Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 103 / 4 / b

Put $m=n-k$ and take the subgroup $S_m\times S_k$ permuting the first $m$ and last $k$ letters. Define the skew representation of a symmetric group as the multiplicity spaceThe last-letter copy of $S_k$ commutes with $S_m$, so it acts on a map $f$ by $g\cdot f={\rho }_{\lambda }(g)\circ f$. This gives a genuine $\mathbb{C}{S}_k$-module, without claiming that the whole restricted module is itself the skew representation. Equivalently,

Fix a prefix tableau of shape $\mu$ and complete it to shape $\lambda$. Iterated restriction branching rule for a symmetric group identifies the multiplicity-space orthonormal basis with standard skew Young tableaux, using labels $1,\dots ,k$ for the last $k$ cells. The Young orthogonal form restricts to this basis. When $R=s_{i}T$ is standard and $d=c_T(i+1)-c_T(i)$,An admissible interchange has $|d|\ge 2$, so its off-diagonal coefficient is nonzero. Consequently $w_R$ belongs to the group algebra span of $w_T$. The preceding reduced-path argument reaches every standard skew tableau, so this span is the whole module. Every $w_T$ is a cyclic vector for a group representation. The inherited invariant inner product makes this a unitary representation.