I'm trying to understand a calculation in a paper in which a composition of $\text{GL}_n(\mathbb{C})$ maps are applied to a vector in $\wedge^2 V \otimes V^*$ (where $V$ is the standard $\text{GL}_n(\mathbb{C})$ representation and $V^*$ is its dual).
Letting $E_{3n} \in \mathfrak{gl}(n, \mathcal{C})$ be the matrix with 1 in the 3rd row, $n$th column (I hope I don't have that backwards), the evaluation of a map I'm trying to understand is:
$$E_{3n} ((e_1 \wedge e_2 \otimes e_2^* + e_1 \wedge e_n \otimes e_n^*) \wedge (e_2 \wedge e_n \otimes e_n^*))$$
I'm sure I'm missing something stupid, but I can't see how $E_{3n}$ acts on this vector (it looks to me like it should kill it entirely!). For example, is not $E_{3n}(e_1) = \vec{0}$?
The induced action of $\mathfrak{gl}(n, \Bbb C)$ on $V$ is $E_{ij} \cdot e_k = \delta_{jk} e_i$, so the dual action on $V^*$ is $E_{ij} \cdot e_k^* = -\delta_{ik} e_j^*$.
So, for example, \begin{align*} E_{ij} \cdot (e_a \wedge e_b \otimes e_c^*) &= (E_{ij} \cdot e_a) \wedge e_b \otimes e_c + e_a \wedge (E_{ij} \cdot e_b) \otimes e_c^* + e_a \wedge e_b \otimes (E_{ij} \cdot e_c^*) \\ &= (\delta_{j a} e_i) \wedge e_b \otimes e_c^* + e_a \wedge (\delta_{jb} \cdot e_i) \otimes e_c^* + e_a \wedge e_b \otimes (-\delta_{ic} \cdot e_j^*) \\ &= \delta_{ja} e_i \wedge e_b \otimes e_c^* + \delta_{jb} e_a \wedge e_i \otimes e_c^* -\delta_{ic} e_a \wedge e_b \otimes e_j^* .\\ \end{align*}