\draw[dots] (11.2,4)--(12.8,4);
\draw[dots] (11.2,0)--(12.8,0);
\draw (13,4)node[above]{$i-1$}--(14,2)--(13,0);
\draw (14,4)node[above]{$i$}--(9,2)--(4,0);
\node[greendot] at (11,0){};
\end{braid}\\
& \overset{\substack{(\ref{dia:psipsi})\\(\ref{dia:ii3})}}= &
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(11.5,-0.5)node{$\underbrace{\hspace*{20mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$i-1$}--(3,0);
\draw (4,4)node[above]{$i$}--(7,2)--(8,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$i-1$}--(6,2)--(7,0);
\draw (8,4)node[above]{$i$}--(7,2)--(4,0);
\draw (9,4)node[above]{$i+1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-2$}--(11,0);
\draw[dots] (11.2,4)--(12.8,4);
\draw[dots] (11.2,0)--(12.8,0);
\draw (13,4)node[above]{$i-1$}--(13,0);
\draw (14,4)node[above]{$i$}--(14,0);
\node[greendot] at (11,0){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(11.5,-0.5)node{$\underbrace{\hspace*{20mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$i-1$}--(3,0);
\draw (4,4)node[above]{$i$}--(9,2)--(14,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$i-1$}--(6,2)--(7,0);
\draw (8,4)node[above]{$i$}--(7,2)--(8,0);
\draw (9,4)node[above]{$i+1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)[densely dotted] node[above]{$e-2$}--(12,2)--(11,0);
\draw[dots] (11.2,4)--(12.8,4);
\draw[dots] (11.2,0)--(12.8,0);
\draw (13,4)node[above]{$i-1$}--(14,2)--(13,0);
\draw (14,4)node[above]{$i$}--(9,2)--(4,0);
\node[greendot] at (11,0){};
\draw[->] (11,0) -- (12,2);
\end{braid}\\
& \overset{(\ref{dia:y-psi com})}= &
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(11.5,-0.5)node{$\underbrace{\hspace*{20mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$i-1$}--(3,0);
\draw (4,4)node[above]{$i$}--(7,2)--(8,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$i-1$}--(6,2)--(7,0);
\draw (8,4)node[above]{$i$}--(7,2)--(4,0);
\draw (9,4)node[above]{$i+1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-2$}--(11,0);
\draw[dots] (11.2,4)--(12.8,4);
\draw[dots] (11.2,0)--(12.8,0);
\draw (13,4)node[above]{$i-1$}--(13,0);
\draw (14,4)node[above]{$i$}--(14,0);
\node[greendot] at (11,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(11.5,-0.5)node{$\underbrace{\hspace*{20mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$i-1$}--(3,0);
\draw (4,4)node[above]{$i$}--(4,3)--(8,2)--(9,1)--(14,0);
\draw (5,4)node[above]{$e-1$}--(5,3)--(4,2)--(5,1)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$i-1$}--(7,3)--(6,2)--(7,1)--(7,0);
\draw (8,4)node[above]{$i$}--(8,3)--(7,2)--(8,1)--(8,0);
\draw (9,4)node[above]{$i+1$}--(10,3)--(10,1)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-2$}--(12,3)--(12,1)--(11,0);
\draw[dots] (11.2,4)--(12.8,4);
\draw[dots] (11.2,0)--(12.8,0);
\draw (13,4)node[above]{$i-1$}--(14,3)--(14,1)--(13,0);
\draw (14,4)node[above]{$i$}--(9,3)--(8,2)--(4,1)--(4,0);
\node[greendot] at (12,2){};
\draw[loosely dashed,red] (-0.5,3) -- (9.5,3)--(9.5,1)--(-0.5,1)--(-0.5,3);
\end{braid}.
\end{eqnarray*}

Then by induction and <ref>, we have

\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(11.5,-0.5)node{$\underbrace{\hspace*{20mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$i-1$}--(3,0);
\draw (4,4)node[above]{$i$}--(4,3)--(8,2)--(9,1)--(14,0);
\draw (5,4)node[above]{$e-1$}--(5,3)--(4,2)--(5,1)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$i-1$}--(7,3)--(6,2)--(7,1)--(7,0);
\draw (8,4)node[above]{$i$}--(8,3)--(7,2)--(8,1)--(8,0);
\draw (9,4)node[above]{$i+1$}--(10,3)--(10,1)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-2$}--(12,3)--(12,1)--(11,0);
\draw[dots] (11.2,4)--(12.8,4);
\draw[dots] (11.2,0)--(12.8,0);
\draw (13,4)node[above]{$i-1$}--(14,3)--(14,1)--(13,0);
\draw (14,4)node[above]{$i$}--(9,3)--(8,2)--(4,1)--(4,0);
\node[greendot] at (12,2){};
\draw[loosely dashed,red] (-0.5,3) -- (9.5,3)--(9.5,1)--(-0.5,1)--(-0.5,3);
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(11.5,-0.5)node{$\underbrace{\hspace*{20mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$i-1$}--(3,0);
\draw (4,4)node[above]{$i$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$i-1$}--(7,0);
\draw (8,4)node[above]{$i$}--(8,0);
\draw (9,4)node[above]{$i+1$}--(10,3)--(10,1)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-2$}--(12,3)--(12,1)--(11,0);
\draw[dots] (11.2,4)--(12.8,4);
\draw[dots] (11.2,0)--(12.8,0);
\draw (13,4)node[above]{$i-1$}--(14,3)--(14,1)--(13,0);
\draw (14,4)node[above]{$i$}--(9,3)--(9,1)--(14,0);
\node[greendot] at (12,2){};
\end{braid} = 0,

which implies that

\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(11.5,-0.5)node{$\underbrace{\hspace*{20mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$i-1$}--(3,0);
\draw (4,4)node[above]{$i$}--(9,2)--(14,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$i-1$}--(6,2)--(7,0);
\draw (8,4)node[above]{$i$}--(7,2)--(8,0);
\draw (9,4)node[above]{$i+1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-2$}--(12,2)--(11,0);
\draw[dots] (11.2,4)--(12.8,4);
\draw[dots] (11.2,0)--(12.8,0);
\draw (13,4)node[above]{$i-1$}--(14,2)--(13,0);
\draw (14,4)node[above]{$i$}--(9,2)--(4,0);
\node[greendot] at (12,2){};
\end{braid}
\in \Blam[\gamma].

Hence by induction and <ref>

\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(11.5,-0.5)node{$\underbrace{\hspace*{20mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$i-1$}--(3,0);
\draw (4,4)node[above]{$i$}--(13,2)--(14,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$i-1$}--(6,2)--(7,0);
\draw (8,4)node[above]{$i$}--(7,2)--(8,0);
\draw (9,4)node[above]{$i+1$}--(8,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-2$}--(10,2)--(11,0);
\draw[dots] (11.2,4)--(12.8,4);
\draw[dots] (11.2,0)--(12.8,0);
\draw (13,4)node[above]{$i-1$}--(12,2)--(13,0);
\draw (14,4)node[above]{$i$}--(13,2)--(4,0);
\node[greendot] at (10,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(11.5,-0.5)node{$\underbrace{\hspace*{20mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$i-1$}--(3,0);
\draw (4,4)node[above]{$i$}--(7,2)--(8,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$i-1$}--(6,2)--(7,0);
\draw (8,4)node[above]{$i$}--(7,2)--(4,0);
\draw (9,4)node[above]{$i+1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-2$}--(11,0);
\draw[dots] (11.2,4)--(12.8,4);
\draw[dots] (11.2,0)--(12.8,0);
\draw (13,4)node[above]{$i-1$}--(13,0);
\draw (14,4)node[above]{$i$}--(14,0);
\node[greendot] at (11,2){};
\end{braid}
=_\gamma e_\gamma y_\gamma.

Case <ref>b: $i = e-2$.

\begin{eqnarray}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(11,2)--(12,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(8,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(10,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(11,2)--(4,0);
\node[greendot] at (7,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(8,3)--(11,2)--(12,1)--(12,0);
\draw (5,4)node[above]{$e-1$}--(4,3)--(4,1)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,3)--(6,1)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,3)--(7,1)--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,3)--(8,2)--(9,1)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,3)--(10,2)--(11,1)--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,3)--(11,2)--(8,1)--(4,0);
\node[greendot] at (7,2){};
\draw[loosely dashed,red] (7.5,3) -- (12.5,3)--(12.5,1)--(7.5,1)--(7.5,3);
\end{braid} \notag \hspace*{11mm}\text{by \autoref{I-problem: short}}\\
& = &
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(8,2)--(4,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)[densely dotted] node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (7,2){};
\node[greendot] at (12,2){};
\draw[->] (7,2) -- (7.5,3);
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(8,2)--(4,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)[densely dotted] node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (7,2){};
\node[greendot] at (11,2){};
\draw[->] (7,2) -- (7.5,3);
\end{braid}\notag \\
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(8,2)--(4,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)[densely dotted] node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (7,2){};
\node[greendot] at (9,2){};
\draw[->] (7,2) -- (7.5,3);
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)[densely dotted] node[above]{$e-2$}--(8,2)--(4,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)[densely dotted] node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (7,2){};
\node[greendot] at (8,2){};
\draw[->] (7,2) -- (7.5,3);
\draw[->] (8,2) -- (6.5,2.75);
\end{braid}\notag \\
&& +
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(4,0);
\node[greendot] at (7,2){};
\end{braid}\notag \\
& \overset{\substack{(\ref{dia:y-psi com})\\(\ref{dia:psipsi})}}= &
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(4,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (12,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(4,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (11,2){};
\end{braid} \label{I-problem: induction2: example2: equation1}\\
&& +
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(4,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (9,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(4,0);
\node[greendot] at (7,2){};
\end{braid}\notag.
\end{eqnarray}

By induction and <ref>, the second and the third terms of (<ref>) are both in $\Blam[\gamma]$. Now for the last term.

\begin{eqnarray}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)[densely dotted] node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(4,0);
\node[greendot] at (7,2){};
\draw[->] (7,2) -- (7.5,3);
\end{braid}\notag\\
& \overset{(\ref{dia:y-psi com})}= &
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(4,0);
\node[greendot] at (8,4){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(4,0);
\end{braid} \label{I-problem: induction2: example2: equation2}.
\end{eqnarray}

Substitute (<ref>) to (<ref>), let $n = \gamma_1 + \gamma_2$, we have

\begin{eqnarray}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(11,2)--(12,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(8,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(10,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(11,2)--(4,0);
\node[greendot] at (7,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(4,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (12,2){};
\end{braid}\notag \\
&& +
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(4,0);
\node[greendot] at (8,4){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(4,0);
\end{braid} \notag\\
& = &
(y_n +
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(8,0);
\end{braid}){\cdot}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(4,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\end{braid} \label{I-problem: induction2: example2: equation3}\\
&& +
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(4,0);
\node[greendot] at (8,4){};
\end{braid} \notag,
\end{eqnarray}

where by <ref>

\begin{eqnarray}
&&y_n +
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(8,0);
\end{braid}\notag\\
& = &
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (12,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(11,2)--(12,0);
\draw (9,4)node[above]{$e-1$}--(8,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(10,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(11,2)--(8,0);
\end{braid}\notag
\end{eqnarray}

\begin{eqnarray}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (12,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (11,2){};
\end{braid}\notag \\
&& +
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (9,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (8,2){};
\end{braid}\notag\\
& = &
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(11,2)--(12,0);
\draw (9,4)node[above]{$e-1$}--(8,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(10,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(11,2)--(8,0);
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (11,2){};
\end{braid} \label{I-problem: induction2: example2: equation4} \\
&& +
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (9,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (8,2){};
\end{braid}\notag.
\end{eqnarray}

Then by $\lambda\in\mathscr S_n^\Lambda$ and <ref>, for the first term of (<ref>),

\begin{eqnarray}
(y_n +
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(8,0);
\end{braid}){\cdot}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(4,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\end{braid} \notag  \hspace*{4mm}\text{by induction}\\
& =_\gamma &
(y_n +
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(9,2)--(12,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(9,2)--(8,0);
\end{braid}){\cdot}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (8,2){};
\end{braid} \notag \hspace*{4mm} \text{by (\ref{I-problem: induction2: example2: equation4})}\\
& = &
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(11,3)--(12,1)--(12,0);
\draw (9,4)node[above]{$e-1$}--(8,3)--(8,1)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(10,3)--(10,1)--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,3)--(11,1)--(8,0);
\node[greendot] at (8,0){};
\draw[loosely dashed,red] (-0.5,3) -- (8.5,3)--(8.5,1)--(-0.5,1)--(-0.5,3);
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (8,2){};
\node[greendot] at (11,2){};
\end{braid} \notag \text{ by \autoref{send}} \\
&& +
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (8,2){};
\node[greendot] at (9,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (8,3){};
\node[greendot] at (8,1){};
\end{braid} \in \Blam[\gamma] \label{I-problem: induction2: example2: equation5}.
\end{eqnarray}

For the second term of (<ref>), by induction, <ref>, $\lambda\in \mathscr S_n^\Lambda$ and <ref>,

\begin{eqnarray}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)[densely dotted] node[above]{$e-2$}--(8,3)--(9,2)--(12,1)--(12,0);
\draw (5,4)node[above]{$e-1$}--(4,3)--(4,1)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,3)--(6,1)--(7,0);
\draw (8,4)[densely dotted] node[above]{$e-2$}--(7,3)--(7,1)--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,3)--(10,2)--(9,1)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,3)--(12,2)--(11,1)--(11,0);
\draw (12,4)[densely dotted] node[above]{$e-2$}--(12,3)--(9,2)--(8,1)--(4,0);
\node[greendot] at (8,4){};
\end{braid}
\overset{(\ref{dia:psipsi})}=
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,3)--(7,2)--(8,1)--(12,0);
\draw (5,4)node[above]{$e-1$}--(5,3)--(4,2)--(5,1)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,3)--(6,2)--(7,1)--(7,0);
\draw (8,4)node[above]{$e-2$}--(9,3)--(9,1)--(8,0);
\draw (9,4)node[above]{$e-1$}--(10,3)--(10,1)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,3)--(12,1)--(11,0);
\draw (12,4)node[above]{$e-2$}--(8,3)--(7,2)--(4,1)--(4,0);
\node[greendot] at (8,4){};
\draw[loosely dashed,red] (-0.5,3) -- (8.5,3)--(8.5,1)--(-0.5,1)--(-0.5,3);
\end{braid} \hspace*{4mm} \text{by induction} \notag \\
& =_\gamma &
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)[densely dotted] node[above]{$e-2$}--(9,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(10,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(12,2)--(11,0);
\draw (12,4)[densely dotted] node[above]{$e-2$}--(8,2)--(12,0);
\node[greendot] at (8,4){};
\node[greendot] at (8,2){};
\draw[->] (8,2) -- (9.5,2.75);
\end{braid}
\overset{\substack{(\ref{dia:y-psi com})\\ (\ref{dia:psipsi})}}=
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,3)--(9,2)--(12,1)--(12,0);
\draw (9,4)node[above]{$e-1$}--(9,3)--(10,2)--(9,1)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,3)--(12,2)--(11,1)--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,3)--(9,2)--(8,1)--(8,0);
\node[greendot] at (8,4){};
\draw[loosely dashed,red] (7.5,3) -- (12.5,3)--(12.5,1)--(7.5,1)--(7.5,3);
\end{braid} \hspace*{4mm} \text{by \autoref{I-problem: short}} \notag\\
& = &
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (8,2){};
\node[greendot] at (12,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (8,2){};
\node[greendot] at (11,2){};
\end{braid} \notag \\
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (8,2){};
\node[greendot] at (9,2){};
\end{braid}
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (8,1){};
\node[greendot] at (8,3){};
\end{braid} \notag \\
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(11,3)--(12,1)--(12,0);
\draw (9,4)node[above]{$e-1$}--(8,3)--(8,1)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(10,3)--(10,1)--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,3)--(11,1)--(8,0);
\node[greendot] at (8,4){};
\draw[loosely dashed,red] (-0.5,3) -- (8.5,3)--(8.5,1)--(-0.5,1)--(-0.5,3);
\end{braid} \notag \hspace*{4mm}\text{by \autoref{send}}\\
& =_\gamma &
\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(7,0);
\draw (8,4)node[above]{$e-2$}--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(11,0);
\draw (12,4)node[above]{$e-2$}--(12,0);
\node[greendot] at (8,2){};
\node[greendot] at (12,2){};
\end{braid}
= e_\gamma y_\gamma \label{I-problem: induction2: example2: equation6}.
\end{eqnarray}

Substitute the results of (<ref>) and (<ref>) to (<ref>), we have

\begin{braid}
\draw(2,-0.5)node{$\underbrace{\hspace*{16mm}}_{\gamma_1}$};
\draw(6.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{\gamma_2-e}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{12mm}}_{e}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-3$}--(3,0);
\draw (4,4)node[above]{$e-2$}--(11,2)--(12,0);
\draw (5,4)node[above]{$e-1$}--(4,2)--(5,0);
\draw[dots] (5.2,4)--(6.8,4);
\draw[dots] (5.2,0)--(6.8,0);
\draw (7,4)node[above]{$e-3$}--(6,2)--(7,0);
\draw (8,4)node[above]{$e-2$}--(7,2)--(8,0);
\draw (9,4)node[above]{$e-1$}--(8,2)--(9,0);
\draw[dots] (9.2,4)--(10.8,4);
\draw[dots] (9.2,0)--(10.8,0);
\draw (11,4)node[above]{$e-3$}--(10,2)--(11,0);
\draw (12,4)node[above]{$e-2$}--(11,2)--(4,0);
\node[greendot] at (7,2){};
\end{braid} =_\gamma e_\gamma y_\gamma.

Case <ref>c: $i = e-1$.
The method to prove this is the same as for Case <ref>a so it is left as an exercise. Then by induction, this completes the proof.

Finally, we can use (<ref>) to prove our main result of this subsection.

Suppose $m$ is a positive integer, $\lambda = (m,m,1) \in \mathscr S_n^\Lambda$ and $\gamma = (\gamma_1,\gamma_2) = (m,m+1)$. Recall $\lambda_- = (m,m)$. Write $\bi_{\lambda_-} = (i_1, i_2, \ldots, i_{n-1})$. If $k = i_{n-1}+1 \in I$, we have

e_\gamma y_\gamma = e(\bi_{\lambda_-} \vee k) y_{\lambda_-} y_n^{b_k^{\lambda_-}} \in \Blam.

Without loss of generally we assume $\Lambda = \Lambda_0$. When $m = 1$, then $\gamma = (1,2)$ and

e_\gamma y_\gamma =
\begin{braid}
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$e-1$}--(1,0);
\draw (2,4)node[above]{$0$}--(2,0);
\end{braid}
\overset{\substack{(\ref{dia:braid})\\(\ref{dia:ii3})}}=
\draw (0,4)node[above]{$0$}--(1,2)--(0,0);
\draw (1,4)node[above]{$e-1$}--(2,2)--(1,0);
\draw (2,4)node[above]{$0$}--(0,2)--(2,0);
\node[greendot] at (0,2){};
\end{braid}
\begin{braid}
\draw (0,4)node[above]{$0$}--(2,0);
\draw (1,4)node[above]{$e-1$}--(0,2)--(1,0);
\draw (2,4)node[above]{$0$}--(0,0);
\end{braid} \in N_3^{\Lambda_0} \subseteq \Blam.

When $m > 1$, write $\bi_{(\gamma_1)} = (i_1, i_2, \ldots, i_m)$. Set $\sigma = (\gamma_1-1,\gamma_2) = (m-1,m)$. Then by <ref>, we have

e_\gamma y_\gamma =
\begin{braid}
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$i_m-1$}--(3,0);
\draw (4,4)node[above]{$i_m$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(5,0);
\draw (6,4)node[above]{$0$}--(6,0);
\draw[dots] (6.2,4)--(7.8,4);
\draw[dots] (6.2,0)--(7.8,0);
\draw (8,4)node[above]{$i_m-1$}--(8,0);
\draw (9,4)node[above]{$i_m$}--(9,0);
\end{braid}
\begin{cases}
\begin{braid}
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$i_m-1$}--(3,0);
\draw (4,4)node[above]{$i_m$}--(8,3)--(9,1)--(9,0);
\draw (5,4)node[above]{$e-1$}--(4,3)--(4,1)--(5,0);
\draw (6,4)node[above]{$0$}--(5,3)--(5,1)--(6,0);
\draw[dots] (6.2,4)--(7.8,4);
\draw[dots] (6.2,0)--(7.8,0);
\draw (8,4)node[above]{$i_m-1$}--(7,3)--(7,1)--(8,0);
\draw (9,4)node[above]{$i_m$}--(9,3)--(8,1)--(4,0);
\draw[loosely dashed,red] (-0.5,3) -- (7.5,3)--(7.5,1)--(-0.5,1)--(-0.5,3);
\end{braid}, & \text{if $i_m \neq e-1$},\\
\begin{braid}
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw[dots] (1.2,4)--(2.8,4);
\draw[dots] (1.2,0)--(2.8,0);
\draw (3,4)node[above]{$e-2$}--(3,0);
\draw (4,4)node[above]{$e-1$}--(4,0);
\draw (5,4)node[above]{$e-1$}--(8,3)--(9,1)--(9,0);
\draw (6,4)node[above]{$0$}--(5,3)--(5,1)--(6,0);
\draw[dots] (6.2,4)--(7.8,4);
\draw[dots] (6.2,0)--(7.8,0);
\draw (8,4)node[above]{$e-2$}--(7,3)--(7,1)--(8,0);
\draw (9,4)node[above]{$e-1$}--(9,3)--(8,1)--(5,0);
\draw[loosely dashed,red] (-0.5,3) -- (7.5,3)--(7.5,1)--(-0.5,1)--(-0.5,3);
\end{braid}, & \text{if $i_m = e-1$}.
\end{cases}

In both cases, the parts bounded by square are both $e_\sigma y_\sigma$. As $|\sigma| = n-2$ and $\lambda \in \mathscr S_n^\Lambda$, by induction, $e_\sigma y_\sigma \in\Blam[\sigma]$. By the definition of $\sigma$ and <ref>, it forces that $e_\sigma y_\sigma \in \Blam[\lambda|_{n-2}]$. Then by <ref>, we have $e_\gamma y_\gamma \in  \Blam $.

§.§ Final part of $y$-problem

In the last subsection we have proved that if $\lambda = (m,m,1) \in \mathscr S_n^\Lambda$, then

e(\bi_{\lambda_-} \vee k) y_{\lambda_-} y_n^{b_k^{\lambda_-}} \in \Blam

with $k = i_n + 1$. In this subsection we will gradually remove the restrictions on $\lambda$ and $k$. First we are going to introduce a useful homomorphism and use it to prove a few more properties of $\mathscr R_n$ and $\R$. After that we are going to show that if $\lambda \in \mathscr P_I^\Lambda$, then we can extend $\lambda$ to a $\l+1$ multipartition by adding an $\emptyset$ at the end and thus the new multipartition is in $\mathscr P_I^{\Lambda+\Lambda_i}$ for any $i\in I$. Analogous results are also true for $\mathscr P_y^\Lambda$ and $\mathscr P_\psi^\Lambda$. These will allow us to extend the result to an arbitrary multipartition $\lambda$.

For any $\bj \in I^m$, we can define a linear map $\hat\theta_\bj \map{\mathscr{R}_n}{\mathscr{R}_{n+m}^\Lambda}$ sending $e(\bi)$ to $e(\bj\vee \bi)$, $y_r$ to $y_{r+m}$ and $\psi_r$ to $\psi_{r+m}$. This map $\hat\theta_{\bj}$ works as embedding from $\mathscr R_n$ to $\mathscr R_{n+m}$ followed by the projection onto $\mathscr R_{n+m}^\Lambda$.

For $\bj\in I^m$, the map $\hat\theta_\bj$ is a homomorphism.

The map is defined to be linear. Hence we only have to check the relations. Since the relations of $\mathscr{R}_n$ and $\mathscr{R}_{n+m}^\Lambda$ from <ref> are independent of the value of $r$, we see that $\hat\theta_\bj$ is a homomorphism.

It will be necessary to cut a multicomposition $\lambda$ into one multicomposition $\mu$ and a composition $\gamma$ for our later work. Note that in our work we will mainly set $\mu$ to be a multipartition and $\gamma$ to be a partition, but generally we don't have such restriction.

Fix $e = 4$, $\Lambda = 2\Lambda_0 + \Lambda_1$, $\kappa_\Lambda = (0,1,0)$. Suppose $\lambda = (4,2|2^2,1|3^2,2)$. So

[\lambda] = \Bigg(\hspace*{1mm}\ydiag(4,2)\hspace*{1mm}\Bigg|\hspace*{1mm}%
\ydiag(2,2,1)\hspace*{1mm}\Bigg|\hspace*{1mm}\ydiag(3,3,2)\hspace*{1mm}\Bigg).

We want to divide the last partition of $\lambda$ after the first row. This is called the cut row of $\lambda$. This gives us a multipartition $\mu$ with Young diagram

[\mu] = \Bigg(\hspace*{1mm}\ydiag(4,2)\hspace*{1mm}\Bigg|\hspace*{1mm}%
\ydiag(2,2,1)\hspace*{1mm}\Bigg|\hspace*{1mm}\ydiag(3)\hspace*{1mm}\Bigg),

and a partition $\gamma$ with diagram

[\gamma] = \ydiag(3,2).

We call $\mu$ and $\gamma$ the cut part and the remaining part, respectively.

Moreover we want to preserve the following data. The value $|\mu|$ is called the cut of $\lambda$ which is 14 in this case.  The residue of the top left node of $\gamma$ as a subdiagram of $\lambda^{(3)}$ is called the cut residue, which in this case is 3.

Now we give a formal definition.

Suppose $\lambda= (\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(\l)}) \in \mathscr C_n^\Lambda$ with $\lambda^{(\l)} = (\lambda^{(\l)}_1,\ldots,\lambda^{(\l)}_{k_\l})$ and $a$ is an integer such that $0\leq a < k_\l$. We call $m$ a cut of $\lambda$ and $a$ the cut row associated to $m$ where $m = \sum_{i = 1}^{\l-1}|\lambda^{(i)}| + \sum_{j = 1}^a \lambda_j^{(\l)}$. Define $\Lambda' = \Lambda_s$, where $s = \kappa_\l + 1 - (a+1) = \kappa_\l - a$, the residue of the node at position $(a+1,1,\l)$. We call $s$ to be cut residue associated to $m$ and $\Lambda'$ to be cut weight associated to $m$. We then define $\mu = \lambda|_m \in \mathscr C^\Lambda_m$ and $\gamma = (\lambda_{a+1}^{(\l)},\lambda_{a+2}^{(\l)},\ldots,\lambda_{k_\l}^{(\l)})\in \mathscr C^{\Lambda'}_{n-m}$ and call $\mu$ and $\gamma$ to be cut part and remaining part of $\lambda$ associated to $m$, respectively.

Note we can either remove a portion of the last tableau, or cut out the whole partition.

We will start to work with $\hat\theta_{\bi}$, which involving elements in both $\Rn$ and $\R$. Recall that $\hat e(\bi)$, $\hat y_r$, $\hat \psi_s$ and $\hat\psi_{\s\t}$ are elements from $\mathscr R_n$ and $e(\bi)$, $y_r$, $\psi_s$ and $\psi_{\s\t}$ are elements from $\R$.

Suppose $\lambda \in \mathscr S_n^\Lambda$. Let $m$ be a cut of $\lambda$ with $m < n-1$, $\nu = \lambda|_m$ and $\Lambda'$ be the cut weight associated to $m$. Consider $N_{n-m}^{\Lambda'} \subseteq \mathscr{R}_{n-m}$. If $\hat\theta_{\bi_\nu}\map{\mathscr{R}_{n-m}}{\mathscr{R}_n^\Lambda}$, then $\hat\theta_{\bi_\nu}(N_{n-m}^{\Lambda'}) y_\nu \subseteq \Blam$.

Consider $r \in N_{n-m}^{\Lambda'}$. Then by (<ref>),

r = \sum_{\bj = (j_1,j_2,\ldots,j_{n-m}) \in I^{n-m}} c_\bj R'_\bj\hat e(\bj) \hat y_1^{(\Lambda',\alpha_{j_1})} R_\bj,

where $R_\bj$ and $R'_\bj$ are some elements in $\mathscr{R}_{n-m}$ and $c_\bj \in \Z$. Therefore

\begin{eqnarray*}
\hat\theta_{\bi_\nu}(r)y_\nu & = & \sum_{\bj = (j_1,j_2,\ldots,j_{n-m}) \in I^{n-m}} c_\bj \hat\theta_{\bi_\nu}(R'_\bj)\hat\theta_{\bi_\nu}(\hat e(\bj) \hat y_1^{(\Lambda',\alpha_{j_1})}) \hat\theta_{\bi_\nu}(R_\bj) y_\nu\\
& = & \sum_{\bj = (j_1,j_2,\ldots,j_{n-m}) \in I^{n-m}} c_\bj  \hat\theta_{\bi_\nu}(R'_\bj)e(\bi_\nu\vee j_1 \vee j_2 j_3 \ldots j_{n-m}) y_\nu y_{m+1}^{(\Lambda',\alpha_{j_1})} \hat\theta_{\bi_\nu}(R_\bj)\\
& = & \sum_{\bj = (j_1,j_2,\ldots,j_{n-m}) \in I^{n-m}} c_\bj  \hat\theta_{\bi_\nu}(R'_\bj)\theta_{(j_2,j_3,\ldots,j_{n-m})}(e(\bi_\nu\vee j_1) y_\nu y_{m+1}^{(\Lambda',\alpha_{j_1})}) \hat\theta_{\bi_\nu}(R_\bj).
\end{eqnarray*}

Next we consider $e(\bi_\nu\vee j_1) y_\nu y_{m+1}^{(\Lambda',\alpha_{j_1})} \in \mathscr R_{m+1}^\Lambda$.

Recall that we can write $\nu = (\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(\l-1)},\nu^{(\l)})$ and $\nu^{(\l)} = (\nu^{(\l)}_1,\ldots,\nu^{(\l)}_l)$. Let $\mu = \lambda|_{m+1}$. As $m < n-1$, $|\mu| = m+1 < n = |\lambda|$, and $\lambda\in \mathscr S_n^\Lambda$, we have $\mu \in \P_I$. Write $\bi_\nu = (i_1,i_2,\ldots,i_m)$. Notice that $(\Lambda',\alpha_{j_1}) = |\mathscr A_{\t^\nu}^{j_1}|$. We consider two cases.

Suppose ${j_1} = i_m +1$ and $\nu^+$ is a multipartition. By <ref> we have $|\mathscr A_{\t^\nu}^{j_1}| = b_{j_1}^\nu - 1$. Then by <ref> we have $e(\bi_\nu \vee {j_1})y_\nu y_{m+1}^{(\Lambda',\alpha_{j_1})} = e(\bi_\nu \vee {j_1})y_\nu y_{m+1}^{b_{j_1}^\nu - 1} = e_{\nu^+}y_{\nu^+}$. Because $m$ is a cut of $\lambda$ and $\nu = \lambda|_m$, $\mu = \lambda|_{m+1}$, we must have $\nu^+ > \mu$. Therefore $e_{\nu^+}y_{\nu^+} \in \Blam[\mu]$. So

e(\bi_\nu \vee {j_1})y_\nu y_{m+1}^{(\Lambda',\alpha_{j_1})} \in \Blam[\mu].

Otherwise, by <ref> we have $|\mathscr A_{\t^\nu}^{j_1}| = b_{j_1}^\nu$. Then by $\mu \in \P_I$ and the definition of $\P_I$, for any ${j_1}\in I$, we have $e(\bi_\nu \vee {j_1})y_\nu y_{m+1}^{b_{j_1}^\nu} \in \Blam[\mu]$ because $\nu = \mu_-$. Therefore

e(\bi_\nu \vee {j_1})y_\nu y_{m+1}^{(\Lambda',\alpha_{j_1})} = e(\bi_\nu \vee {j_1})y_\nu y_{m+1}^{b_{j_1}^\nu} \in \Blam[\mu].

Therefore for any ${j_1}\in I$ we have $e(\bi_\nu \vee {j_1})y_\nu y_{m+1}^{(\Lambda',\alpha_{j_1})} \in \Blam[\mu]$. Hence by <ref> and <ref>,

\hat\theta_{\bi_\nu}(R'_\bj)\theta_{(j_2,j_3,\ldots,j_{n-m})}(e(\bi_\nu\vee j_1) y_\nu y_{m+1}^{(\Lambda',\alpha_{j_1})}) \hat\theta_{\bi_\nu}(R_\bj) \subseteq \Blam.

Therefore $\hat\theta_{\bi_\nu}(r)y_\nu = \sum_{\bj = (j_1,j_2,\ldots,j_{n-m}) \in I^{n-m}} c_\bj  \hat\theta_{\bi_\nu}(R'_\bj)\theta_{(j_2,j_3,\ldots,j_{n-m})}(e(\bi_\nu\vee j_1) y_\nu y_{m+1}^{(\Lambda',\alpha_{j_1})}) \hat\theta_{\bi_\nu}(R_\bj) \subseteq \Blam$.

Suppose $\lambda$ is a multicomposition of $m$ and $\mu$ is a composition. If there exists a multicomposition $\gamma$ such that $\lambda$ and $\mu$ are cut part and remaining part of $\gamma$ associated $m$, we write $\gamma = \lambda \vee \mu$ and say $\gamma$ is the concatenation of $\lambda$ and $\mu$.

For example, suppose $\lambda = (2^2,1|3^3|2)$ and $\mu = (4,2)$, then $\gamma = \lambda\vee\mu = (2^2,1|3^3|2,4,2)$. Notice that in general $\gamma$ is not a multipartition.

The following Corollaries follows by the definition of $\lambda\vee\mu$.

Suppose $\lambda$ is a multipartition of $n$ and $\mu, \gamma$ are partitions of $m$. Then $\mu > \gamma$ if and only if $\lambda\vee\mu > \lambda\vee\gamma$.

Suppose $\lambda$ is a multipartition of $n$ and $\mu$ is a partition of $m$. If $\gamma = \lambda\vee\mu$, $\hat\theta_{\bi_\lambda}(\hat e_\mu \hat y_\mu)y_\lambda = e_\gamma y_\gamma$.

Suppose $\lambda$ and $\mu$ are multipartitions and $\gamma$ is a partition such that $\lambda = \mu \vee \gamma$. If $\u$ and $$̌ are standard $\gamma$-tableaux, there exist standard $\lambda$-tableaux $\dot\u$ and $\dot\v$ such that $\hat\theta_{\bi_\mu}(\hat\psi_{\u\v})y_\mu = \psi_{\dot\u\dot\v}$.

Suppose $\lambda\in\P_n$ and $\mu\in\P_m$. By <ref>, $\mu$ is the cut part of $\lambda$ associated to $m$. Let $a$ be the cut row associated to $m$. Define $\dot\u$ to be the standard $\lambda$-tableau such that $\dot\u|_m = \t^\mu$, and for any $k > m$, if $\dot\u^{-1}(k) = (r_1,c_1,\l_1)$ and $\u^{-1}(k-m) = (r_2,c_2,1)$, then

\begin{array}{rl}
&c_1  =  c_2,\\
&r_1 = r_2 + a.
\end{array}

Define $\dot\v$ similarly. It is trivial that $\hat\theta_{\bi_\mu} (\hat\psi_{d(\u)}) = \psi_{d(\dot\u)}$ and $\hat\theta_{\bi_\mu} (\hat\psi_{d(\v)}) = \psi_{d(\dot\v)}$. Therefore by <ref>,

\hat\theta_{\bi_\mu}(\hat\psi_{\u\v}) = \hat\theta_{\bi_\mu}(\hat\psi_{d(\u)}^*)\hat\theta_{\bi_\mu}(\hat e_\gamma \hat y_\gamma) \hat\theta_{\bi_\mu}(\hat \psi_{d(\v)}) = \psi_{d(\dot\u)}^* e_\lambda y_\lambda \psi_{d(\dot\v)} = \psi_{\dot\u\dot\v}.

Suppose $\lambda \in \mathscr S_n^\Lambda$ and $\mu \in \mathscr C_n^\Lambda$ with $\mu > \lambda$. If $\mu_- \neq \lambda_-$, then $e_\mu y_\mu \in \Blam$.

As $\mu > \lambda$ and $\mu_- \neq \lambda_-$, there exists $m < n$ such that $\mu|_m > \lambda|_m$ and $\mu|_{m-1} = \lambda|_{m-1}$. Set $\nu = \mu|_m$. If $\nu \in \P_m$, we have $e_\nu y_\nu = \psi_{\t^\nu \t^\nu} \in \Blam[\lambda|_m]$, so by <ref> we have $e_\mu y_\mu \in \Blam$.

If $\nu \not\in \P_m$, because $\lambda\in \mathscr S_n^\Lambda$ and $|\nu| = m < n$, we have $\nu \in \P_I$. Notice that if we write $\nu = (\nu^{(1)},\ldots,\nu^{(l)},\emptyset,\ldots,\emptyset)$ with $\nu^{(l)} = (\nu^{(l)}_1,\ldots,\nu^{(l)}_{k-1},\nu^{(l)}_k)$, because $\nu|_{m-1} = \mu|_{m-1} = \lambda|_{m-1} \in \P_{m-1}$ and $\nu \not\in \P_m$, we must have $\nu_{k-1}^{(l)} + 1 = \nu_k^{(l)}$. Therefore if we write $\bi_\nu = (i_1,i_2,\ldots,i_m)$, we have

e_\nu y_\nu = e(\bi_{\nu_-}\vee i_m) y_{\nu_-} y_m^{b_{i_m}^{\nu_-}} \in \Blam[\nu] \subseteq \Blam[\lambda|_m].

Then by <ref>, we have $e_\mu y_\mu \in \Blam$. This completes the proof.

Now we are ready to start proving the main result of this section. We start by proving two more specialized Propositions. After that we will introduce a Proposition which removes these restrictions and leads to the main Theorem of this section.

Suppose $\lambda\in\mathscr S_n^\Lambda$ and $\lambda_- = (\lambda^{(1)},\ldots,\lambda^{(\l)}_-)$ with $\lambda^{(\l)}_- = (\lambda^{(\l)}_1,\ldots,\lambda^{(\l)}_{l-1},\lambda^{(\l)}_l) \neq \emptyset$. Write $\bi_{\lambda_-} = (i_1,i_2,\ldots,i_{n-1})$. For $k\in I$, if $k \neq i_{n-1}+1$, or $k = i_{n-1}+1$ and $\lambda^{(\l)}_{l-1} > \lambda^{(\l)}_l$, we have

e(\bi_{\lambda_-}\vee k)y_{\lambda_-} y_n^{b_k^{\lambda_-}} \in \Blam.

For convenience set $m = \lambda^{(\l)}_l$ and $\mu = \lambda|_{n-m-1}$. Therefore $\mu = (\lambda^{(1)},\ldots,\lambda^{(\l-1)},\mu^{(\l)})$ where

\mu^{(\l)} = \begin{cases}
(\lambda^{(\l)}_1,\ldots,\lambda^{(\l)}_{l-1}), & \text{ if $l > 1$,}\\
\emptyset, & \text{ if $l = 1$.}
\end{cases}

Suppose $i$ is the residue of node $(l,1,\l)$ in $\lambda$ and $\Lambda' = \Lambda_i$. Define $\gamma = (m+1)\in \mathscr P^{\Lambda'}_{m+1}$. Notice that $\lambda_- = \mu\vee\gamma_-$. Because $k\neq i_{n-1}+1$ or $k = i_{n-1}+1$ and $\lambda_{k-1}^{(\l)} > \lambda_k^{(\l)}$, we have $b_k^{\gamma_-} = b_k^{\lambda_-}$. By <ref>, in $\mathscr R_{m+1}^{\Lambda'}$ we have $e(\bi_{\gamma_-}\vee k)y_{\gamma_-}y_{m+1}^{b_k^{\gamma_-}} \in \Blam[\gamma]$. This implies that in $\mathscr R_{m+1}$, $\hat e(\bi_{\gamma_-}\vee k)\hat y_{\gamma_-}\hat y_{m+1}^{b_k^{\gamma_-}} \in N_{m+1}^{\Lambda'}$. Then let $\hat\theta_{\bi_\mu}\map{\mathscr R_{m+1}}{\R}$, by <ref>,

e(\bi_{\lambda_-}\vee k)y_{\lambda_-} y_n^{b_k^{\lambda_-}} = e(\bi_\mu \vee \bi_{\gamma_-}\vee k)y_{\lambda_-} y_n^{b_k^{\gamma_-}} = \hat\theta_{\bi_\mu}(\hat e(\bi_{\gamma_-}\vee k) \hat y_{\gamma_-}\hat y_{m+1}^{b_k^{\gamma_-}})y_\mu \in \hat\theta_{\bi_\mu}(N_{m+1}^{\Lambda'}) y_\mu \subseteq\Blam,

which completes the proof.

Suppose $\lambda = (\lambda^{(1)},\ldots,\lambda^{(\l)})\in\mathscr S_n^\Lambda$ with $\lambda^{(\l)} = (\lambda^{(\l)}_1,\ldots,\lambda^{(\l)}_{l-1},\lambda^{(\l)}_l,1)$ and $l \geq 2$, where $\lambda^{(\l)}_{l-1} = \lambda^{(\l)}_l$. Write $\bi_{\lambda_-} = (i_1,i_2,\ldots,i_{n-1})$. Suppose $k\in I$ and $k \equiv i_{n-1}+1\pmod{e}$. Then

e(\bi_{\lambda_-}\vee k)y_{\lambda_-} y_n^{b_k^{\lambda_-}}\in \Blam.

For convenience set $m = \lambda^{(\l)}_{l-1} = \lambda^{(\l)}_l$, and $\mu = \lambda|_{n-2m-1}$. Therefore $\mu = (\lambda^{(1)},\ldots,\lambda^{(\l-1)},\mu^{(\l)})$ where

\mu^{(\l)} = \begin{cases}
(\lambda^{(\l)}_1,\ldots,\lambda^{(\l)}_{l-2}), & \text{ if $l > 2$,}\\
\emptyset, & \text{ if $l = 2$.}
\end{cases}

Suppose $i$ is the residue of node $(l-1,1,\l)$ in $\lambda$ and $\Lambda' = \Lambda_i$. Define $\gamma = (m,m+1)\in \mathscr P^{\Lambda'}_{2m+1}$. Notice that $\lambda_- = \mu\vee \gamma_-$. Because $k \equiv i_{n-1}+1\pmod{e}$, we have $b_k^{\gamma_-} = b_k^{\lambda_-}$ and $e(\bi_{\gamma_-}\vee k)y_{\gamma_-} y_{2m+1}^{b_k^{\gamma_-}} = e_\gamma y_\gamma$. By <ref>, we have $e_\gamma y_\gamma \in \Blam[\gamma]$. Therefore we can write $e_\gamma y_\gamma = \sum_{\substack{\u,\v\in \Std(\sigma)\\ \sigma > \gamma}} c_{\u\v}\psi_{\u\v}$ with $\sigma = (\sigma_1,\sigma_2)$ where $\sigma_2 \geq 0$ and $\sigma_1 > \gamma_1 = m$. Therefore in $\mathscr R_{2m+1}$, we have

\hat e_\gamma \hat y_\gamma = \sum_{\substack{\u,\v\in \Std(\sigma)\\ \sigma > \gamma}} c_{\u\v}\hat\psi_{\u\v} + r,

with $r \in N_{2m+1}^{\Lambda'}$ and $c_{\u\v}\in\Z$. Therefore

\begin{align}
e(\bi_{\lambda_-}\vee k)y_{\lambda_-}y_n^{b_k^{\lambda_-}} & = e(\bi_\mu\vee \bi_{\gamma_-}\vee k)y_{\lambda_-} y_n^{b_k^{\gamma_-}} = \hat\theta_{\bi_\mu} (\hat e(\bi_{\gamma_-}\vee k)\hat y_{\gamma_-}\hat y_{2m+1}^{b_k^{\gamma_-}})y_\mu = \hat\theta_{\bi_\mu}(\hat e_\gamma \hat y_\gamma) y_\mu \notag \\
& = \sum_{\substack{\u,\v\in\Std(\sigma)\\ \sigma > \gamma}} c_{\u\v}\hat\theta_{\bi_\mu}(\hat \psi_{\u\v})y_\mu + \hat\theta_{\bi_\mu}(r) \label{I-problem: no empty tail: part2: help}.
\end{align}

For the first term of (<ref>), define $\alpha = \mu\vee\sigma \in \mathscr C_n^\Lambda$. Because $\sigma > \gamma$, by <ref> we have $\alpha = \mu\vee\sigma > \mu\vee\gamma > \lambda$. Therefore

\hat\theta_{\bi_\mu}(\hat\psi_{\u\v})y_\mu = \hat\theta_{\bi_\mu}(\hat\psi_{d(\u)}^*)\hat\theta_{\bi_\mu}(\hat e_\sigma \hat y_\sigma) y_\mu \hat\theta_{\bi_\mu}(\hat\psi_{d(\v)}) = \hat\theta_{\bi_\mu}(\hat\psi_{d(\u)}^*)e_{\mu\vee\sigma}y_{\mu\vee\sigma}\hat\theta_{\bi_\mu}(\hat\psi_{d(\v)}) = \hat\theta_{\bi_\mu}(\hat\psi_{d(\u)}^*)e_\alpha y_\alpha \hat\theta_{\bi_\mu}(\hat\psi_{d(\v)})

Because $\sigma = (\sigma_1,\sigma_2) > \gamma = (m,m+1)$, we must have $\sigma_1 > m$. Therefore $\alpha_- = \mu\vee\sigma_- \neq \mu\vee\gamma_- = \lambda_-$. Then by <ref>, $e_\alpha y_\alpha \in \Blam$. By <ref>, we have $\hat\theta_{\bi_\mu}(\hat\psi_{\u\v})y_\mu = \hat\theta_{\bi_\mu}(\hat\psi_{d(\u)}^*)e_\alpha y_\alpha \hat\theta_{\bi_\mu}(\hat\psi_{d(\v)})\in \Blam$ which yields $\sum_{\substack{\u,\v\in\Std(\sigma)\\ \sigma > \gamma}} c_{\u\v}\hat\theta_{\bi_\mu}(\hat \psi_{\u\v})y_\mu\in \Blam$.

For the second term of (<ref>), by <ref>, we have $\hat\theta_{\bi_\mu}(r) \in \Blam$. Therefore
$$e(\bi_{\lambda_-}\vee k)y_{\lambda_-} y_n^{b_k^{\lambda_-}}\in \Blam.$$

Suppose $\lambda \in \mathscr S_n^\Lambda$. If $\lambda_- = (\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(\l-1)},\lambda_-^{(\l)})$ with $\lambda_-^{(\l)} \neq \emptyset$ by <ref> and <ref> we have $\lambda\in\P_I$. In the rest of the subsection we are going to prove the result is still true if $\lambda_-^{(\l)} = \emptyset$.

Suppose $\mu = (\mu^{(1)},\ldots,\mu^{(\l)}) \in \P_n$ and $\kappa_\Lambda = (\kappa_1,\ldots,\kappa_\l)$, if $\mu^{(\l)} = \emptyset$, we define $\bar\Lambda = \Lambda - \Lambda_{\kappa_\l}$, $\kappa_{\bar\Lambda} = (\kappa_1,\ldots,\kappa_{\l-1})$ and $\bar\mu = (\mu^{(1)},\ldots,\mu^{(\l-1)}) \in \mathscr P^{\bar\Lambda}_n$. Suppose $\u,\v$ are two standard $μ$-tableaux, define $̆̅$ and $̌̅$ to be standard $μ̅$-tableaux obtained by removing the $∅$ at the end of $$̆ and $\v$ respectively. Write $k = κ_ł$ for convenience. If $= (i_1,…, i_n) ∈I^n$, define
\begin{equation*} \label{notation: y i k}
y_{\bi,k} = y_1^{\delta_{i_1,k}}y_2^{\delta_{i_2,k}}\ldots y_n^{\delta_{i_n,k}}.
\end{equation*}

\begin{Lemma} \label{remove empty tableaux: help}
Suppose the notations are defined as above and $\bi_\v$ is the residue sequence of $\v$, then
\psi_{\u\v} = \psi_{\bar \u \bar \v} y_{\bi_\nu,k},
where $\psi_{\u\v}$ is an element in $\mathscr{R}_n^\Lambda$ and $\psi_{\bar \u \bar \v}$ is an element in $\mathscr{R}_n^{\bar\Lambda}$.
\end{Lemma}

\proof Without loss of generality, assume $=̆ ^μ$. By the definition of $μ$ and $μ̅$, writing $_μ= (i_1, i_2, …, i_n)$, we have $y_μ= y_μ̅ y_1^δ_i_1,ky_2^δ_i_2,k…y_n^δ_i_n,k = y_μ̅ y__μ,k$.

Now for any residue sequence $= (i_1,i_2,…, i_n) ∈I^n$ and any $r$, If $i_r ≠i_r+1$
\begin{eqnarray*}
e(\bi)y_1^{\delta_{i_1,k}} y_2^{\delta_{i_2,k}}\ldots y_n^{\delta_{i_n,k}}\psi_r & = & (e(\bi)y_r^{\delta_{i_r,k}}y_{r+1}^{\delta_{i_{r+1},k}} \psi_r) y_1^{\delta_{i_1,k}} \ldots y_{r-1}^{\delta_{i_{r-1},k}}y_{r+2}^{\delta_{i_{r+1},k}}\ldots y_n^{\delta_{i_n,k}}\\
& = &e(\bi) \psi_r y_r^{\delta_{i_{r+1},k}} y_{r+1}^{\delta_{i_r,k}} y_1^{\delta_{i_1,k}} \ldots y_{r-1}^{\delta_{i_{r-1},k}}y_{r+2}^{\delta_{i_{r+1},k}}\ldots y_n^{\delta_{i_n,k}}\\
& = & e(\bi) \psi_r y_1^{\delta_{s_r(i_1),k}} \ldots y_{n}^{\delta_{s_r(i_{n}),k}} = e(\bi) \psi_r y_{\bi{\cdot}s_r,k}.
\end{eqnarray*}

If $i_r = i_r+1$, then by relation (\ref{alg:ypsi}), as $δ_i_r,k = δ_i_r+1,k$, we have the same result.

\begin{eqnarray*}
\psi_{\t^\mu\v} & = & e_\mu y_\mu \psi_{d(\v)} = e(\bi_\mu)y_{\bar\mu} y_{\bi_\mu,k} \psi_{d(\v)}\\
& = & e(\bi_\mu)y_{\bar\mu}\psi_{d(\v)} y_{\bi_\mu{\cdot}d(\v),k} = e_\mu y_{\bar\mu} \psi_{d(v)} y_{\bi_\v,k}.
\end{eqnarray*}

As $e_μ= e_μ̅$ and $ψ_d()̌ = ψ_d(̌̅)$, this completes the proof. \endproof

\begin{Proposition} \label{remove empty tableaux}

Suppose $\mu$, $\kappa_\Lambda$, $\bar\mu$, and $\kappa_{\bar\Lambda}$ are defined as above. Then $\bar\mu \in \mathscr P_I^{\bar\Lambda}\cap \mathscr P_y^{\bar\Lambda}\cap  \mathscr P_\psi^{\bar\Lambda}$ implies $\mu \in \P_I \cap \P_y \cap \P_\psi$.

\end{Proposition}

\proof We are only going to prove that $μ̅∈𝒫_I^Λ̅$ implies $μ∈_I$. The other two cases are similar.

Suppose $μ̅∈𝒫^Λ̅_I$. Then for any $s ∈I$, by the definition of $𝒫^Λ̅_I$,
e(\bi_{\bar\mu_-} \vee s)y_{\bar\mu_-} y_n^{b_s^{\bar\mu_-}}  = \sum_{\bar\u,\bar\v \in \Std( > \bar\mu)} c_{\bar\u \bar\v} \psi_{\bar\u \bar \v},
where $_̌̅ = _μ̅_-∨s = _μ_-∨s$ and $c_̆̅̌̅∈$.

Also we have $e(_μ̅_- ∨s)y_μ̅_- y_n^b_s^μ̅_- = θ_s(ψ_t^μ̅_- t^μ̅_-)  y_n^b_s^μ̅_-$. Therefore we have
\theta_s(\psi_{t^{\bar\mu_-} t^{\bar\mu_-}})  y_n^{b_s^{\bar\mu_-}} = \sum_{\bar\u,\bar\v \in \Std( > \bar\mu)} c_{\bar\u \bar\v} \psi_{\bar\u \bar \v}.

Notice that $^μ̅_- = ^μ_-$. Recall $k = κ_ł$, the last term of the multicharge $κ_Λ$. We consider two cases, $s ≠k$ and $s = k$ in the rest of the proof.

If $s ≠k$, then $b_s^μ_- = b_s^μ̅_-$. Hence by \autoref{remove empty tableaux: help}
\begin{eqnarray*}
e(\bi_{\mu_-} \vee s)y_{\mu_-} y_n^{b_s^{\mu_-}} & = & \theta_s(\psi_{t^{\mu_-} t^{\mu_-}})  y_n^{b_s^{\mu_-}}\\
& = & \theta_s(\psi_{\bar \t^{\mu_-} \bar \t^{\mu_-}} y_{\bi_{\mu_-},k}) y_n^{b_s^{\bar\mu_-}}\\
& = & \theta_s(\psi_{\t^{\bar\mu_-} \t^{\bar\mu_-}}) y_n^{b_s^{\bar\mu_-}}  y_{\bi_{\mu_-},k}\\
& = & \sum_{\bar\u,\bar\v \in \Std( > \bar\mu)} c_{\bar\u \bar\v} \psi_{\bar\u \bar\v} y_{\bi_{\mu_-},k},
\end{eqnarray*}
and as $s ≠k$, $δ_s,k = 0$. Hence $y__μ_-,k = y__μ_-∨s, k = y__̌̅,k = y__,̌k$. By \autoref{remove empty tableaux: help},
e(\bi_{\mu_-} \vee s)y_{\mu_-} y_n^{b_s^{\mu_-}} = \sum_{\bar\u,\bar\v \in \Std( > \bar\mu)} c_{\bar\u \bar\v} \psi_{\bar\u \bar\v} y_{\bi_\v,k} = \sum_{\bar\u,\bar\v \in \Std( > \bar\mu)} c_{\bar\u \bar\v} \psi_{\u\v} \in \Blam,
because $̆̅,̌̅∈( > μ̅)$ implies $,̆( > μ)$.\\

If $s = k$, then $b_s^μ_- = b_s^μ̅_- + 1$. Hence by \autoref{remove empty tableaux: help}
\begin{eqnarray*}
e(\bi_{\mu_-} \vee s)y_{\mu_-} y_n^{b_s^{\mu_-}} & = & \theta_s(\psi_{t^{\mu_-} t^{\mu_-}})  y_n^{b_s^{\mu_-}}\\
& = & \theta_s(\psi_{\bar \t^{\mu_-} \bar \t^{\mu_-}} y_{\bi_{\mu_-},k}) y_n^{b_s^{\bar\mu_-}}y_n\\
& = & \theta_s(\psi_{\t^{\bar\mu_-} \t^{\bar\mu_-}}) y_n^{b_s^{\bar\mu_-}}  y_{\bi_{\mu_-},k}y_n\\
& = & \sum_{\bar\u,\bar\v \in \Std( > \bar\mu)} c_{\bar\u \bar\v} \psi_{\bar\u \bar\v} y_{\bi_{\mu_-},k}y_n,
\end{eqnarray*}
and as $s = k$, $δ_s,k = 1$. Hence $y__μ_-,ky_n = y__μ_-∨s, k = y__̌̅,k = y__,̌k$. By \autoref{remove empty tableaux: help}
e(\bi_{\mu_-} \vee s)y_{\mu_-} y_n^{b_s^{\mu_-}} = \sum_{\bar\u,\bar\v \in \Std( > \bar\mu)} c_{\bar\u \bar\v} \psi_{\bar\u \bar\v} y_{\bi_\v,k} = \sum_{\bar\u,\bar\v \in \Std( > \bar\mu)} c_{\bar\u \bar\v} \psi_{\u\v} \in \Blam.

These implies that $μ∈𝒫_I$. \endproof

Now we are ready to prove \autoref{I-problem: final}.\\

\hspace*{-6mm}\textbf{Proof of \autoref{I-problem: final}.} Write $μ= λ_- = (λ^(1),λ^(2),…,λ^(ł-1),μ^(ł))$. If $μ^(ł) ≠∅$, by \autoref{I-problem: no empty tail: part 1} and \autoref{I-problem: no empty tail: part 2}, we have $λ∈_I$.

If $μ^(ł) = ∅$, write $λ^(ł-1) = (λ^(ł-1)_1,λ^(ł-1)_2,…,λ^(ł-1)_k_ł-1)$ and define $γ= (λ^(1),…,λ^(ł-2),γ^(ł-1),∅) ∈𝒫_n^Λ$ with $γ^(ł-1) = (λ^(ł-1)_1,λ^(ł-1)_2,…,λ^(ł-1)_k_ł-1,1)$ and $γ̅= (λ^(1),…,λ^(ł-2),γ^(ł-1))$. As $l(γ̅) < l(λ) = ł$, by the definition of $𝒮_n^Λ$, $γ̅∈𝒫_I^Λ̅$. Then by \autoref{remove empty tableaux} we have $γ∈_I$. Since $γ_- = μ= λ_-$ and $γ> λ$, for any $k∈I$,
e(\bi_{\lambda_-}\vee k) y_{\lambda_-} y_n^{b_k^{\lambda_-}} = e(\bi_{\gamma_-}\vee k) y_{\gamma_-} y_n^{b_k^{\gamma_-}} \in \Blam[\gamma] \subseteq \Blam,
which yields that $λ∈𝒫_I$. This completes the proof. \endproof

The following Corollary is directly implied by \autoref{I-problem: final}. It will contribute to proving the $ψ$-problem.

\begin{Corollary} \label{composition in Blam}

Suppose $\lambda \in \mathscr S_n^\Lambda$ and $\mu \in \mathscr C_n^\Lambda$ where $\mu > \lambda$. Then we have $e_\mu y_\mu \in \Blam$.

\end{Corollary}

\proof If $μ_- ≠λ_-$, using \autoref{concatenation: cor4} we have $e_μy_μ∈$. Suppose then that $μ_- = λ_-$. If $μ∈_n$, then $e_μy_μ= ψ_^μ^μ ∈$. Finally, suppose that $μ∉_n$. If we write $μ= (μ^(1),…,μ^(l),∅,…,∅)$ with $μ^(l) = (μ^(l)_1,…,μ^(l)_k-1,μ^(l)_k)$, we must have $μ^(l)_k-1 + 1 = μ^(l)_k$. If we write $_μ= (i_1,i_2,…,i_n)$, we have $e_μy_μ= e(_μ_-∨i_n) y_μ_- y_n^b_i_n^μ_-$. By \autoref{I-problem: final}, as $λ∈𝒮_n^Λ$, we have $λ∈_I$. Since $λ_- = μ_-$,
e_\mu y_\mu = e(\bi_{\mu_-}\vee i_n) y_{\mu_-} y_n^{b_{i_n}^{\mu_-}} = e(\bi_{\lambda_-}\vee i_n) y_{\lambda_-} y_n^{b_{i_n}^{\lambda_-}}  \in \Blam.

\section{Integral Basis Theorem II}

In this section our main purpose is to prove that $= $ by proving that $ψ_y_r$ and $ψ_ψ_r$ are both in $$. We first define an integer $m_λ$ such that if $∈(λ)$ and $l(d()) < m_λ$, we have $ψ_y_r ∈$ and $ψ_ψ_r ∈$. Our first step is to show that $m_λ> 0$. Then we prove if $l(d()) ≤m_λ$, we will have $ψ_y_r ∈$ and $ψ_ψ_r ∈$ as well. By induction we will show that for any $∈(λ)$, $l(d()) < m_λ$, which indicates that $ψ_y_r ∈$ and $ψ_ψ_r ∈$ for any $,∈(λ)$. Finally combining the results from the last section, we can prove that $= $.

\subsection{Base case of induction}
In this section we fix $λ∈𝒮_n^Λ$. First we will give a proper definition of $m_λ$.

\begin{Definition}\label{notation: W lambda}
Define $m_\lambda$ to be the smallest nonnegative integer such that for any standard $\lambda$-tableau $\t$ with $l(d(\t)) < m_\lambda$ we have
\begin{eqnarray*}
\psi_{\s\t} y_r & = & \sum_{(\u,\v) \rhd (\s,\t)} c_{\u\v} \psi_{\u\v},\\
\psi_{\s\t} \psi_r & = & \begin{cases}
\psi_{\s \w} + \sum_{(\u,\v)\rhd (\s,\t)}c_{\u\v}\psi_{\u\v},& \text{ if $\w = \t{\cdot}s_r$ is standard and $d(\u){\cdot}s_r$ is reduced,}\\
\sum_{(\u,\v)\rhd (\s,\t)}c_{\u\v} \psi_{\u\v}, & \text{ if $\u{\cdot}s_r$ is not standard or $d(\u){\cdot}s_r$ is not reduced.}
\end{cases}
\end{eqnarray*}
for some $c_{\u\v} \in \Z$.
\end{Definition}

We will use induction to prove that for any $∈(λ)$, $l(d()) < m_λ$ in this section. In this subsection we will prove that $m_λ> 0$, which is the base case of the induction.

\begin{Lemma} \label{y-problem: induction}

Suppose $\lambda\in \mathscr S_n^\Lambda$. For any $1 \leq r \leq n$, $e_\lambda y_\lambda y_r = \psi_{\t^\lambda \t^\lambda}y_r \in \Blam$.

\end{Lemma}

\proof If $ r < n$, write $μ= λ|_r$. As $λ∈𝒮_n^Λ$ we have $μ∈_y$. Therefore $e_μy_μy_r ∈[μ]$. By \autoref{send}, we have $e_λy_λy_r ∈$.

If $r = n$, write $_λ= (i_1,i_2,…,i_n)$. There exists a positive integer $b$ such that $e_λy_λ= e(_λ_-∨i_n)y_λ_-y_n^b$. By the definition of $b_i_n^λ_-$ we have $b < b_i_n^λ_-$. If $b + 1 < b_i_n^λ_-$, by \autoref{I-problem: downstair cases} there exists $ν∈_n$ such that
e_\lambda y_\lambda y_n = e(\bi_{\lambda_-}\vee i_n) y_{\lambda_-} y_n^{b+1} = e_\nu y_\nu,
and it is trivial that $ν> λ$. Therefore $e_λy_λy_n ∈$. If we have $b + 1 = b_i_n^λ_-$, by \autoref{I-problem: final} we have
e_\lambda y_\lambda y_n = e(\bi_{\lambda_-}\vee i_n) y_{\lambda_-} y_n^{b+1} = e(\bi_{\lambda_-}\vee i_n) y_{\lambda_-} y_n^{b_{i_n}^{\lambda_-}} \in \Blam,
which completes the proof. \endproof

\begin{Lemma} \label{psi-problem: induction}

Suppose $\lambda\in\mathscr S_n^\Lambda$. For any $1\leq r < n$, $e_\lambda y_\lambda \psi_r = \psi_{\t^\lambda \t^\lambda}\psi_r \in \Bgelam$.

\end{Lemma}

\proof Suppose $^λ·s_r = $ is standard, we have $e_λy_λψ_r = ψ_^λ ∈$. So we consider the case that $^λ·s_r = $ is not standard. If $r < n-1$, as $λ∈𝒮_n^Λ$, we have $μ= λ_- ∈_ψ$. Because $^μ·s_r = |_n-1$ which is not standard, $e_μy_μψ_r ∈[μ]$. Then by \autoref{send}, we have $e_λy_λψ_r ∈$. If $r = n-1$ and write $λ= (λ^(1),…,λ^(l),∅,…,∅)$ with $λ^(l) = (λ^(l)_1,…,λ^(l)_k-1,λ^(l)_k)$, we must have either $λ^(l)_k ≥2$ or $λ^(l)_k-1 = λ^(l)_k = 1$. Then set $μ= (λ^(1),…,μ^(l),∅,…,∅)$ with
\mu^{(l)} = \begin{cases}
(\lambda^{(l)}_1,\ldots,\lambda^{(l)}_{k-1}), & \text{ if $\lambda^{(l)}_k \geq 2$ and $k > 1$,}\\
(\lambda^{(l)}_1,\ldots,\lambda^{(l)}_{k-2}), & \text{ if $\lambda^{(l)}_{k-1} = \lambda^{(l)}_k = 1$ and $k > 2$,}\\
\emptyset, & \text{ otherwise.}
\end{cases}

Suppose $i$ is the residue of node $(k,1,l)$ in $λ$ or the residue of node $(k-1,1,l)$ in $λ$, $Λ' = Λ_i$, $m = λ^(l)_k$ or $m = 2$ and $γ= (m)∈𝒫^Λ'_m$ or $γ= (1,1)∈𝒫^Λ'_m$ if $λ^(l)_k ≥2$ or $λ^(l)_k-1 = λ^(l)_k = 1$ respectively. Therefore $λ= μ∨γ$. Because $λ∈𝒮_n^Λ$, we have $γ∈_ψ$. Hence because $^γ·s_m-1$ is not standard, we have $e_γy_γψ_m-1 ∈[γ] = N_m^Λ'$. Then by \autoref{theta_j: go up},
e_\lambda y_\lambda \psi_r = e(\bi_\mu \vee \bi_\gamma) y_\lambda \psi_r = \hat\theta_{\bi_\mu}(e_\gamma y_\gamma \psi_{m-1}) y_\mu \in \Blam,
which completes the proof. \endproof

\begin{Corollary} \label{m_lambda is positive}
For $\lambda \in \mathscr S_n^\Lambda$, we have $m_\lambda > 0$.

\end{Corollary}

\proof Combining the above two Lemmas, \autoref{residue sequence has to be the same} and \autoref{basis to hecke}, the Corollary follows.\endproof

\subsection{Completion of the $y$-problem}

In this subsection we are going to prove that for any $∈(λ)$, if $l(d()) ≤m_λ$, then for any $1≤r≤n-1$ and any $∈(λ)$, if $·s_r$ is standard and $d()·s_r$ is reduced, $ψ_ψ_r ∈$ and for any $1≤r≤n$, $ψ_y_r ∈$.

First we introduce the following Lemma.

\begin{Lemma} \label{length of psi}
Suppose $m$ is a positive integer such that $m \leq m_\lambda$, then
\begin{eqnarray*}
e_\lambda y_\lambda \psi_{r_1}\psi_{r_2}\ldots \psi_{r_m}
& =_\lambda &\sum_{\substack{\v \in \text{Std}(\lambda)\\ l(d(\v)) \leq m}} c_{\t^\lambda\v} \psi_{t^\lambda \v}.
\end{eqnarray*}
\end{Lemma}

\proof We apply induction on $m$. Suppose $m = 0$ then there is nothing to prove. Assume for any $m' < m$ the Lemma holds. Therefore $e_λy_λψ_r_1ψ_r_2…ψ_r_m-1 =_λ∑_∈̆Std(λ)
l(d()̆) ≤m-1 c_^λ ψ_t^λ$ which yields
e_\lambda y_\lambda\psi_{r_1}\psi_{r_2}\ldots \psi_{r_{m-1}}\psi_{r_m} =_\lambda\sum_{\substack{\u \in \text{Std}(\lambda)\\ l(d(\u)) \leq m-1}} c_{\t^\lambda \u} \psi_{t^\lambda \u}\psi_{r_m}.

For $∈̆(λ)$ and $l(d()̆) ≤m-1 < m_λ$, if $·̆s_r$ is standard and $s_d()̌·s_r_m$ is reduced, by the definition of $m_λ$,
\psi_{\t^\lambda \u}\psi_{r_m} = \psi_{\t^\lambda, \u{\cdot}s_r} + \sum_{(\x,\y)\rhd(\t^\lambda,\u)} c_{\x \y} \psi_{\x \y} =_\lambda \psi_{\t^\lambda, \u{\cdot}s_r} + \sum_{\v \rhd \u} c_{\t^\lambda \v} \psi_{\t^\lambda \v},
where $l(d(·̆s_r)) = 1 + l(d()̆) ≤m$ and $l(d()̌) < l(d()̆) < m$ as $$̆. Hence

\psi_{\t^\lambda \u}\psi_{r_m} =_\lambda \sum_{\substack{\v \in \text{Std}(\lambda)\\ l(d(\v)) \leq m}} c_\v \psi_{t^\lambda \v}.

If $\u{\cdot}s_r$ is not standard or $s_{d(\v)}{\cdot}s_{r_m}$ is not reduced, we have

\psi_{\t^\lambda \u} \psi_{r_m} = \sum_{(\x,\y)\rhd(\t^\lambda,\u)} c_{\x \y} \psi_{\x \y} =_\lambda \sum_{\v \rhd \u} c_{\t^\lambda \v} \psi_{\t^\lambda \v},

where $l(d(\v)) < l(d(\u)) \leq m-1 < m$ as $\v \rhd \u$. Hence
\psi_{\t^\lambda \u}\psi_{r_m} =_\lambda \sum_{\substack{\v \in \text{Std}(\lambda)\\ l(d(\v)) \leq m}} c_{\t^\lambda\v} \psi_{t^\lambda \v},
which completes the proof. \endproof

Now we can start to prove that $ψ_ψ_r∈$ when $·s_r$ is standard and $d()·s_r$ is reduced.

\begin{Lemma}\label{basis change}
Suppose $\t$ is a standard $\lambda$-tableau with $d(\t) = s_{r_1}s_{ r_2} \ldots s_{r_l}$ where $l \leq m_\lambda$, and $d'(t) = s_{t_1} s_{t_2} \ldots s_{t_l}$ is another reduced decomposition of $d(\t)$, then
e_\lambda y_\lambda \psi_{d(\t)} - e_\lambda y_\lambda \psi_{d'(\t)} = \sum_{(\u,\v)\rhd (\t^\lambda \t)} c_{\u\v} \psi_{\u\v}.
\end{Lemma}

\proof By \cite[Proposition 2.5]{BKW:GradedSpecht}, we have
y_\lambda e_\lambda \psi_{d(\t)} - y_\lambda e_\lambda \psi_{d'(\t)} = \sum_{u < d(\t)}c_{u,f}y_\lambda e_\lambda f(y) \psi_u,
where $f(y)$ is a polynomial in $y_r$'s and $u$ is a word in $𝔖_n$. If $f(y)≠1$, by \autoref{y-problem: induction} we have $e_λy_λf(y)∈$. Hence $y_λe_λf(y) ψ_u ∈$. If $f(y) = 1$, as $u <  d()$ then $l(u) <l ≤m_λ$, by \autoref{length of psi} we have $e_λy_λψ_u ∈$. Henceforth
y_\lambda e_\lambda \psi_w - y_\lambda e_\lambda \psi_{w'} \in \Bgelam.

Then by \autoref{basis to hecke} and \cite[Lemma 5.7]{HuMathas:GradedCellular}, we complete the proof.\endproof

The following Corollary is straightforward by \autoref{basis change} which explains the action of $ψ_r$ to $ψ_$ from right when $·s_r$ is standard and $d()·s_r$ is reduced.

\begin{Corollary}\label{psi problem: part2}
Suppose $\t$ is a standard $\lambda$-tableau with $l(d(\t)) \leq m_\lambda$, if $\w = \t{\cdot}s_r$ is standard and $d(\t){\cdot}s_r$ is reduced,
\psi_{\s\t} \psi_r = \psi_{\s\w} + \sum_{(\u,\v)\rhd (\s,\t)}c_{\u\v} \psi_{\u\v}.
\end{Corollary}

Now we start to prove that $ψ_y_r ∈$.

\begin{Lemma} \label{yup: help}
Suppose $\t$ is a standard $\lambda$-tableau with $l(d(\t)) < m_\lambda$. For any $1\leq k \leq n$, $1\leq r \leq n-1$ and any standard $\lambda$-tableau $\s$, we have
\psi_{\s\t}y_k\psi_r \in \Bgelam.

\proof As $l(d(\t)) < m_\lambda$, we have
\psi_{\s \t}y_k = \sum_{(\u,\v)\rhd (\s,\t)}c_{\u\v} \psi_{\u\v} = \sum_{\v\rhd \t}c_{\s \v} \psi_{\s \v} + \sum_{\u,\v\in\Std(>\lambda)}c_{\u\v} \psi_{\u\v}.

For $\Shape(\v) = \lambda$, since $\v \rhd \t$, $l(d(\v)) < l(d(\t)) < m_\lambda$. Then we have $\psi_{\s\v}\psi_{r} \in \Bgelam$.

For $\u,\v\in\Std(>\lambda)$, $\psi_{\u\v}\in \Blam $. As $\lambda \in \mathscr S_n^\Lambda$, $\Blam $ is an ideal by \autoref{B_lambda is an ideal}. Hence $\psi_{\u\v}\psi_{r_l} \in \Blam$ and this completes the proof. \endproof

\end{Lemma}

\begin{Proposition}\label{yup}
Suppose $\t$ is a standard $\lambda$-tableau with $l(d(\t)) \leq m_\lambda$, for any $1\leq r\leq n$ and any standard $\lambda$-tableau $\s$, we have
\psi_{\s\t} y_r = \sum_{(\u,\v)\rhd (\s,\t)}c_{\u\v} \psi_{\u\v}.
\end{Proposition}

\proof Write $d() = s_r_1s_r_2…s_r_l-1s_r_l$ and $= ·s_r_l = s_r_1s_r_2…s_r_l-1$. We prove this Proposition by considering different values of $r$.

If $r∉{r_l,r_l+1}$, then $ψ_r_l$ and $y_r$ commute. Hence
\psi_{\s \t} y_r =\psi_{d(\s)}^* e_\lambda y_\lambda \psi_{d(\t)}y_r = \psi_{d(\s)}^* e_\lambda y_\lambda \psi_{d(\w)}y_r\psi_{r_l} = \psi_{\s\w}y_r\psi_{r_l}.

As $l(d()) = l(d()) -1 < m_λ$, by \autoref{yup: help} we have $ψ_y_r ∈$.\\

If $r = r_l$, let $$ be a sequence such that $e(_λ)ψ_d() = ψ_d()e()ψ_r_l$. We separate this case further into $j_r_l ≠j_r_l+1$ and $j_r_l = j_r_l+1$. First suppose $j_r_l ≠j_r_l+1$, then
\psi_{\s \t} y_r =\psi_{d(\s)}^* e_\lambda y_\lambda \psi_{d(\t)}y_r = \psi_{d(\s)}^* e_\lambda y_\lambda \psi_{d(\w)}y_{r+1}\psi_{r_l} = \psi_{\s\w}y_{r+1}\psi_{r_l}.

Hence as $l(d()) < m_λ$, by \autoref{yup: help} we have $ψ_y_r ∈$ when $j_r_l ≠j_r_l+1$. Now suppose $j_r_l = j_r_l+1$, we have
\psi_{\s\t}y_r = \psi_{d(\s)}^* e_\lambda y_\lambda \psi_{d(\t)}\y_r = \psi^*_{d(\s)}e_\lambda y_\lambda \psi_{d(\w)} + \psi_{d(\s)}^* e_\lambda y_\lambda \psi_{d(\w)}y_{r+1}\psi_{r_l} = \psi_{\s\w} + \psi_{\s\w}y_{r+1}\psi_{r_l}.

As $l(d()) < m_λ$, by \autoref{yup: help} we have $ψ_y_r+1ψ_r_l ∈$. As $ψ_∈$ as well, we have $ψ_y_r ∈$. So for $r = r_l$, we have $ψ_y_r ∈$.\\

If $r = r_l + 1$, the method is the same as $r = r_l$.

Therefore in all the cases, we have $ψ_y_r ∈$. So
\psi_{\s\t}y_r = \sum_{\u,\v\in\Std(\geq\lambda)} c_{\u\v}\psi_{\u\v},
and by \autoref{basis to hecke} we complete the proof.\endproof

\subsection{Properties of $m_\lambda$}

In the rest of this section we will prove that if $∈(λ)$ and $l(d()) ≤m_λ$, then for any $1≤r≤n-1$ and any $∈(λ)$, we have $ψ_ ψ_r∈$. In this subsection we will give some properties for $m_λ$ which will be used in proving the above argument.

\begin{Lemma} \label{does not change by psi}
Suppose $\lambda\in \mathscr{S}_n^\Lambda$. For any permutation $w \in \mathfrak{S}_n$ with reduced expression $w = s_{r_1}s_{r_2}\ldots s_{r_{m-1}}s_{r_m}$ and $r = \text{min}\{r_1,r_2,\ldots,r_m\}$, if we write
e_\lambda y_\lambda \psi_w = e_\lambda y_\lambda \psi_{r_1}\psi_{r_2}\ldots \psi_{r_m} = \sum_{\u,\v\in\Std(\geq\lambda)} c_{\u \v} \psi_{\u \v},
then $c_{\u \v} \neq 0$ implies $\v|_k \unrhd \t^{\lambda|_k}$ for any $k < r$.

\end{Lemma}

\proof We prove the Lemma by induction. If $m = 1$, then $r_1 = r$.
e_\lambda y_\lambda \psi_r = \begin{cases}
\psi_{\t^\lambda \v}, & \text{ if $\v = \t^\lambda{\cdot}s_r$ is standard,}\\
\sum_{(\u,\v) \rhd (\t^\lambda,\t^\lambda)} c_{\u\v} \psi_{\u\v}, & \text{ otherwise.}
\end{cases}

If $=̌ ^λ·s_r$ is standard, then by the definition of $$̌, $\v|_k = \t^\lambda|_k = \t^{\lambda|_k}$ for $k < r$. If it is the other case, as $\v \rhd \t^\lambda$, then $\v|_k \unrhd \t^\lambda|_k = \t^{\lambda|_k}$.

Assume for any $m' < m$ the Corollary holds. Then

e_\lambda y_\lambda \psi_{r_1}\psi_{r_2}\ldots \psi_{r_{m-1}} = \sum_{\u_1,\v_1 \in \Std(\geq\lambda)} c_{\u_1\v_1}\psi_{\u_1\v_1},

where $\v_1|_k \unrhd \t^{\lambda|_k}$ for $k < r$. Then

e_\lambda y_\lambda \psi_{r_1}\psi_{r_2}\ldots \psi_{r_{m-1}}\psi_{r_m} = \sum_{\u_1,\v_1 \in \Std(\geq\lambda)}c_{\u_1\v_1}\psi_{\u_1\v_1}\psi_{r_m}.

\psi_{\u_1\v_1}\psi_{r_m} = \begin{cases}
\psi_{\u_1\v} + \sum_{(\u,\v_2) \rhd (\u_1,\v_1)} c_{\u\v_2}\psi_{\u\v_2}, & \text{ if $\v = \v_1{\cdot}s_{r_m}$ is standard and $d(\v_1){\cdot}s_{r_m}$ is reduced,}\\
\sum_{(\u,\v) \rhd (\u_1,\v_1)} c_{\u\v}\psi_{\u\v}, & \text{ otherwise.}
\end{cases}

If $\v = \v_1{\cdot}s_{r_m}$ is standard and $d(\v_1){\cdot}s_{r_m}$ is reduced, recall $\v_1|_k \unrhd \t^{\lambda|_k}$ for $k < r$, as $\v = \v_1{\cdot}s_{r_m}$, $\v|_k = \v_1|_k \unrhd \t^{\lambda|_k}$ for $k < r \leq r_m$. For $\v_2 \rhd \v_1$, we have $\v_2|_k \rhd \v_1|_k \unrhd \t^{\lambda|_k}$ for $k < r$.

If it is of the other case, as $\v \rhd \v_1$, $\v|_k \rhd \v_1|_k \unrhd \t^{\lambda|_k}$. Therefore

e_\lambda y_\lambda \psi_w = e_\lambda y_\lambda \psi_{r_1}\psi_{r_2}\ldots \psi_{r_m} = \sum_{\u,\v\in\Std(\geq\lambda)} c_{\u \v} \psi_{\u \v},

and $c_{\u \v} \neq 0$ implies $\v|_k \unrhd \t^{\lambda|_k}$ for any $k < r$. This completes the proof.

Suppose $\lambda \in \mathscr{S}_n^\Lambda \cap (\P_I \cap \P_y \cap \P_\psi)$. Then for any $1 \leq r_1,r_2,\ldots,r_m \leq n-1$

e_\lambda y_\lambda \psi_{r_1}\psi_{r_2} \ldots \psi_{r_m} = \sum_{\v\in\Std(\lambda)} c_{\t^\lambda\v}\psi_{\t^\lambda\v} + \sum_{\substack{\u,\v\in\Std(>\lambda)\\\u \rhd \t^\lambda}} c_{\u\v}\psi_{\u\v}.

When $m = 1$, we have

e_\lambda y_\lambda \psi_{r_1} = \begin{cases}
\psi_{\t^\lambda \v}, & \text{ if $\v = \t^\lambda{\cdot}s_{r_1}$ is standard,}\\
\sum_{(\u,\v) \rhd (\t^\lambda, \t^\lambda)} c_{\u\v} \psi_{\u\v} = \sum_{\substack{\u,\v\in\Std(>\lambda)\\ \u \rhd \t^\lambda}} c_{\u\v}\psi_{\u\v}, & \text{ if $\v = \t^\lambda{\cdot}s_{r_1}$ is not standard.}
\end{cases}

which follows the Lemma.

Suppose for $m' < m$ the Lemma holds. Then by induction

\begin{equation}\label{psi_n-1 is not involved: help1}
e_\lambda y_\lambda \psi_{r_1}\ldots\psi_{r_{m-1}} \psi_{r_m} = \sum_{\v_1 \in \Std(\lambda)} c_{\t^\lambda\v_1}\psi_{\t^\lambda\v_1}\psi_{r_m} + \sum_{\substack{\u_1,\v_1 \in \Std(>\lambda)\\ \u_1 \rhd \t^\lambda}} c_{\u_1\v_1}\psi_{\u_1\v_1}\psi_{r_m}.
\end{equation}

For $\v_1 \in \Std(\lambda)$, as $\lambda\in \P_\psi$,

\psi_{\t^\lambda\v_1}\psi_{r_m} = \begin{cases}
\psi_{\t^\lambda \v_2} + \sum_{(\u_2,\v_2)\rhd (\t^\lambda,\v_1)}c_{\u_2\v_2}\psi_{\u_2\v_2}, & \text{ if $\v_2 = \v_1{\cdot}s_{r_m}$ is standard}\\
& \hspace*{10mm}\text{and $d(\v_2) = d(\v_1){\cdot}s_{r_m}$ is reduced},\\
\sum_{(\u_2,\v_2)\rhd (\t^\lambda,\v_1)}c_{\u_2\v_2}\psi_{\u_2\v_2}, & \text{ if $\v_2 = \v_1{\cdot}s_{r_m}$ is not standard}\\
& \hspace*{10mm}\text{or $d(\v_2) = d(\v_1){\cdot}s_{r_m}$ is not reduced}.
\end{cases}

where in both cases, we can write

\begin{equation}\label{psi_n-1 is not involved: help2}
\psi_{\t^\lambda\v_1}\psi_{r_m} = \sum_{\v_2 \in \Std(\lambda)} c_{\t^\lambda\v_2}\psi_{\t^\lambda\v} + \sum_{\substack{\u_2,\v_2 \in \Std(>\lambda)\\ \u_2 \rhd \t^\lambda}} c_{\u_2\v_2}\psi_{\u_2\v_2}.
\end{equation}

For $\u_1,\u_2 \in \Std(>\lambda)$,

\psi_{\u_1\v_1}\psi_{r_m} = \begin{cases}
\psi_{\u_1 \v_2} + \sum_{(\u_2,\v_2)\rhd (\u_1,\v_1)}c_{\u_2\v_2}\psi_{\u_2\v_2}, & \text{ if $\v_2 = \v_1{\cdot}s_{r_m}$ is standard}\\
& \hspace*{10mm}\text{and $d(\v_2) = d(\v_1){\cdot}s_{r_m}$ is reduced},\\
\sum_{(\u_2,\v_2)\rhd (\u_1,\v_1)}c_{\u_2\v_2}\psi_{\u_2\v_2}, & \text{ if $\v_2 = \v_1{\cdot}s_{r_m}$ is not standard}\\
&\hspace*{10mm}\text{or $d(\v_2) = d(\v_1){\cdot}s_{r_m}$ is not reduced}.
\end{cases}

where since $\u_1 \rhd \t^\lambda$, we can always write

\begin{equation} \label{psi_n-1 is not involved: help3}
\psi_{\u_1\v_1}\psi_{r_m} = \sum_{\substack{\u_2,\v_2 \in \Std(>\lambda)\\\u_2 \rhd \t^\lambda}} c_{\u_2\v_2}\psi_{\u_2\v_2}.
\end{equation}

Therefore, substitute (<ref>) and (<ref>) back to (<ref>), we have

e_\lambda y_\lambda \psi_{r_1}\psi_{r_2} \ldots \psi_{r_m} = \sum_{\v\in\Std(\lambda)} c_{\t^\lambda\v}\psi_{\t^\lambda\v} + \sum_{\substack{\u,\v\in\Std(>\lambda)\\ \u \rhd \t^\lambda}} c_{\u\v}\psi_{\u\v},

which completes the proof.

Suppose $\lambda \in \mathscr S_n^\Lambda$ and $r_1,r_2,\ldots,r_m$ are positive integers such that $r_1,\ldots,r_m < n-1$. Then

e_\lambda y_\lambda \psi_{r_1}\psi_{r_2} \ldots \psi_{r_m} \in \Bgelam.

Define $\mu = \lambda|_{n-1}$. As $\lambda \in \mathscr S_n^\Lambda$, $\mu \in \mathscr S_{n-1}^\Lambda\cap(\P_I \cap \P_y \cap \P_\psi)$. Define $i \in I$ such that $\bi_\lambda = \bi_\mu\vee i$. As $r_1,r_2,\ldots,r_m < n-1$, we have

e_\lambda y_\lambda \psi_{r_1}\psi_{r_2}\ldots \psi_{r_m} = \theta_i(e_\mu y_\mu \psi_{r_1}\psi_{r_2} \ldots \psi_{r_m}),

e_\mu y_\mu \psi_{r_1}\psi_{r_2} \ldots \psi_{r_m} = \sum_{\dot\v\in\Std(\mu) } c_{\t^\mu \dot\v} \psi_{\t^\mu \dot\v} + \sum_{\dot\u,\dot\v\in\Std(>\mu)} c_{\dot\u\dot\v} \psi_{\dot\u\dot\v}.

As $\sum_{\dot\u,\dot\v\in\Std(>\mu)} c_{\dot\u\dot\v} \psi_{\dot\u\dot\v} \in\Blam[\mu] = \Blam[\lambda|_{n-1}]$, by <ref>, $\theta_i(\sum_{\dot\u,\dot\v\in\Std(>\mu)} c_{\dot\u\dot\v} \psi_{\dot\u\dot\v}) \in \Blam $.

For $\dot\v \in \Std(\mu) = \Std(\lambda|_{n-1})$ and $\bi_\mu\vee i = \bi_\lambda$, define $\v$ to be the standard $λ$-tableau with $|̌_n-1 = ̌̇$. Hence $θ_i(ψ_^μ̌̇) = ψ_^λ$. Therefore
\theta_i(\sum_{\dot\v\in\Std(\mu) } c_{\t^\mu \dot\v} \psi_{\t^\mu \dot\v}) = \sum_{\v\in\Std(\lambda) } c_{\t^\mu \dot\v} \psi_{\t^\lambda \v} \in \Bgelam.

e_\mu y_\mu \psi_{r_1}\psi_{r_2} \ldots \psi_{r_m} = \theta_i(\sum_{\dot\v\in\Std(\mu) } c_{\t^\mu \dot\v} \psi_{\t^\mu \dot\v}) + \theta_i(\sum_{\dot\u,\dot\v\in\Std(>\mu)} c_{\dot\u\dot\v} \psi_{\dot\u\dot\v})  \in \Bgelam.
\endproof

\subsection{Garnir tableaux}

In the following subsections we will prove that $ψ_ψ_r ∈$ for $l(d()) ≤m_λ$. Generally, if $·s_r$ is standard and $d()·s_r$ is reduced or $l(d())·s_r$ is not reduced, it is comparatively easy to prove that $ψ_ψ_r∈$. Our main difficulty is to prove that when $·s_r$ is not standard then $ψ_ψ_r∈$. In order to prove this we consider different types of $$. Among these cases the hardest part is that when $$ is a special kind of tableaux which is called the Garnir tableau and $·s_r$ is not standard. In this subsection we will prove that in such case $ψ_ψ_r ∈$.

The method of proving the argument in this subsection is first assuming that $()$ is a partition of two rows, and using the similar argument we used in the last section to extend the result to general multipartitions. First we give a detailed definition of garnir tableaux.

We introduce a special kind of tableaux, the Garnir tableaux, which was first introduced by Murphy~[22]. Let $(a,b,m)$ be a node of $λ$ such that $(a+1,b,m)$ is also a node of $λ$. The \textbf{$(a,b,m)$-Garnir belt} of $λ$ consists of the nodes $(a,c,m)$ for $b ≤c≤λ_a^(m)$ and the nodes $(a+1,g,m)$ for $1≤g≤b$. For example here is a picture of the $(2,3,2)$-Garnir belt for $λ= (3,1|7,6,5,2)$.
\begin{eqnarray*}
&&\tab(\ \ \ ,\ )\\
&&\tab(\ \ \ \ \ \ \ ,\ \ \times\times\times\times,\times\times\times\ \ ,\ \ )
\end{eqnarray*}

The \textbf{$(a,b,m)$-Garnir tableau} of shape $λ$ is the unique maximal standard $λ$-tableau with respect to the Bruhat order ($$) among the standard $λ$-tableaux which agree with $^λ$ outside the $(a,b,m)$-Garnir belt. For example the following is the $(2,3,2)$-Garnir tableau for $λ= (3,1|7,6,5,2)$.
\Bigg(\hspace*{1mm}\tab(123,4)\hspace*{1mm}\Bigg|\hspace*{1mm}
\tab(56789\ten\eleven,\twelve\thirteen\sixteen\eighteen\nineteen\twenty,\fourteen\fifteen\seventeen\twentyone\twentytwo,\twentythree,\twentyfour)\hspace*{1mm}\Bigg)

Suppose $λ= (λ^(1),…,λ^(ł))$ and $λ^(ł) = (λ_1^(ł),…,λ_k^(ł))$. Let $(a,b,m) = (k-1,λ_k^(ł),ł)$ and $$ be the $(a,b,m)$-Garnir tableau. Let the entry in node $(a,b,m)$ of $$ be $r$. Then $·s_r$ is not standard.

\begin{Definition}

Suppose $\lambda \in \P_n$ with $\lambda = (\lambda^{(1)},\ldots,\lambda^{(\l)})$ and $\lambda^{(\l)} = (\lambda^{(\l)}_1,\ldots,\lambda^{(\l)}_k)$. If $k \geq 2$, and $\t$ is the $(k-1,\lambda^{(\l)}_k,\l)$-Garnir tableau, then we call $\t$ the \textbf{last Garnir tableau} of shape $\lambda$, and $r = \t(k-1,\lambda^{(\l)}_k,\l)$ the \textbf{last Garnir entry} of $\t$.

\end{Definition}

For example
\Bigg(\hspace*{1mm}\tab(123,4)\hspace*{1mm}\Bigg|\hspace*{1mm}
\tab(56789\ten\eleven,\twelve\thirteen\sixteen\eighteen\nineteen\twenty,\fourteen\fifteen\seventeen)\hspace*{1mm}\Bigg)
is the last $(2,3,2)$-Garnir tableau, and
\Bigg(\hspace*{1mm}\tab(123,4)\hspace*{1mm}\Bigg|\hspace*{1mm}
\tab(56789\ten\eleven,\twelve\thirteen\sixteen\eighteen\nineteen\twenty,\fourteen\fifteen\seventeen\twentyone\twentytwo,\twentythree\twentyfour)\hspace*{1mm}\Bigg)
is not the last one. Notice that $·s_r$ is not standard.\\

Because we are going to play around with $ψ_d()$ a lot, we introduce more detailed notation for these elements in the next Lemma.

\begin{Lemma} \label{standard expression of garnir tableau}
Suppose $\lambda = (\lambda^{(1)},\ldots,\lambda^{(\l)})$. Let $\t$ be a $(a,b,m)$-Garnir tableau of shape $\lambda$ and $\lambda^{(m)} = (\lambda_1^{(m)},\ldots,\lambda_k^{(m)})$. Suppose
\begin{cases}
\t^\lambda(a,b,m) & = l,\\
\t^\lambda(a,\lambda_a^{(m)},m) & = s,\\
\t^\lambda(a+1,b,m) & = t.
\end{cases}

Then $l \leq s < t$. Write $t - s = c$,
\psi_{d(\t)} = \psi_s\psi_{s+2} \ldots \psi_{t-1}{\cdot} \psi_{s-1}\psi_s\ldots \psi_{t-2}{\cdot}\ldots{\cdot}\psi_{l+1}\psi_{l+2}\ldots\psi_{l+c}{\cdot}\psi_l\psi_{l+1}\ldots\psi_{l+c-2}
l(\psi_s\psi_{s+1}\ldots \psi_{t-1}) = l(\psi_{s-1}\psi_{s}\ldots\psi_{t-2}) = \ldots = l(\psi_{l+1}\psi_{l+2}\ldots\psi_{l+c}) = c
l(\psi_l\psi_{l+1}\ldots\psi_{l+c-2}) = c-1.
\end{Lemma}

\proof The Lemma follows by direct calculation. \endproof

\begin{Example}
Suppose $\lambda = (3,1|7,6,5,2)$ and $(a,b,m) = (2,3,2)$. Let $\t$ be the $(2,3,2)$-Garnir tableau of shape $\lambda$. Then
\t =
\Bigg(\hspace*{1mm}\tab(123,4)\hspace*{1mm}\Bigg|\hspace*{1mm}
\tab(56789\ten\eleven,\twelve\thirteen\sixteen\eighteen\nineteen\twenty,\fourteen\fifteen\seventeen\twentyone\twentytwo,\twentythree\twentyfour)\hspace*{1mm}\Bigg),
\t^\lambda =
\Bigg(\hspace*{1mm}\tab(123,4)\hspace*{1mm}\Bigg|\hspace*{1mm}
\tab(56789\ten\eleven,\twelve\thirteen\fourteen\fifteen\sixteen\seventeen,\eighteen\nineteen\twenty\twentyone\twentytwo,\twentythree\twentyfour)\hspace*{1mm}\Bigg).

\begin{array}{lll}
\t^\lambda(a,b,m) & = \t^\lambda(2,3,2) & = 14,\\
\t^\lambda(a,\lambda_a^{(m)},m) & = \t^\lambda(2,6,2) & = 17,\\
\t^\lambda(a+1,b,m) & = \t^\lambda(3,3,2) & = 20,
\end{array}
and $c = t - s = 3$. Therefore
\psi_{d(\t)} = \psi_{17}\psi_{18}\psi_{19}{\cdot}\psi_{16}\psi_{17}\psi_{18}{\cdot}\psi_{15}\psi_{16}\psi_{17}{\cdot}\psi_{14}\psi_{15}
l(\psi_{17}\psi_{18}\psi_{19}) = l(\psi_{16}\psi_{17}\psi_{18}) = l(\psi_{15}\psi_{16}\psi_{17}) = 3 = c
l(\psi_{14}\psi_{15}) = 2 = c-1.
\end{Example}

\begin{Remark}
For $a \leq b-1$, we will write $\psi_{a,b} = \psi_a \psi_{a+1} \psi_{a+2} \ldots \psi_{b-2} \psi_{b-1}$ and $\psi_{b,a} = \psi^*_{a,b}$ in order to simplify our notations.
\end{Remark}

Our first step is to prove that when $λ$ is a partition with two rows and $$ is the last Garnir tableau of shape $λ$ with $r$ as its last Garnir entry, then $ψ_ψ_r ∈$ for any $∈(λ)$. We set $λ= (λ_1,λ_2)$ and without loss of generality, set $Λ= Λ_0$. Therefore $λ∈_n$ with $n = λ_1 + λ_2$. Also we set $μ= (λ_1,λ_2-1,1)$, $λ̇= (λ_1-1,λ_2)$ and $μ̇= (λ_1-1,λ_2-1,1)$. Furthermore, let $i = (γ_1)$, $j = (γ_2)$, where $γ_1 = (1,λ_1,1)$ and $γ_2 = (2,λ_2,1)$.

First we prove a few useful Lemmas.
\begin{Lemma}\label{swap}
Suppose $\lambda$, $\dot\lambda$, $\dot\mu$, $i$ and $j$ are defined as above, we have
e_\lambda y_\lambda \psi_{\lambda_1}\psi_{\lambda_1+1}\ldots \psi_{n-2}\psi_{n-1} =_\lambda \begin{cases}
\psi_{\lambda_1}\psi_{\lambda_1+1}\ldots\psi_{n-2}\psi_{n-1} e(\bi_{\dot{\lambda}}\vee i)y_{\dot{\lambda}}y_n\\
\hspace*{14mm} - \psi_{\lambda_1+1}\ldots\psi_{n-2}\psi_{n-1} e(\bi_{\dot{\lambda}}\vee i)y_{\dot{\lambda}}, & \text{ if $i = e-1$, $j\neq e-1$,}\\
\psi_{\lambda_1}\psi_{\lambda_1+1}\ldots\psi_{n-2}\psi_{n-1} e(\bi_{\dot{\lambda}}\vee i)y_{\dot{\lambda}}y_n\\
\hspace*{14mm} - \psi_{\lambda_1+1}\ldots\psi_{n-2}\psi_{n-1} e(\bi_{\dot{\lambda}}\vee i)y_{\dot{\lambda}}\\
\hspace*{28mm} - \psi_{\lambda_1}\ldots\psi_{n-2} e(\bi_{\dot{\lambda}}\vee i)y_{\dot{\lambda}}, & \text{ if $i = j = e-1$,}\\
\psi_{\lambda_1}\psi_{\lambda_1+1}\ldots\psi_{n-2}\psi_{n-1} e(\bi_{\dot{\lambda}}\vee i)y_{\dot{\lambda}}\\
\hspace*{14mm} + \psi_{\lambda_1}\psi_{\lambda_1+1}\ldots\psi_{n-2} e(\bi_{\dot{\mu}}\vee i)y_{\dot{\mu}}, & \text{ if $i = j = e-2$,}\\
\psi_{\lambda_1}\psi_{\lambda_1+1}\ldots\psi_{n-2}\psi_{n-1} e(\bi_{\dot{\lambda}}\vee i)y_{\dot{\lambda}}, & \text{ otherwise.}
\end{cases}
\end{Lemma}

\proof In order to make the notation and diagrams clearer we set $e = 4$. For the other choices of $e$ one can check that the argument is the same because the proof does not depend on the value of $e$.

By using the diagrammatic notation we have
e_\lambda y_\lambda \psi_{\lambda_1}\psi_{\lambda_1+1}\ldots \psi_{n-2}\psi_{n-1}
\begin{braid}
\draw(3,-0.5)node{$\underbrace{\hspace*{24mm}}_{\lambda_1-1}$};
\draw(10.5,-0.5)node{$\underbrace{\hspace*{27mm}}_{\lambda_2-1}$};
\draw(3.5,5.2)node{$\overbrace{\hspace*{27mm}}^{\lambda_1}$};
\draw(12,5.2)node{$\overbrace{\hspace*{31mm}}^{\lambda_2}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$i-1$}--(6,0);
\draw (7,4)node[above]{$i$}--(16,0);
\draw (8,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw[dots] (13.2,4)--(14.8,4);
\draw[dots] (12.2,0)--(13.8,0);
\draw (15,4)node[above]{$j-1$}--(14,0);
\draw (16,4)node[above]{$j$}--(15,0);
\node[greendot] at (3,4){};
\node[greendot] at (7,4){};
\node[below,scale = 1] at (7,4){$\delta_{i,3}$};
\node[greendot] at (11,4){};
\node[greendot] at (16,4){};
\node[below right,scale = 1] at (16,4){$\delta_{j,2}$};
\end{braid}.

Therefore, the Lemma is equivalent to moving all of the dots from the top to the bottom of the diagram. In order to do this we have to consider several cases.

\textbf{Case ~\ref{swap}a:} $i ≠e-2,e-1$.

Because we set $e = 4$ so in this case we have $i ≠2,3$. As $i ≠3$, $δ_i,3 = 0$. Therefore there are no dots on the strand labelled by $i$. And as $i ≠2$, by relation~\ref{dia:y-psi com}, we have
\begin{eqnarray*}
&&e_\lambda y_\lambda \psi_{\lambda_1}\psi_{\lambda_1+1}\ldots \psi_{n-2}\psi_{n-1}\\
& = &
\begin{braid}
\draw(3.5,5.2)node{$\overbrace{\hspace*{27mm}}^{\lambda_1}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$i-1$}--(6,0);
\draw (7,4)node[above]{$i$}--(16,0);
\draw (8,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)[densely dotted] node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw[dots] (13.2,4)--(14.8,4);
\draw[dots] (12.2,0)--(13.8,0);
\draw (15,4)node[above]{$j-1$}--(14,0);
\draw (16,4)[densely dotted] node[above]{$j$}--(15,0);
\node[greendot] at (3,4){};
\node[greendot] at (11,4){};
\node[greendot] at (16,4){};
\node[below right] at (16,4){$\delta_{j,2}$};
\draw[->] (11,4) -- (10.5,2);
\draw[->] (16,4) -- (15.02,0.08);
\end{braid}
\begin{braid}
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$i-1$}--(6,0);
\draw (7,4)node[above]{$i$}--(16,0);
\draw (8,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw[dots] (13.2,4)--(14.8,4);
\draw[dots] (12.2,0)--(13.8,0);
\draw (15,4)node[above]{$j-1$}--(14,0);
\draw (16,4)node[above]{$j$}--(15,0);
\node[greendot] at (3,0){};
\node[greendot] at (10,0){};
\node[greendot] at (15,0){};
\node[below] at (15,0){$\delta_{j,2}$};
\end{braid}\\
& = &
\psi_{\lambda_1}\psi_{\lambda_1+1}\ldots\psi_{n-2}\psi_{n-1} e(\bi_{\dot{\lambda}}\vee i)y_{\dot{\lambda}}.
\end{eqnarray*}

\textbf{Case ~\ref{swap}b:} $i = e-1$ and $j ≠e-1$.

Because we set $e = 4$ so in this case we have $i = 3$ and $j ≠3$. Then $δ_i,3 = 1$. Hence
\begin{eqnarray*}
&&e_\lambda y_\lambda \psi_{\lambda_1}\psi_{\lambda_1+1}\ldots \psi_{n-2}\psi_{n-1}
\begin{braid}
\draw(3.5,5.2)node{$\overbrace{\hspace*{27mm}}^{\lambda_1}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (7,4)[densely dotted] node[above]{$3$}--(20,0);
\draw (8,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)node[above]{$j$}--(19,0);
\node[greendot] at (3,4){};
\node[greendot] at (7,4){};
\node[greendot] at (11,4){};
\node[greendot] at (15,4){};
\node[greendot] at (20,4){};
\node[below right] at (20,4){$\delta_{j,2}$};
\draw[->] (7,4) -- (19.5, 4-3.815);
\end{braid}\\
&\hspace*{-5mm} \overset{(\ref{dia:y-psi com})}=&
\draw(3.5,5.2)node{$\overbrace{\hspace*{27mm}}^{\lambda_1}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (8,4)node[above]{$3$}--(20,0);
\draw (7,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)node[above]{$j$}--(19,0);
\node[greendot] at (3,4){};
\node[greendot] at (11,4){};
\node[greendot] at (15,4){};
\node[greendot] at (20,4){};
\node[below right] at (20,4){$\delta_{j,2}$};
\end{braid}
\begin{braid}
\draw(3.5,5.2)node{$\overbrace{\hspace*{27mm}}^{\lambda_1}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (12,4)node[above]{$3$}--(20,0);
\draw (8,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (7,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)node[above]{$j$}--(19,0);
\node[greendot] at (3,4){};
\node[greendot] at (11,4){};
\node[greendot] at (15,4){};
\node[greendot] at (20,4){};
\node[below right] at (20,4){$\delta_{j,2}$};
\end{braid}\\
&\hspace*{-5mm} - &
\begin{braid}
\draw(3.5,5.2)node{$\overbrace{\hspace*{27mm}}^{\lambda_1}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (16,4)node[above]{$3$}--(20,0);
\draw (8,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (7,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)node[above]{$j$}--(19,0);
\node[greendot] at (3,4){};
\node[greendot] at (11,4){};
\node[greendot] at (15,4){};
\node[greendot] at (20,4){};
\node[below right] at (20,4){$\delta_{j,2}$};
\end{braid}
-  \ldots  +
\begin{braid}
\draw(3.5,5.2)node{$\overbrace{\hspace*{27mm}}^{\lambda_1}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (7,4)node[above]{$3$}--(20,0);
\draw (8,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)node[above]{$j$}--(19,0);
\node[greendot] at (3,4){};
\node[greendot] at (20,0){};
\node[greendot] at (11,4){};
\node[greendot] at (15,4){};
\node[greendot] at (20,4){};
\node[below right] at (20,4){$\delta_{j,2}$};
\end{braid}\\
& \hspace*{-5mm} \overset{(\ref{dia:ii3})}= &
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (8,4)node[above]{$3$}--(20,0);
\draw (7,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)node[above]{$j$}--(19,0);
\node[greendot] at (3,4){};
\node[greendot] at (11,4){};
\node[greendot] at (15,4){};
\node[greendot] at (20,4){};
\node[below right] at (20,4){$\delta_{j,2}$};
\end{braid} +
\begin{braid}
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (7,4)node[above]{$3$}--(8,3)--(8,1)--(7,0);
\draw (8,4)node[above]{$3$}--(7,3)--(7,1)--(11,0);
\draw (9,4)node[above]{$0$}--(9,1)--(8,0);
\draw (10,4)node[above]{$1$}--(10,1)--(9,0);
\draw (11,4)node[above]{$2$}--(11,1)--(10,0);
\draw (12,4)node[above]{$3$}--(20,0);
\draw (13,4)node[above]{$0$}--(12,3)--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)node[above]{$j$}--(19,0);
\node[greendot] at (3,2){};
\node[greendot] at (7,2){};
\node[greendot] at (11,2){};
\node[greendot] at (15,4){};
\node[greendot] at (20,4){};
\node[below right] at (20,4){$\delta_{j,2}$};
\draw[dashed, color = red] (-0.5,3) -- (12.5,3) -- (12.5,1) -- (-0.5,1) -- (-0.5,3);
\end{braid}\\
&\hspace*{-5mm} + &
\begin{braid}
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (7,4)node[above]{$3$}--(8,3)--(8,1)--(7,0);
\draw (8,4)node[above]{$3$}--(7,3)--(7,1)--(15,0);
\draw (9,4)node[above]{$0$}--(9,1)--(8,0);
\draw (10,4)node[above]{$1$}--(10,1)--(9,0);
\draw (11,4)node[above]{$2$}--(11,1)--(10,0);
\draw (12,4)node[above]{$3$}--(12,1)--(11,0);
\draw (13,4)node[above]{$0$}--(13,1)--(12,0);
\draw (14,4)node[above]{$1$}--(14,1)--(13,0);
\draw (15,4)node[above]{$2$}--(15,1)--(14,0);
\draw (16,4)node[above]{$3$}--(20,0);
\draw (17,4)node[above]{$0$}--(16,3)--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)node[above]{$j$}--(19,0);
\node[greendot] at (3,2){};
\node[greendot] at (7,2){};
\node[greendot] at (11,2){};
\node[greendot] at (15,2){};
\node[greendot] at (20,4){};
\node[below right] at (20,4){$\delta_{j,2}$};
\draw[dashed, color = red] (-0.5,3) -- (16.5,3) -- (16.5,1) -- (-0.5,1) -- (-0.5,3);
\end{braid} +\ \ \ \   \ldots\ldots\ldots\ldots \ \ \ \text{by \autoref{send}}\\
&\hspace*{-5mm} + &
\begin{braid}
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (7,4)node[above]{$3$}--(20,0);
\draw (8,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)node[above]{$j$}--(19,0);
\node[greendot] at (3,4){};
\node[greendot] at (20,0){};
\node[greendot] at (11,4){};
\node[greendot] at (15,4){};
\node[greendot] at (20,4){};
\node[below right] at (20,4){$\delta_{j,2}$};
\end{braid}
\end{eqnarray*}

\begin{eqnarray*}
&\hspace*{-5mm} =_\lambda &
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (8,4)node[above]{$3$}--(20,0);
\draw (7,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)[densely dotted] node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)[densely dotted] node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)[densely dotted] node[above]{$j$}--(19,0);
\node[greendot] at (3,4){};
\node[greendot] at (11,4){};
\node[greendot] at (15,4){};
\node[greendot] at (20,4){};
\node[below right] at (20,4){$\delta_{j,2}$};
\draw[->] (11,4) -- (10.5,2);
\draw[->] (15,4) -- (14.25,1);
\draw[->] (20,4) -- (19.0,0);
\end{braid} +
\begin{braid}
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (7,4)node[above]{$3$}--(20,0);
\draw (8,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)[densely dotted] node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)[densely dotted] node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)[densely dotted] node[above]{$j$}--(19,0);
\node[greendot] at (3,4){};
\node[greendot] at (20,0){};
\node[greendot] at (11,4){};
\node[greendot] at (15,4){};
\node[greendot] at (20,4){};
\node[below right] at (20,4){$\delta_{j,2}$};
\draw[->] (11,4) -- (10.5,2);
\draw[->] (15,4) -- (14.25,1);
\draw[->] (20,4) -- (19,0);
\end{braid}\\
&\hspace*{-5mm} \overset{(\ref{dia:y-psi com})}= &
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (8,4)node[above]{$3$}--(20,0);
\draw (7,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)node[above]{$j$}--(19,0);
\node[greendot] at (3,0){};
\node[greendot] at (10,0){};
\node[greendot] at (14,0){};
\node[greendot] at (19,0){};
\node[below right] at (19,0){$\delta_{j,2}$};
\end{braid} +
\begin{braid}
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (7,4)node[above]{$3$}--(20,0);
\draw (8,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$j-1$}--(18,0);
\draw (20,4)node[above]{$j$}--(19,0);
\node[greendot] at (3,0){};
\node[greendot] at (10,0){};
\node[greendot] at (14,0){};
\node[greendot] at (19,0){};
\node[below right] at (19,0){$\delta_{j,2}$};
\node[greendot] at (20,0){};
\end{braid}\\
&\hspace*{-5mm} = & -\psi_{\lambda_1+1}\ldots\psi_{n-2}\psi_{n-1} e(\bi_{\dot{\lambda}}\vee i)y_{\dot{\lambda}} + \psi_{\lambda_1}\psi_{\lambda_1+1}\ldots\psi_{n-2}\psi_{n-1} e(\bi_{\dot{\lambda}}\vee i)y_{\dot{\lambda}}y_n.
\end{eqnarray*}

\textbf{Case ~\ref{swap}c:} $i = j = e-1$.

Because we set $e = 4$ so in this case we have $i = j = 3$. Similarly as in Case \ref{swap}b, we have
\begin{eqnarray*}
&&e_\lambda y_\lambda \psi_{\lambda_1}\psi_{\lambda_1+1}\ldots \psi_{n-2}\psi_{n-1}\\
&\hspace*{-10mm} = &
\draw(3.5,5.2)node{$\overbrace{\hspace*{27mm}}^{\lambda_1}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (8,4)node[above]{$3$}--(20,0);
\draw (7,4)node[above]{$3$}--(7,0);
\draw (9,4)node[above]{$0$}--(8,0);
\draw (10,4)node[above]{$1$}--(9,0);
\draw (11,4)node[above]{$2$}--(10,0);
\draw (12,4)node[above]{$3$}--(11,0);
\draw (13,4)node[above]{$0$}--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$2$}--(18,0);
\draw (20,4)node[above]{$3$}--(19,0);
\node[greendot] at (3,4){};
\node[greendot] at (11,4){};
\node[greendot] at (15,4){};
\node[greendot] at (19,4){};
\end{braid} -
\begin{braid}
\draw(3.5,5.2)node{$\overbrace{\hspace*{27mm}}^{\lambda_1}$};
\draw (0,4)node[above]{$0$}--(0,0);
\draw (1,4)node[above]{$1$}--(1,0);
\draw (2,4)node[above]{$2$}--(2,0);
\draw (3,4)node[above]{$3$}--(3,0);
\draw (4,4)node[above]{$0$}--(4,0);
\draw[dots] (4.2,4)--(5.8,4);
\draw[dots] (4.2,0)--(5.8,0);
\draw (6,4)node[above]{$2$}--(6,0);
\draw (7,4)node[above]{$3$}--(8,3)--(8,1)--(7,0);
\draw (8,4)node[above]{$3$}--(7,3)--(7,1)--(11,0);
\draw (9,4)node[above]{$0$}--(9,1)--(8,0);
\draw (10,4)node[above]{$1$}--(10,1)--(9,0);
\draw (11,4)node[above]{$2$}--(11,1)--(10,0);
\draw (12,4)node[above]{$3$}--(20,0);
\draw (13,4)node[above]{$0$}--(12,3)--(12,0);
\draw (14,4)node[above]{$1$}--(13,0);
\draw (15,4)node[above]{$2$}--(14,0);
\draw (16,4)node[above]{$3$}--(15,0);
\draw (17,4)node[above]{$0$}--(16,0);
\draw[dots] (17.2,4)--(18.8,4);
\draw[dots] (16.2,0)--(17.8,0);
\draw (19,4)node[above]{$2$}--(18,0);
\draw (20,4)node[above]{$3$}--(19,0);
\node[greendot] at (3,2){};
\node[greendot] at (7,2){};