phisms from $S^\la$ to $S^{\mu'}$ is one-dimensional, spanned by the homomorphism $\sigma=\sum_{T\in{\calu}}{\hat\Theta_{T}}$. On the other hand, the space of homomorphisms from $S^\mu$ to $S^\la$ has dimension one or two, each homomorphism being a linear combination of the homomorphisms ${\hat\Theta_{A}}$ and ${\hat\Theta_{B}}$. So it suffices to compute the compositions $\sigma\circ{\hat\Theta_{A}}$ and $\sigma\circ{\hat\Theta_{B}}$.

Let $D$ be the $\mu$-tableau $$D=\gyoung(;1;2;3_4{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(4*.25,0);\end{tikzpicture}}};u,;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v)$$ of type $\mu'$. Then we have the following.

\[uab\] Suppose $T\in{\calu}$, and let $x$ be the entry in the $(2,2)$-position of $T$. Then $${\hat\Theta_{T}}\circ{\hat\Theta_{A}}={\hat\Theta_{D}},
\qquad
{\hat\Theta_{T}}\circ{\hat\Theta_{B}}=\begin{cases}
{\hat\Theta_{D}}&(x\ls v)\\
0&(x>v).
\end{cases}$$ Furthermore, ${\hat\Theta_{D}}\neq0$.

The fact that ${\hat\Theta_{D}}\neq0$ is a simple application of Lemma \[lemma7\]. To show that the compositions of homomorphisms are as claimed, take $T\in{\calu}$ and recall the notation of Proposition \[tabcomp\], with $S$ equal to either $A$ or $B$.

Suppose $X\in\calx$. Since each $T^i$ is a proper set, each $X^{ij}$ must be as well. This means that if some integer $i$ appears in two sets $X^{kj},X^{lj}$, then the multinomial coefficient $\mbinom{X^{1j}_j+X^{2j}_j+X^{3j}_j+\dots}{X^{1j}_j,X^{2j}_j,X^{3j}_j,\dots}$ from Proposition \[tabcomp\] will include a factor $\binom21$, which gives $0$.

So in order to get a non-zero coefficient in Proposition \[tabcomp\], we must have $X^{1j},X^{2j},X^{3j},\dots$ pairwise disjoint for each $j$, which means that we will have $$X^{11}\sqcup X^{21}\sqcup\dots=\{1,\dots,u\},\qquad X^{12}\sqcup X^{22}=\{1,\dots,v\};\tag*{($\dagger$)}$$ so $U_X$ will equal $D$.

If $S=A$, the only way to achieve this is to have $$X^{11}=T^1\setminus\{1,\d