= Characteristic-three sign quotient of the Specht module of shape three two
{title2=$S^{(3,2)}/D^{(4,1)}\cong\operatorname{sgn}\quad(\operatorname{char}F=3)$}
For the symmetric group on five letters, $S^{(4,1)}$ is the four-dimensional augmentation representation. In characteristic three it is simple, since its only possible proper nonzero submodule would be the constant line, which is excluded by $5\ne0$ in the field. Thus it is $D^{(4,1)}$. If this module occurs as a submodule of $S^{(3,2)}$, the quotient has dimension one, since the standard polytabloid basis gives $\dim S^{(3,2)}=5$.
A transposition has trace one on $S^{(3,2)}$, by the <Murnaghan–Nakayama rule>, and trace two on $S^{(4,1)}$, by subtracting the trivial line from the five-point permutation representation. These integral traces reduce to the same elements in characteristic three. The quotient trace is therefore minus one, so its one-dimensional action is the <sign representation>. In particular an arbitrary permutation acts on it by its sign. This identifies the quotient action without asserting that the original short exact sequence splits.
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