What are defining & fundamental representations?

Question

In physics terminology, one hears of the fundamental & defining representations of lie algebras or groups - are these the same as irreducible representations?

Answer 1

I'm assuming that the physics terminology is consistent with the mathematics terminology (which maybe is a big assumption). I will also assume you are familiar with highest weight theory.

When mathematicians say fundamental representations, they're usually referring to representations whose highest weight takes the value 1 on a single coroot and 0 everywhere else. Thus the number of fundamental representations is the rank of the Lie algebra and the collection of weights corresponding to the fundamental representations span the weight lattice.

These are useful because by highest weight theory for semisimple Lie algebras, every irreducible representation of such a Lie algebra is contained in a tensor product of fundamental representations.

On the other hand, the defining representations refer to Lie algebras that are defined as matrix subalgebras of $\mathfrak{gl}(n)$ (in which case the defining rep is on $\mathbb C^n$). For example, the defining representation of $\mathfrak{so}(n)$ is on $\mathbb C^n$ because $\mathfrak{so}(n)$ is defined as the space of skew-symmetric $n\times n$ matrices and these, by definition, act on $\mathbb C^n$.

For $\mathfrak{so}(n), \mathfrak{su}(n)$ and $\mathfrak{sp}(n)$, the defining representation is a fundamental representation (but of course there will be more fundamental representations).