What is the inclusion of this symmetrized space in $V \otimes \Lambda^2 V$?

Question

This question concerns Exercise 2.8.1(1) in J.M. Landsberg's book Tensors: Geometry and Applications.

Let $V$ be a $\mathbb{C}$-vector space, and consider the vector space which is defined as the image of $V^{\otimes 3}$ under the map where antisymmetrizes the first two vector spaces, and symmetrizes the first and third vector spaces.

For example, the image of $v_1 \otimes v_2 \otimes v_3$ under this composition of maps is $v_1 \otimes v_2 \otimes v_3+v_3 \otimes v_2 \otimes v_1-v_2\otimes v_1 \otimes v_3-v_3 \otimes v_1 \otimes v_2$. In Exercise 2.8.1(1) it is claimed that there is an inclusion of in $V \otimes \Lambda^2 V$, where $\Lambda^2 V \subseteq V \otimes V$ is the anti-symmetric subspace.

What is this inclusion?

Answer 1

The inclusion can be obtained by chasing the isomorphism $ \rho_{23} \rho_{\substack{1\\2}} V^{\otimes 3} \cong \rho_{\substack{1\\2}} \rho_{23} V^{\otimes 3} \subseteq \Lambda^{2} V \otimes V$. This isomorphism is given by $v \mapsto \rho_{\substack{1\\2}} v$. The fact that this is an isomorphism can be seen using irreducibility of the modules $\mathbb{C}[S_3] \rho_{23} \rho_{\substack{1\\2}}$ and $\mathbb{C}[S_3] \rho_{\substack{1\\2}}\rho_{23} $, and employing similar arguments to those used in https://math.stackexchange.com/a/471255/748775.