\begin{figure}[tp]
\centering
\begin{tikzpicture}
\draw[black!45,very thin,step=0.55cm,shift={(0.00,0)}] (0,0) grid (2.20,-1.65);
\draw (0.00,0) rectangle (2.20,-1.65);
\node[left,font=\scriptsize] at (0.00,-0.28) {$A_1$};
\node[left,font=\scriptsize] at (0.00,-0.83) {$A_2$};
\node[left,font=\scriptsize] at (0.00,-1.38) {$A_3$};
\node[above,font=\scriptsize] at (0.28,0) {$B_1$};
\node[above,font=\scriptsize] at (0.83,0) {$B_2$};
\node[above,font=\scriptsize] at (1.38,0) {$B_3$};
\node[above,font=\scriptsize] at (1.93,0) {$B_4$};
\node[font=\footnotesize] at (0.28,-0.28) {$+$};
\node[font=\footnotesize] at (0.83,-0.28) {$+$};
\node[font=\footnotesize] at (1.38,-0.28) {$+$};
\node[font=\footnotesize] at (1.93,-0.28) {$+$};
\node[font=\footnotesize] at (0.28,-0.83) {$+$};
\node[font=\footnotesize] at (0.83,-0.83) {$+$};
\node[font=\footnotesize] at (1.38,-0.83) {$+$};
\node[font=\footnotesize] at (1.93,-0.83) {$+$};
\node[below,font=\small] at (1.10,-1.70) {step 1};
\draw[black!45,very thin,step=0.55cm,shift={(4.20,0)}] (0,0) grid (2.20,-1.65);
\draw (4.20,0) rectangle (6.40,-1.65);
\node[left,font=\scriptsize] at (4.20,-0.28) {$A_1$};
\node[left,font=\scriptsize] at (4.20,-0.83) {$A_2$};
\node[left,font=\scriptsize] at (4.20,-1.38) {$A_3$};
\node[above,font=\scriptsize] at (4.48,0) {$B_1$};
\node[above,font=\scriptsize] at (5.03,0) {$B_2$};
\node[above,font=\scriptsize] at (5.58,0) {$B_3$};
\node[above,font=\scriptsize] at (6.12,0) {$B_4$};
\node[font=\footnotesize] at (5.58,-0.28) {$+$};
\node[font=\footnotesize] at (6.12,-0.28) {$+$};
\node[font=\footnotesize] at (4.48,-0.83) {$-$};
\node[font=\footnotesize] at (5.03,-0.83) {$-$};
\node[font=\footnotesize] at (6.12,-0.83) {$+$};
\node[font=\footnotesize] at (4.48,-1.38) {$-$};
\node[font=\footnotesize] at (5.03,-1.38) {$-$};
\node[font=\footnotesize] at (5.58,-1.38) {$-$};
\node[below,font=\small] at (5.30,-1.70) {step 2};
\fill[black!12] (8.95,0.00) rectangle (9.50,-0.55);
\fill[black!12] (9.50,-0.55) rectangle (10.05,-1.10);
\fill[black!12] (10.05,-1.10) rectangle (10.60,-1.65);
\draw[black!45,very thin,step=0.55cm,shift={(8.40,0)}] (0,0) grid (2.20,-1.65);
\draw (8.40,0) rectangle (10.60,-1.65);
\node[left,font=\scriptsize] at (8.40,-0.28) {$A_1$};
\node[left,font=\scriptsize] at (8.40,-0.83) {$A_2$};
\node[left,font=\scriptsize] at (8.40,-1.38) {$A_3$};
\node[above,font=\scriptsize] at (8.68,0) {$B_1$};
\node[above,font=\scriptsize] at (9.22,0) {$B_2$};
\node[above,font=\scriptsize] at (9.78,0) {$B_3$};
\node[above,font=\scriptsize] at (10.33,0) {$B_4$};
\node[font=\footnotesize] at (8.68,-0.28) {$-$};
\node[font=\footnotesize] at (9.78,-0.28) {$+$};
\node[font=\footnotesize] at (10.33,-0.28) {$+$};
\node[font=\footnotesize] at (8.68,-0.83) {$-$};
\node[font=\footnotesize] at (9.22,-0.83) {$-$};
\node[font=\footnotesize] at (10.33,-0.83) {$+$};
\node[font=\footnotesize] at (8.68,-1.38) {$-$};
\node[font=\footnotesize] at (9.22,-1.38) {$-$};
\node[font=\footnotesize] at (9.78,-1.38) {$-$};
\node[below,font=\small] at (9.50,-1.70) {step 3};
\begin{scope}[yshift=-3.1cm]
\fill[black!12] (0.27,-0.55) rectangle (0.82,-1.10);
\fill[black!12] (0.82,-1.10) rectangle (1.37,-1.65);
\fill[black!12] (1.37,-1.65) rectangle (1.92,-2.20);
\draw[black!45,very thin,step=0.55cm,shift={(0.27,0)}] (0,0) grid (1.65,-2.20);
\draw (0.27,0) rectangle (1.92,-2.20);
\node[left,font=\scriptsize] at (0.27,-0.28) {$A'_1$};
\node[left,font=\scriptsize] at (0.27,-0.83) {$A'_2$};
\node[left,font=\scriptsize] at (0.27,-1.38) {$A'_3$};
\node[left,font=\scriptsize] at (0.27,-1.93) {$A'_4$};
\node[above,font=\scriptsize] at (0.55,0) {$B'_1$};
\node[above,font=\scriptsize] at (1.10,0) {$B'_2$};
\node[above,font=\scriptsize] at (1.65,0) {$B'_3$};
\node[font=\footnotesize] at (0.55,-0.28) {$+$};
\node[font=\footnotesize] at (1.10,-0.28) {$+$};
\node[font=\footnotesize] at (1.65,-0.28) {$+$};
\node[font=\footnotesize] at (1.10,-0.83) {$+$};
\node[font=\footnotesize] at (1.65,-0.83) {$+$};
\node[font=\footnotesize] at (0.55,-1.38) {$-$};
\node[font=\footnotesize] at (1.65,-1.38) {$+$};
\node[font=\footnotesize] at (0.55,-1.93) {$-$};
\node[font=\footnotesize] at (1.10,-1.93) {$-$};
\node[below,font=\small] at (1.10,-2.25) {step 4, first stage};
\draw[black!45,very thin,step=0.55cm,shift={(4.47,0)}] (0,0) grid (1.65,-2.20);
\draw (4.47,0) rectangle (6.12,-2.20);
\node[left,font=\scriptsize] at (4.47,-0.28) {$A'_1$};
\node[left,font=\scriptsize] at (4.47,-0.83) {$A'_2$};
\node[left,font=\scriptsize] at (4.47,-1.38) {$A'_3$};
\node[left,font=\scriptsize] at (4.47,-1.93) {$A'_4$};
\node[above,font=\scriptsize] at (4.75,0) {$B'_1$};
\node[above,font=\scriptsize] at (5.29,0) {$B'_2$};
\node[above,font=\scriptsize] at (5.84,0) {$B'_3$};
\node[font=\footnotesize] at (4.75,-0.28) {$+$};
\node[font=\footnotesize] at (5.29,-0.28) {$+$};
\node[font=\footnotesize] at (5.84,-0.28) {$+$};
\node[font=\footnotesize] at (4.75,-0.83) {$+$};
\node[font=\footnotesize] at (5.29,-0.83) {$+$};
\node[font=\footnotesize] at (5.84,-0.83) {$+$};
\node[font=\footnotesize] at (4.75,-1.38) {$+$};
\node[font=\footnotesize] at (5.29,-1.38) {$+$};
\node[font=\footnotesize] at (5.84,-1.38) {$+$};
\node[below,font=\small] at (5.29,-2.25) {step 4, second stage};
\draw[black!45,very thin,step=0.55cm,shift={(8.67,0)}] (0,0) grid (1.65,-2.20);
\draw (8.67,0) rectangle (10.32,-2.20);
\node[left,font=\scriptsize] at (8.67,-0.28) {$A_1$};
\node[left,font=\scriptsize] at (8.67,-0.83) {$A_2$};
\node[left,font=\scriptsize] at (8.67,-1.38) {$A_3$};
\node[left,font=\scriptsize] at (8.67,-1.93) {$A_4$};
\node[above,font=\scriptsize] at (8.95,0) {$B_1$};
\node[above,font=\scriptsize] at (9.49,0) {$B_2$};
\node[above,font=\scriptsize] at (10.04,0) {$B_3$};
\node[font=\footnotesize] at (8.95,-0.28) {$+$};
\node[font=\footnotesize] at (9.49,-0.28) {$+$};
\node[font=\footnotesize] at (10.04,-0.28) {$+$};
\node[font=\footnotesize] at (8.95,-0.83) {$+$};
\node[font=\footnotesize] at (9.49,-0.83) {$+$};
\node[font=\footnotesize] at (10.04,-0.83) {$+$};
\node[font=\footnotesize] at (8.95,-1.38) {$+$};
\node[font=\footnotesize] at (9.49,-1.38) {$+$};
\node[font=\footnotesize] at (10.04,-1.38) {$+$};
\node[font=\footnotesize] at (8.95,-1.93) {$+$};
\node[font=\footnotesize] at (9.49,-1.93) {$+$};
\node[font=\footnotesize] at (10.04,-1.93) {$+$};
\node[below,font=\small] at (9.49,-2.25) {step 6};
\end{scope}
\end{tikzpicture}
\caption{The differences between the two chains in
Table~\ref{rw:tab:char2-route}, for $n=4$, with the conventions of
Figure~\ref{rw:fig:interleave}: the sign $+$ stands for a root (row)
minus (column) and the sign $-$ for its opposite. In step~3 and in the
first stage of step~4 the signs are those of the weights for the
grading of step~3. A shaded cell marks two members of the chains which
have the same weight for that grading: these are the blocks
$(A_i,B_{i+1})$ in which $g_{\rm pair}$ acts in step~4, and after the
change of basis the pairs $(A'_{i+1},B'_i)$. In step~6 the
chains have been renamed, and the terminal rows of the $B_j$ are
restricted to $L_{c_0}$.}
\label{rw:fig:char2-cross}
\end{figure}