<s>[INST] $\mathcal{D}$-classes Let $$\alpha = \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
1&3&3
\end{array}\right) \in \mathcal{T}_3\text{.}$$
(a) Show that the $\mathcal{D}$-class of $\alpha$ contains all those elements of $\mathcal{T}_3$ which have the same rank (cardinality of their image) as $\alpha$.
(b) Show that the $\mathcal{D}$-class of $\alpha$ contains 3 $\mathcal{R}$-classes and 3 $\mathcal{L}$-classes and that it has 2 element $\mathcal{H}$-classes.
How to do (b)?
From (a) we have that $D_{\alpha} = \{ \beta \in \mathcal{T}_3 : | \text{Im}( \beta )| = 2 \}$.
Explicitly then:
$$D_{\alpha} = \left\{ \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
1&1&2
\end{array}\right), \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
1&2&1
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
2&1&1
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
2&2&1
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
2&1&2
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
1&2&2
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
3&3&1
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
3&1&3
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
1&3&3
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
1&1&3
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
1&3&1
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
3&1&1
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
2&2&3
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
2&3&2
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
3&2&2
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
3&3&2
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
3&2&3
\end{array}\right) , \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
2&3&3
\end{array}\right) \right\}$$
 [/INST] I've had to change accounts as I can't access my email account so I can't reply to any posts because I'm not using the account I originally asked the question from!
So:
$\beta \in R_{\alpha} \iff \text{Ker}(\beta)$ has classes $\{1\},\{2,3\}$ and $\beta \in L_{\alpha} \iff \text{Im}(\beta) = \{1,3\}$
$$R_{\alpha} = \left\{ \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
1&2&2
\end{array}\right), \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
2&1&1
\end{array}\right), \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
1&3&3
\end{array}\right), \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
3&1&1
\end{array}\right), \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
2&3&3
\end{array}\right), \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}}
1&2&3\\
3&2&2
\end{array}\right) \right\}$$
</s>