0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;\star;\star),$$ which do not occur in any other ${\psi_{2,1}}\circ\tau_i$ (note these tableaux do occur when we compute ${\psi_{2,1}}\circ{\hat\Theta_{T}}$ for $T$ of type $4$, but each one occurs twice when we sum over tableaux of type $4$, so the contributions cancel). So $\tau_5$ does not appear in $\theta$.

Next we consider ${\psi_{3,1}}\circ\tau_i$ for each $i$. For $i=3,6$ or $8$, we find that ${\psi_{3,1}}\circ\tau_i$ involves semistandard tableaux which do not occur in any other ${\psi_{3,1}}\circ\tau_i$; so $\tau_3,\tau_6,\tau_8$ cannot occur in $\theta$. Moreover, we find that ${\psi_{3,1}}\circ\tau_1={\psi_{3,1}}\circ\tau_7$ and ${\psi_{3,1}}\circ\tau_2={\psi_{3,1}}\circ\tau_4$, and that these two homomorphisms are linearly independent. So $\theta$ must be a linear combination of $\tau_1+\tau_7$ and $\tau_2+\tau_4$.

Finally we return once more to ${\psi_{2,1}}\circ\theta$. We find that ${\psi_{2,1}}\circ\tau_1={\psi_{2,1}}\circ\tau_4\neq0$, while ${\psi_{2,1}}\circ\tau_2={\psi_{2,1}}\circ\tau_7=0$. So $\tau_1$ and $\tau_4$ must appear with the same coefficient in $\theta$; so $\theta$ must be a scalar multiple of $\tau_1+\tau_2+\tau_4+\tau_7$, and so the homomorphism space has dimension at most $1$.

Homomorphisms from $S^\mu$ to $S^\la$
-------------------------------------

Now we consider homomorphisms from $S^\mu$ to $S^\la$. We begin by constructing a non-zero homomorphism. Let $C$ be the $\mu$-tableau $${\text{\footnotesize$\gyoungx(1.2,;1;1_{3.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3.5*.25,0);\end{tikzpicture}}};1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};{b\!\!+\!\!2},;1;1_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};1,;2;2)$}}$$ of type $\la$.

\[cd2chom\] With $C$ as above, we have ${\psi_{d,t}}\circ{\hat\Theta_{C}}=0$ for all $d,t$, and ${\hat\The