= Smallest non-linear ordinary degree of an alternating group
For $A_n$, the least non-linear ordinary degree is $3$ at $n=4,5$ and $n-1$ for $n\ge6$. For $n\le3$ no non-linear <irreducible> exists. The degree-$n-1$ <representation> is unique for $n\ge2$, except $n=3$ where it does not exist and $n=6$ where two distinct degree-five <irreducibles> occur. The proof uses tableau branching to bound the degrees of nonstandard shapes and of the half-degree constituents of self-conjugate shapes, with the small cases checked by the <hook-length formula>.
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