\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[font=\tiny,anchor=north east,inner sep=0.8pt] at (0.55,0.00) {4};
\node[font=\tiny,anchor=north east,inner sep=0.8pt] at (1.10,0.00) {5};
\node[font=\tiny,anchor=north east,inner sep=0.8pt] at (0.55,-0.55) {1};
\node[font=\tiny,anchor=north east,inner sep=0.8pt] at (1.10,-0.55) {2};
\node[font=\tiny,anchor=north east,inner sep=0.8pt] at (1.65,-0.55) {3};
\node[below,font=\small] at (1.10,-1.70) {before};
\draw[black!45,very thin,step=0.55cm,shift={(3.60,0)}] (0,0) grid (2.20,-1.65);
\draw (3.60,0) rectangle (5.80,-1.65);
\node[left,font=\scriptsize] at (3.60,-0.28) {$A_1$};
\node[left,font=\scriptsize] at (3.60,-0.83) {$A_2$};
\node[left,font=\scriptsize] at (3.60,-1.38) {$A_3$};
\node[above,font=\scriptsize] at (3.88,0) {$B_1$};
\node[above,font=\scriptsize] at (4.42,0) {$B_2$};
\node[above,font=\scriptsize] at (4.97,0) {$B_3$};
\node[above,font=\scriptsize] at (5.53,0) {$B_4$};
\node[font=\footnotesize] at (4.97,-0.28) {$+$};
\node[font=\footnotesize] at (5.53,-0.28) {$+$};
\node[font=\footnotesize] at (3.88,-0.83) {$-$};
\node[font=\footnotesize] at (4.42,-0.83) {$-$};
\node[font=\footnotesize] at (5.53,-0.83) {$+$};
\node[font=\footnotesize] at (3.88,-1.38) {$-$};
\node[font=\footnotesize] at (4.42,-1.38) {$-$};
\node[font=\footnotesize] at (4.97,-1.38) {$-$};
\node[below,font=\small] at (4.70,-1.70) {after};
\draw[black!45,very thin,step=0.55cm,shift={(8.00,0)}] (0,0) grid (1.65,-2.20);
\draw (8.00,0) rectangle (9.65,-2.20);
\node[left,font=\scriptsize] at (8.00,-0.28) {$A_1$};
\node[left,font=\scriptsize] at (8.00,-0.83) {$A_2$};
\node[left,font=\scriptsize] at (8.00,-1.38) {$A_3$};
\node[left,font=\scriptsize] at (8.00,-1.93) {$A_4$};
\node[above,font=\scriptsize] at (8.28,0) {$B_1$};
\node[above,font=\scriptsize] at (8.82,0) {$B_2$};
\node[above,font=\scriptsize] at (9.38,0) {$B_3$};
\node[font=\footnotesize] at (8.28,-0.28) {$+$};
\node[font=\footnotesize] at (8.82,-0.28) {$+$};
\node[font=\footnotesize] at (9.38,-0.28) {$+$};
\node[font=\footnotesize] at (8.28,-0.83) {$+$};
\node[font=\footnotesize] at (8.82,-0.83) {$+$};
\node[font=\footnotesize] at (9.38,-0.83) {$+$};
\node[font=\footnotesize] at (8.28,-1.38) {$+$};
\node[font=\footnotesize] at (8.82,-1.38) {$+$};
\node[font=\footnotesize] at (9.38,-1.38) {$+$};
\node[font=\tiny,anchor=north east,inner sep=0.8pt] at (8.55,-0.55) {3};
\node[font=\tiny,anchor=north east,inner sep=0.8pt] at (8.55,-1.10) {1};
\node[font=\tiny,anchor=north east,inner sep=0.8pt] at (9.10,-1.10) {2};
\node[below,font=\small] at (8.82,-2.25) {before};
\draw[black!45,very thin,step=0.55cm,shift={(11.00,0)}] (0,0) grid (1.65,-2.20);
\draw (11.00,0) rectangle (12.65,-2.20);
\node[left,font=\scriptsize] at (11.00,-0.28) {$A_1$};
\node[left,font=\scriptsize] at (11.00,-0.83) {$A_2$};
\node[left,font=\scriptsize] at (11.00,-1.38) {$A_3$};
\node[left,font=\scriptsize] at (11.00,-1.93) {$A_4$};
\node[above,font=\scriptsize] at (11.28,0) {$B_1$};
\node[above,font=\scriptsize] at (11.82,0) {$B_2$};
\node[above,font=\scriptsize] at (12.38,0) {$B_3$};
\node[font=\footnotesize] at (11.28,-0.28) {$+$};
\node[font=\footnotesize] at (11.82,-0.28) {$+$};
\node[font=\footnotesize] at (12.38,-0.28) {$+$};
\node[font=\footnotesize] at (11.82,-0.83) {$+$};
\node[font=\footnotesize] at (12.38,-0.83) {$+$};
\node[font=\footnotesize] at (11.28,-1.38) {$-$};
\node[font=\footnotesize] at (12.38,-1.38) {$+$};
\node[font=\footnotesize] at (11.28,-1.93) {$-$};
\node[font=\footnotesize] at (11.82,-1.93) {$-$};
\node[below,font=\small] at (11.82,-2.25) {after};
\end{tikzpicture}
\caption{The cross roots of Lemma~\ref{fl:lem:uniform-interleave}.
A cell in the row of $A_i$ and the column of $B_j$ contains the sign
$+$ if the root subgroup of $A_i-B_j$ belongs to the group, and the
sign $-$ if that of $B_j-A_i$ does. Left: $r=3$, $b=4$, $d=-1$, the
case of step~2 of the proof of
Proposition~\ref{fl:prop:uniform-b-nonzero} for $n=4$. Right: $r=4$,
$b=3$, $d=1$, the case of step~2 of the proof of
Lemma~\ref{fl:lem:even-period-to-generic} for $n=4$. The small numbers
give the order of the exchanges; each replaces the sign $+$ of a cell
by the sign $-$ in the cell below it. At the end the signs are those
of the degrees of the roots $A_i-B_j$ for a grading which decreases by
$2$ along each chain and in which $A_i$ and $B_{i-d}$ have the same
degree, except that for $d=-1$ the root $B_1-A_1$ is absent.}
\label{rw:fig:interleave}
\end{figure}