Fundamental representations and weight space dimension

Question

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?

Answer 1

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 $0 \leq d < p$).

Answer 2

Such representations are actually quite rare and are superset of the minuscule representations.