rent from $S$. If $i>2$ then we apply Lemma \ref{lemma7} to move the $2$ from row $3$ to row $2$; neglecting the tableau not dominated by $S$ and (in the case $j>2$) neglecting the tableau with two rows equal to $\young(j)$, the only tableau we get is}}
U''[i,j]&={\text{\footnotesize$\gyoungx(1.2,;1;1;2;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{\hat\jmath};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!3};{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;2;j;{b\!\!+\!\!4},;i,;3,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}};\end{aligned}$$]{} $i-3$ more applications of Lemma \[lemma7\] show that ${\hat\Theta_{U''[i,j]}}$ equals a semistandard homomorphism different from ${\hat\Theta_{S}}$.

We conclude that ${\hat\Theta_{U[i]}}$ equals ${\hat\Theta_{S}}$ plus a linear combination of homomorphisms indexed by tableaux which are either not dominated by $S$ or semistandard and different from $S$. The homomorphism ${\hat\Theta_{V[i]}}$ is analysed in exactly the same way, interchanging $b+3$ and $b+4$.

Putting these cases together, we find that the coefficient of ${\hat\Theta_{S}}$ in $\sigma$ is the total number of tableaux of the form $T[i]$, $U[i]$ or $V[i]$, i.e. $(b+2-v)+2(b+1)$, which is odd.

It remains to consider the case $v=b+3$. In this case only the tableaux $V[i]$ appear, but the analysis of these tableaux is exactly the same, so the coefficient of ${\hat\Theta_{S}}$ in $\sigma$ is the number of tableaux $V[i]$, i.e. $b+1$, which again is odd.

It turns out that up to scaling, $\sigma$ is the only homomorphism from $S^\la$ to $S^\mu$.

\[cdhomdim1\] With $\la,\mu$ as above, $$\dim_\bbf{