\begin{figure}[tp]
\centering
\begin{tikzpicture}
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\node[font=\footnotesize] at (0.83,-0.28) {$+$};
\draw[black!45,very thin] (1.10,0.00) rectangle (1.65,-0.55);
\node[font=\footnotesize] at (1.38,-0.28) {$+$};
\draw[black!45,very thin] (1.65,0.00) rectangle (2.20,-0.55);
\node[font=\footnotesize] at (1.93,-0.28) {$+$};
\draw[black!45,very thin] (2.20,0.00) rectangle (2.75,-0.55);
\node[font=\footnotesize] at (2.48,-0.28) {$+$};
\draw[black!45,very thin] (0.55,-0.55) rectangle (1.10,-1.10);
\node[font=\footnotesize] at (0.83,-0.83) {$+$};
\draw[black!45,very thin] (1.10,-0.55) rectangle (1.65,-1.10);
\node[font=\footnotesize] at (1.38,-0.83) {$+$};
\draw[black!45,very thin] (1.65,-0.55) rectangle (2.20,-1.10);
\node[font=\footnotesize] at (1.93,-0.83) {$+$};
\draw[black!45,very thin] (2.20,-0.55) rectangle (2.75,-1.10);
\node[font=\footnotesize] at (2.48,-0.83) {$+$};
\draw[black!45,very thin] (1.10,-1.10) rectangle (1.65,-1.65);
\node[font=\footnotesize] at (1.38,-1.38) {$+$};
\draw[black!45,very thin] (1.65,-1.10) rectangle (2.20,-1.65);
\node[font=\footnotesize] at (1.93,-1.38) {$+$};
\draw[black!45,very thin] (2.20,-1.10) rectangle (2.75,-1.65);
\node[font=\footnotesize] at (2.48,-1.38) {$+$};
\draw[black!45,very thin] (1.65,-1.65) rectangle (2.20,-2.20);
\node[font=\footnotesize] at (1.93,-1.93) {$+$};
\draw[black!45,very thin] (2.20,-1.65) rectangle (2.75,-2.20);
\node[font=\footnotesize] at (2.48,-1.93) {$+$};
\draw[black!45,very thin] (2.20,-2.20) rectangle (2.75,-2.75);
\node[font=\footnotesize] at (2.48,-2.48) {$+$};
\node[left,font=\scriptsize] at (0.00,-0.28) {$1$};
\node[left,font=\scriptsize] at (0.55,-0.83) {$2$};
\node[left,font=\scriptsize] at (1.10,-1.38) {$3$};
\node[left,font=\scriptsize] at (1.65,-1.93) {$4$};
\node[left,font=\scriptsize] at (2.20,-2.48) {$5$};
\node[above,font=\scriptsize] at (0.28,0) {$2$};
\node[above,font=\scriptsize] at (0.83,0) {$3$};
\node[above,font=\scriptsize] at (1.38,0) {$4$};
\node[above,font=\scriptsize] at (1.93,0) {$5$};
\node[above,font=\scriptsize] at (2.48,0) {$6$};
\node[below,font=\small] at (1.38,-2.80) {start};
\draw[black!45,very thin] (3.60,0.00) rectangle (4.15,-0.55);
\node[font=\footnotesize] at (3.88,-0.28) {$+$};
\draw[black!45,very thin] (4.15,0.00) rectangle (4.70,-0.55);
\node[font=\footnotesize] at (4.43,-0.28) {$+$};
\draw[black!45,very thin] (4.70,0.00) rectangle (5.25,-0.55);
\node[font=\footnotesize] at (4.98,-0.28) {$+$};
\draw[black!45,very thin] (5.25,0.00) rectangle (5.80,-0.55);
\node[font=\footnotesize] at (5.53,-0.28) {$+$};
\draw[black!45,very thin] (5.80,0.00) rectangle (6.35,-0.55);
\node[font=\footnotesize] at (6.08,-0.28) {$+$};
\draw[black!45,very thin] (4.15,-0.55) rectangle (4.70,-1.10);
\node[font=\footnotesize] at (4.43,-0.83) {$+$};
\draw[black!45,very thin] (4.70,-0.55) rectangle (5.25,-1.10);
\node[font=\footnotesize] at (4.98,-0.83) {$+$};
\draw[black!45,very thin] (5.25,-0.55) rectangle (5.80,-1.10);
\node[font=\footnotesize] at (5.53,-0.83) {$+$};
\draw[black!45,very thin] (5.80,-0.55) rectangle (6.35,-1.10);
\node[font=\footnotesize] at (6.08,-0.83) {$+$};
\draw[black!45,very thin] (4.70,-1.10) rectangle (5.25,-1.65);
\node[font=\footnotesize] at (4.98,-1.38) {$+$};
\draw[black!45,very thin] (5.25,-1.10) rectangle (5.80,-1.65);
\node[font=\footnotesize] at (5.53,-1.38) {$+$};
\draw[black!45,very thin] (5.80,-1.10) rectangle (6.35,-1.65);
\node[font=\footnotesize] at (6.08,-1.38) {$+$};
\draw[black!45,very thin] (5.25,-1.65) rectangle (5.80,-2.20);
\node[font=\footnotesize] at (5.53,-1.93) {$+$};
\draw[black!45,very thin] (5.80,-1.65) rectangle (6.35,-2.20);
\draw[black!45,very thin] (5.80,-2.20) rectangle (6.35,-2.75);
\node[font=\footnotesize] at (6.08,-2.48) {$-$};
\node[left,font=\scriptsize] at (3.60,-0.28) {$1$};
\node[left,font=\scriptsize] at (4.15,-0.83) {$2$};
\node[left,font=\scriptsize] at (4.70,-1.38) {$3$};
\node[left,font=\scriptsize] at (5.25,-1.93) {$4$};
\node[left,font=\scriptsize] at (5.80,-2.48) {$5$};
\node[above,font=\scriptsize] at (3.88,0) {$2$};
\node[above,font=\scriptsize] at (4.42,0) {$3$};
\node[above,font=\scriptsize] at (4.97,0) {$4$};
\node[above,font=\scriptsize] at (5.53,0) {$5$};
\node[above,font=\scriptsize] at (6.08,0) {$6$};
\node[below,font=\small] at (4.97,-2.80) {after $i=4$};
\draw[black!45,very thin] (7.20,0.00) rectangle (7.75,-0.55);
\node[font=\footnotesize] at (7.48,-0.28) {$+$};
\draw[black!45,very thin] (7.75,0.00) rectangle (8.30,-0.55);
\node[font=\footnotesize] at (8.03,-0.28) {$+$};
\draw[black!45,very thin] (8.30,0.00) rectangle (8.85,-0.55);
\node[font=\footnotesize] at (8.58,-0.28) {$+$};
\draw[black!45,very thin] (8.85,0.00) rectangle (9.40,-0.55);
\node[font=\footnotesize] at (9.12,-0.28) {$+$};
\draw[black!45,very thin] (9.40,0.00) rectangle (9.95,-0.55);
\node[font=\footnotesize] at (9.68,-0.28) {$+$};
\draw[black!45,very thin] (7.75,-0.55) rectangle (8.30,-1.10);
\node[font=\footnotesize] at (8.03,-0.83) {$+$};
\draw[black!45,very thin] (8.30,-0.55) rectangle (8.85,-1.10);
\node[font=\footnotesize] at (8.58,-0.83) {$+$};
\draw[black!45,very thin] (8.85,-0.55) rectangle (9.40,-1.10);
\node[font=\footnotesize] at (9.12,-0.83) {$+$};
\draw[black!45,very thin] (9.40,-0.55) rectangle (9.95,-1.10);
\node[font=\footnotesize] at (9.68,-0.83) {$+$};
\draw[black!45,very thin] (8.30,-1.10) rectangle (8.85,-1.65);
\node[font=\footnotesize] at (8.58,-1.38) {$+$};
\draw[black!45,very thin] (8.85,-1.10) rectangle (9.40,-1.65);
\draw[black!45,very thin] (9.40,-1.10) rectangle (9.95,-1.65);
\draw[black!45,very thin] (8.85,-1.65) rectangle (9.40,-2.20);
\node[font=\footnotesize] at (9.12,-1.93) {$-$};
\draw[black!45,very thin] (9.40,-1.65) rectangle (9.95,-2.20);
\node[font=\footnotesize] at (9.68,-1.93) {$-$};
\draw[black!45,very thin] (9.40,-2.20) rectangle (9.95,-2.75);
\node[font=\footnotesize] at (9.68,-2.48) {$-$};
\node[left,font=\scriptsize] at (7.20,-0.28) {$1$};
\node[left,font=\scriptsize] at (7.75,-0.83) {$2$};
\node[left,font=\scriptsize] at (8.30,-1.38) {$3$};
\node[left,font=\scriptsize] at (8.85,-1.93) {$4$};
\node[left,font=\scriptsize] at (9.40,-2.48) {$5$};
\node[above,font=\scriptsize] at (7.48,0) {$2$};
\node[above,font=\scriptsize] at (8.03,0) {$3$};
\node[above,font=\scriptsize] at (8.57,0) {$4$};
\node[above,font=\scriptsize] at (9.12,0) {$5$};
\node[above,font=\scriptsize] at (9.68,0) {$6$};
\node[below,font=\small] at (8.57,-2.80) {after $i=3$};
\draw[black!45,very thin] (10.80,0.00) rectangle (11.35,-0.55);
\node[font=\footnotesize] at (11.08,-0.28) {$+$};
\draw[black!45,very thin] (11.35,0.00) rectangle (11.90,-0.55);
\node[font=\footnotesize] at (11.63,-0.28) {$+$};
\draw[black!45,very thin] (11.90,0.00) rectangle (12.45,-0.55);
\node[font=\footnotesize] at (12.18,-0.28) {$+$};
\draw[black!45,very thin] (12.45,0.00) rectangle (13.00,-0.55);
\node[font=\footnotesize] at (12.73,-0.28) {$+$};
\draw[black!45,very thin] (13.00,0.00) rectangle (13.55,-0.55);
\node[font=\footnotesize] at (13.28,-0.28) {$+$};
\draw[black!45,very thin] (11.35,-0.55) rectangle (11.90,-1.10);
\node[font=\footnotesize] at (11.63,-0.83) {$+$};
\draw[black!45,very thin] (11.90,-0.55) rectangle (12.45,-1.10);
\node[font=\footnotesize] at (12.18,-0.83) {$+$};
\draw[black!45,very thin] (12.45,-0.55) rectangle (13.00,-1.10);
\node[font=\footnotesize] at (12.73,-0.83) {$+$};
\draw[black!45,very thin] (13.00,-0.55) rectangle (13.55,-1.10);
\draw[black!45,very thin] (11.90,-1.10) rectangle (12.45,-1.65);
\node[font=\footnotesize] at (12.18,-1.38) {$+$};
\draw[black!45,very thin] (12.45,-1.10) rectangle (13.00,-1.65);
\draw[black!45,very thin] (13.00,-1.10) rectangle (13.55,-1.65);
\node[font=\footnotesize] at (13.28,-1.38) {$-$};
\draw[black!45,very thin] (12.45,-1.65) rectangle (13.00,-2.20);
\node[font=\footnotesize] at (12.73,-1.93) {$-$};
\draw[black!45,very thin] (13.00,-1.65) rectangle (13.55,-2.20);
\node[font=\footnotesize] at (13.28,-1.93) {$-$};
\draw[black!45,very thin] (13.00,-2.20) rectangle (13.55,-2.75);
\node[font=\footnotesize] at (13.28,-2.48) {$-$};
\node[left,font=\scriptsize] at (10.80,-0.28) {$1$};
\node[left,font=\scriptsize] at (11.35,-0.83) {$2$};
\node[left,font=\scriptsize] at (11.90,-1.38) {$3$};
\node[left,font=\scriptsize] at (12.45,-1.93) {$4$};
\node[left,font=\scriptsize] at (13.00,-2.48) {$5$};
\node[above,font=\scriptsize] at (11.08,0) {$2$};
\node[above,font=\scriptsize] at (11.62,0) {$3$};
\node[above,font=\scriptsize] at (12.18,0) {$4$};
\node[above,font=\scriptsize] at (12.73,0) {$5$};
\node[above,font=\scriptsize] at (13.28,0) {$6$};
\node[below,font=\small] at (12.18,-2.80) {after $i=2$};
\end{tikzpicture}
\caption{Step~4 of the proof of
Proposition~\ref{fl:prop:uniform-b-nonzero} for $m=6$. The cell in
row $k$ and column $l$, $k<l$, contains the sign $+$ if the root
subgroup of $e_k+e_l$ belongs to the group and the sign $-$ if that of
$-e_k-e_l$ does. At the start all the signs are $+$, as for the
Siegel--Whittaker coefficient. For $i=4$ and $i=3$ the root
$e_{i+1}+e_{i+2}$ is removed first; each exchange then replaces the
sign $+$ of a cell $(i,j)$ by the sign $-$ in the cell $(i+1,j)$
below it. At the end the sign is $+$ for $k+l<8$ and $-$ for $k+l>8$,
and the cells with $k+l=8$ are empty: these are the roots of positive
degree for $D_{\rm SW}$.}
\label{rw:fig:siegel-exchange}
\end{figure}