# Decomposing $k$th exterior powers $\Lambda^kV(\omega_1)$

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?