nsely 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}$$]{} and $i-3$ more applications of Lemma \[lemma7\] show that ${\hat\Theta_{T''[i]}}={\hat\Theta_{S}}$.

Now consider applying Lemma \[lemma7\] to $T'[i,j]$, to move the $2$ from row $3$ to row $2$. If $j=2$, then neither of the two tableaux obtained is dominated by $S$. If $j>2$, then two of the three tableaux obtained are not dominated by $S$; the third has rows $3$ and $j+1$ both equal to $\young(j)$, so the resulting homomorphism is zero by Lemma \[lemma7\].

So we conclude that ${\hat\Theta_{T[i]}}$ equals ${\hat\Theta_{S}}$ plus a linear combination of homomorphisms indexed by tableaux not dominated by $S$.\
Now we consider applying Lemma \[lemma7\] to $U[i]$, moving the $1$ up from row $2$. The tableaux obtained that are dominated by $S$ are $T'[i]$ and the tableaux [$$\begin{aligned}
U'[i,j]&={\text{\footnotesize$\gyoungx(1.2,;1;1;2;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{\hat\jmath};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!3};{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;j;i;{b\!\!+\!\!4},;2,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\tag*{for $2\ls j\ls v$ with $i\neq j$.}\\
\intertext{\parbox{\linewidth}{(Note that if $i<j$ then $i$ and $j$ in the second row should be written the other way round; the case $i=j$ does not occur because the accompanying coefficient would be $\binom21=0$.)\\\hspace*{17pt}If $i=2$, then $U'[i,j]$ is a semistandard tableau diffe