= Characteristic-three Specht module for the two-by-two partition
{title2=$0\to\mathbf1\to S^{(2,2)}\to\operatorname{sgn}\to0$}
Use the standard <polytabloids> for tableaux with rows $12\mid34$ and $13\mid24$, as an ordered <basis> $e_1,e_2$. Adjacent <transpositions> act by
$$
\rho(12)=\rho(34)=\begin{pmatrix}1&0\\-1&-1\end{pmatrix},\qquad\rho(23)=\begin{pmatrix}0&1\\1&0\end{pmatrix}.
$$
In characteristic three their common fixed line is $F(e_1+e_2)$. On its quotient every <transposition> acts by minus one, giving the <sign representation>. Thus the displayed sequence has the <trivial representation> and sign as its <composition factors>, each once. It is nonsplit: the minus-one <eigenspace> of $\rho(12)$ is $Fe_2$, which is not preserved by $\rho(23)$.
The natural <tabloid> form restricts to <Gram matrix> $\begin{pmatrix}4&2\\2&4\end{pmatrix}$. Its radical in characteristic three is exactly that fixed line. Therefore $D^{(2,2)}$ is the sign quotient, while the trivial factor is $D^{(4)}$. This gives an explicit small failure of semisimplicity in modular symmetric-group <representation> theory.
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