For the Lie algebra $\frak{sl}\_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all $1$-dimensional. Is this true for the other series - are the weight spaces of the fundamental representations always $1$-dimensional. If this is true, does it identify the fundamental representations, that is are the fundaental representation precisely those with $1$-dimensional weight spaces? 



Fundamental representations and weight space dimension
Let $\mathfrak{g}$ be a simple Lie algebra over an algebraically closed field of characteristic $0$ and denote the fundamental highest weights by $\varpi\_1, \ldots, \varpi\_l$, where $l$ is the rank of $\mathfrak{g}$ and the ordering of the $\varpi\_i$ is the usual one (Bourbaki).


The cases where an irreducible representation of highest weight $\lambda$ has all weight spaces $1$-dimensional are the following:


* Type $A\_l$: $\lambda = \varpi\_i, c \varpi\_1$, or $c\varpi\_l$.
* Type $B\_l$ ($l \geq 2$): $\lambda = \varpi\_1$ or $\varpi\_l$.
* Type $C\_l$ ($l \geq 3$): $\lambda = \varpi\_1$. For type $C\_3$, also $\lambda = \varpi\_3$.
* Type $D\_l$ ($l \geq 4$): $\lambda = \varpi\_1$, $\varpi\_{l-1}$ or $\varpi\_{l}$.
* Type $G\_2$: $\lambda = \varpi\_1$.
* Type $E\_6$: $\lambda = \varpi\_1$ or $\varpi\_6$.
* Type $E\_7$: $\lambda = \varpi\_7$.


So most fundamental representations do not have $1$-dimensional weight spaces. Also, in type $A$ you have non-fundamental representations which have all weight spaces $1$-dimensional. These can be realized as symmetric powers of the natural irreducible representation and its dual.


For a reference, see Chapter 6 in Seitz, *The maximal subgroups of classical algebraic groups* (Memoirs of the AMS). Seitz works with algebraic groups, but this will of course give you the result for Lie algebras as well. 


Seitz actually proves a result over an algebraically closed field of characteristic $p \geq 0$. In characteristic $p > 0$ we get some additional examples, such as $\lambda = d \varpi\_i + (p-d-1) \varpi\_{i+1}$ in type $A\_l$ (for $1 \leq i < l$ and $0 \leq d < p$).


