Fundamental representation

Remarks

They exist and are unique up to isomorphism by the theorem of the highest weight. Example: for $\mathfrak{sl}_n(\C)$ with diagonal $\mathfrak{h}$, $h_j = E_{jj} - E_{j+1,j+1}$, and $\omega_k = \varepsilon_1 + \dots + \varepsilon_k$, where $\varepsilon_i$ reads the $i$-th diagonal entry. The module $\Lambda^k \C^n$ has highest weight vector $e_1 \wedge \dots \wedge e_k$ of weight $\omega_k$, so $L(\omega_k) \cong \Lambda^k \C^n$. Non-example: the adjoint representation of $\mathfrak{sl}_3(\C)$ is irreducible of highest weight $\varepsilon_1 - \varepsilon_3 = \omega_1 + \omega_2$, which is not a fundamental weight. Every dominant integral weight is $\lambda = \sum_i n_i \omega_i$ with $n_i \in \Z_{\ge 0}$. Then $L(\lambda)$ is the submodule of $\bigotimes_i L(\omega_i)^{\otimes n_i}$ generated by the tensor product of highest weight vectors.