= Skew representation of a symmetric group
{title2=$V^{\lambda/\mu}$}
For $|\lambda|=n$, $|\mu|=m$, and $k=n-m$, define
$$
V^{\lambda/\mu}=\operatorname{Hom}_{S_m}(V^\mu,\operatorname{Res}^{S_n}_{S_m}V^\lambda).
$$
The copy of $S_k$ acting on the last $k$ letters commutes with $S_m$ and acts on this multiplicity space. Iterated <restriction branching rule for a symmetric group> supplies a basis indexed by <standard skew Young tableaux>. Every such basis vector is a <cyclic vector for a group representation>: admissible swaps connect all tableaux, and their off-diagonal coefficients in the <Young orthogonal form> are nonzero.
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