I'm trying to show that $(e_1 \otimes e_2 \otimes e_3 - e_3 \otimes e_1 \otimes e_2) \otimes e_n^*$ is a highest weight vector for the irreducible submodule of $V^{\otimes 3} \otimes V^*$ with highest weight (0,0,1,0,...,0,1), where $V$ is the standard $\mathfrak{sl}_n(\mathbb{C})$-representation and $V^*$ is its dual.
According to this post, it should be the unique vector that satisfies:
- $E_{33}$ scales it by 1
- $E_{(n-1)(n-1)}$ scales it by 1
- $E_{ii}$ for $1 \leq i \leq n-1$ and $i \neq 3,n-1$ kill it
- $E_{ij}$ for $i < j$ kill it
If the $E_{ij}$'s act on my vector in the way given by the answer to this question, I find that $E_{33}$ fixes my vector but $E_{(n-1)(n-1)}$ kills it. So I'm misunderstanding something.
I don't think the linked post says what you claim it does. The highest weight $(0,0,1,\dots,0,1)$ just means $\omega_3 + \omega_{n-1}$ so its weight vectors are scaled by $1$ by the coroots $\alpha_3^\vee$ and $\alpha_{n-1}^\vee$ (fundamental weights are a dual basis to the coroots) but these are not $E_{33}$ and $E_{(n-1)(n-1)}$. Rather, the coroots in $\mathfrak{sl}_n$ look like $\alpha_k^\vee = E_{kk} - E_{(k+1)(k+1)}$. You should think of them as generalising the element $h = \begin{pmatrix}1&0\\0&-1\end{pmatrix} \in \mathfrak{sl}_2$
Let $v=(e_1 \otimes e_2 \otimes e_3 - e_3 \otimes e_1 \otimes e_2) \otimes e_n^*$
Then $E_{11}v = E_{22}v = E_{33}v = v$ so that $\alpha_1^\vee v = (E_{11} - E_{22})v = 0$ and $\alpha_2^\vee v = (E_{22} - E_{33})v = 0$. Then $E_{kk}v = 0$ for $3 < k < n$ so $\alpha_3^\vee v = v$ and $\alpha_k^\vee v = v$ for $3 < k < n-1$. Finally, $E_{nn}v = -v$ so that $\alpha_{n-1}^\vee v = (E_{(n-1)(n-1)} - E_{nn})v = v$.
To conclude $\alpha_k^\vee v = v$ if $k=3,n-1$ and $\alpha_k^\vee v = 0$ otherwise so it is an element of the $\omega_3 + \omega_{n-1}$ weight space.
Edit: Looking at this some more, I am not convinced it is a highest weight vector, however, or that there is even an irreducible submodule of that highest weight in $V^{\otimes3} \otimes V^*$.