Fixed vectors in a hook Specht module in characteristic two ($\dim (S^{(n-2,1,1)}{)}^{S_n}=1\text{ exactly when }n\text{ is odd}$)

Let $n\ge 4$ and let the coefficient field have characteristic two. The subgroup fixing one permutes the standard hook basis $e_{1ij}$ indexed by unordered pairs from $\{2,\dots ,n\}$. Its fixed space is the line generated by $v={\sum }_{2\le i<j\le n}{e}_{1ij}$. Record a hook tabloid by its second- and third-row entries. In $v$, every recorded ordered pair avoiding one occurs once, and every pair involving one occurs $n-2$ times: a polytabloid is the sum of the six ordered pairs from its three column entries. For odd $n$, $v$ is therefore the sum of all tabloids and is fixed by the full symmetric group. For even $n$, it is the sum of the tabloids avoiding one and is moved by $(1,2)$. No other full-group fixed vector is possible because the point-stabilizer fixed space is one-dimensional. In characteristic different from two, swapping $i,j$ negates the $e_{1ij}$ coefficient and forces it to vanish, so even the point-stabilizer fixed space is zero.