Symmetric and alternating-squares of vector space

Definition

Let $V$ be a vector space. We define a linear map $\sigma$ from the tensor product $V\otimes V$ to itself by

$\sigma (v\otimes w)=w\otimes v$

for all $v\otimes w\in V\otimes V$.

Then the symmetric-square of $V$ is

${\mathrm{S}\mathrm{y}\mathrm{m}}^2(V)=S^2(V):=\{a\in V\otimes V:\sigma a=a\}$

and the alternating-square or exterior-square of $V$ is

${\mathrm{A}\mathrm{l}\mathrm{t}}^2(V)={\mathrm{\Lambda }}^2(V):=\{a\in V\otimes V:\sigma a=-a\}$.

These are both eigenspaces of $\sigma$.