= Least non-linear ordinary degree of a symmetric group
For $n\ge3$, the least ordinary <irreducible> <dimension> exceeding one is $n-1$, except at $n=4$, where $S^{(2,2)}$ has <dimension> two. The dimension-$n-1$ shapes are $(n-1,1)$ and its <conjugate partition>, with the additional shapes $(3,3)$ and $(2,2,2)$ at $n=6$. For $n=2$ there is no non-linear <irreducible>. A proof uses the <restriction branching rule for a symmetric group>: a nonstandard shape with at least two corners has two non-linear predecessors, and a rectangular shape has one predecessor which in turn has two non-linear predecessors. This bounds its degree strictly above $n-1$ once $n\ge9$; the smaller cases follow directly from the <hook-length formula>.
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