Characteristic-three sign quotient of the Specht module of shape three two ($S^{(3,2)}/D^{(4,1)}\cong \mathrm{sgn}(\mathrm{char}F=3)$)

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 $5\ne 0$ 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.