaking, it doesn't make sense to even talk about invariant subspaces of a vector space unless you already have the representation in hand.

Hopefully this helps!

Cheers!

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Concerning a Lie algebra representation $\rho: L \to {\rm End}(V,\mathbb{F})$, where $L$ is a Lie algebra, where $\mathbb{F}$ is a field (typically $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$), where $V$ is a $\mathbb{F}$-vector space, and where $\rho$ is a Lie algebra homomorphism, be aware that physicists refer to both the map $\rho$ and the vector space $V$ as "a representation".

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