 Two-subset incidence realization of a two-row Specht module

Over a field of characteristic zero with $n\ge 4$, let $V_2$ be the permutation representation on unordered two-element subsets and $V_1$ the representation on singletons. The equivariant incidence map $D:V_2\to V_1$ sends $e_{ij}$ to $e_i+e_j$. It is surjective: if coefficients $c_i$ annihilate its image, then $c_i+c_j=0$ for all distinct $i,j$, forcing all coefficients to vanish. Thus its kernel has dimension $\left(\genfrac{}{}{0ex}{}{n}{2}\right)-n$.

A polytabloid of shape $(n-2,2)$, identified by its bottom-row pair, is an alternating four-edge vector $e_{cd}-e_{ad}-e_{cb}+e_{ab}$. Its incidence is zero. The hook-length formula gives the same dimension $\left(\genfrac{}{}{0ex}{}{n}{2}\right)-n$ for this Specht module, proving the identity. Maschke's theorem yields $V_2\cong S^{(n-2,2)}\oplus V_1$.

For a Young subgroup with block sizes ${\mu }_1,\dots ,{\mu }_l$, there are $l$ singleton orbits and $\left(\genfrac{}{}{0ex}{}{l}{2}\right)+\#\{i:{\mu }_i\ge 2\}$ pair orbits. Taking fixed vectors and using Frobenius reciprocity therefore gives the multiplicity of $S^{(n-2,2)}$ in $M^{\mu }$ as $\left(\genfrac{}{}{0ex}{}{l}{2}\right)+\#\{i:{\mu }_i\ge 2\}-l$. This derives the multiplicity without Young's rule; block sizes $(3,2,1)$ give exactly two.