Fundamental representation

Let $\mathfrak g$ be a complex semisimple Lie algebra with a fixed Cartan subalgebra $\mathfrak h$ and a choice of positive roots (see positive roots ). Let $\{\alpha_1,\dots,\alpha_r\}$ be the set of simple roots , and let $\{\omega_1,\dots,\omega_r\}\subset \mathfrak h^*$ be the corresponding fundamental weights, characterized by

where $\alpha_j^\vee$ are the simple coroots.

Definition (Fundamental representation).
A fundamental representation of $\mathfrak g$ is a finite-dimensional irreducible highest-weight representation whose highest weight is one of the fundamental weights $\omega_i$.

Example (type $A_{n-1}$).
For $\mathfrak{sl}_n(\mathbb C)$ (see special linear Lie algebra ), the fundamental representations are the exterior powers of the defining representation:

With the standard choice of $\mathfrak h$ as diagonal trace-zero matrices, the highest weight of $\Lambda^k(\mathbb C^n)$ is $\omega_k$ (in the usual convention where weights record the eigenvalues of $\mathfrak h$ on weight vectors; compare weights and weight spaces ). In particular, $\Lambda^1(\mathbb C^n)=\mathbb C^n$ has highest weight $\omega_1$, while $\Lambda^{n-1}(\mathbb C^n)\cong (\mathbb C^n)^*$ has highest weight $\omega_{n-1}$.

Context.
Fundamental representations correspond to the nodes of the Dynkin diagram and generate the representation ring in many settings; the classification of all irreducibles proceeds via the highest-weight theorem .