= Fixed vectors in a hook Specht module in characteristic two
{title2=$\dim(S^{(n-2,1,1)})^{S_n}=1\text{ exactly when }n\text{ is odd}$}
Let $n\geq4$ and let the <coefficient> <field> have <characteristic> two. The <subgroup> fixing one permutes the standard <hook of a Young diagram>[hook] <basis> $e_{1ij}$ indexed by unordered pairs from $\{2,\ldots,n\}$. Its fixed space is the line generated by $v=\sum_{2\leq i<j\leq n}e_{1ij}$. Record a <hook of a Young diagram>[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.
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