Let $\Phi$ be a $G_2$ root system, $\omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V = V(\omega_1)$ of highest weight $\omega_1$.

I am asked to decompose in terms of highest weights the exterior powers $\Lambda^kV$ for all $k$.

Note $\Lambda^kV = V^{\otimes k}/N_n$, where:

$N_n = \{v_1 \otimes \dots \otimes v_k \mid v_i \in V \; \forall i \text{ and } v_i = v_j \text{ for some }i \text{ and } j\}$.

Denote the coset of $u_1 \otimes \dots \otimes u_k$ by $u_1 \wedge \dots \wedge u_k$. It is known that if $V $ has basis $v_1, \dots, v_n$, then $\Lambda^kV$ has basis $\{v_{i_1} \wedge \dots \wedge v_{i_k} \mid i_1 < \dots < i_k \in \{1,\dots, n\} \}$.

In our case, $V$ has dimension $n = 7$ and so, I think that it would be correct to say that $\lambda^kV = 0, \; \forall k \geq 8$ since if we can't have eight of the seven basis vectors wedged together without a repeat, meaning it would land within $N_7$.

Further, I think this means that $\Lambda^7V$ has dimension $1$ and is exactly $span\{v_1\wedge \dots \wedge v_7\}$

Now, regarding the specifics of the question, I think this means that for $k\geq 8$ the $k$th exterior powers of $V$ have highest weight $0$ since it is the $0$-space?

For $k=7$ however, I am less certain of how to proceed. I suppose that we could without loss of generality suppose that $v_1$ is the highest weight vector of weight $\omega_1$ in $V(\omega_1)$, but then we have:

For $m\in \mathbb C$ and $t \in \mathfrak{t}$ the Cartan Subalgebra,

$t \cdot m(v_1\wedge \dots \wedge v_7) = m((t \cdot v_1)\wedge \dots \wedge v_7 + \dots + v_1 \wedge \dots \wedge (t\cdot v_7))$

It is not clear to me that this is a highest weight vector of any weight, because I am not sure how $t$ acts on any of the other basis vectors. Thus I am not sure how to proceed with this case.

Additionally, I am even less sure how to proceed with the cases of $k \leq 6$. In these cases, the representation has dimension greater than 1, but aside from that I am struggling to make any helpful or worthwhile observations.

How might I be able to proceed and decompose these exterior powers?


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