ScalingStacks

Proof. We have

	$\displaystyle d((\varsigma^{\prime}_{i})_{11})$	$\displaystyle=E_{2}\varsigma_{i}\circ d(\tau_{2}E_{2}^{i-1}\circ\cdots\circ E_{2}^{i-1}\tau_{2})F_{1}^{i}\circ\lambda_{(i+1\cdots 2i+1)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}\varsigma_{i}\circ\lambda_{(1\cdots r)(r+1\cdots i+1)}\circ\lambda_{(i+1\cdots 2i+1)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}\varsigma_{i}\circ\lambda_{(1\cdots r)(r+1\cdots i+1)}\circ\lambda_{(i+1\cdots 2i+1)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}\varsigma_{i}\circ\lambda_{(1\cdots r)(2i+1\cdots i+r+1)}\circ\lambda_{(i+1\cdots 2i+1)}$
		$\displaystyle=E_{2}\varsigma_{i}\circ\lambda_{(1\cdots i)}$

$$(\varsigma^{\prime}_{i})_{12}\circ E_{2}^{i}F_{1}^{i}\pi=$$

		$\displaystyle=\sum_{r=1}^{i}E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varsigma_{1}\circ\lambda_{(2i,2i+1)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}F_{1}E_{2}\circ E_{2}^{i}F_{1}^{i}\eta_{1}E_{2}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}\varsigma_{i}\circ\lambda_{(i+1\cdots 2i)}\circ\lambda_{(2i,2i+1)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i}\circ\lambda_{(1\cdots i)}$
		$\displaystyle=d((\varsigma^{\prime}_{i})_{11}).$

	$\displaystyle d((\varsigma^{\prime}_{i})_{22})$	$\displaystyle=E_{1}\varsigma_{i}\circ\lambda_{(1\cdots i+1)}\circ E_{2}^{i}d(\rho F_{1}^{i-1}\circ F_{1}\rho F_{1}^{i-2}\circ\cdots\circ F_{1}^{i-1}\rho)$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{i}\circ\lambda_{(1\cdots i+r)}\circ E_{2}^{i}F_{1}^{r-1}\eta_{1}F_{1}^{i-r}\circ E_{2}^{i}F_{1}^{r-1}\varepsilon_{1}F_{1}^{i-r}\circ\lambda_{(i+r+1\cdots 2i+1)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{i}\circ\lambda_{(i+r+1\cdots 2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{i}\circ\lambda_{(i+r+1\cdots i+2)}\circ\lambda_{(i+2\cdots 2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{i}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(i+2\cdots 2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=E_{1}\varsigma_{i}\circ\lambda_{(i+2\cdots 2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+1)}$

	$\displaystyle\pi\circ(\varsigma^{\prime}_{i})_{12}$	$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{1}\circ\lambda_{(12)}\circ E_{2}\eta_{1}\circ E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{1}\circ\lambda_{(23)}\circ\eta_{1}E_{2}\circ E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{1}\circ E_{1}E_{2}F_{1}\varsigma_{i-1}\circ\lambda_{23}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{i}\circ\lambda_{(i+2\cdots 3)}\circ\lambda_{23}\circ\lambda_{(3\cdots r+2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=E_{1}\varsigma_{i}\circ\lambda_{(i+2\cdots 2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+r)}=d((\varsigma^{\prime}_{i})_{22}).$

We have

$$d((\varsigma^{\prime}_{i})_{12})=A+B$$

where

	$\displaystyle A$	$\displaystyle=\sum_{1\leq s<r\leq i}E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots s)(s+1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{1\leq s<r\leq i}E_{2}\varsigma_{i-1}\circ\lambda_{(s+1\cdots r)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots s)(2i\cdots i+r)}$
		$\displaystyle=\sum_{1\leq s<r\leq i}E_{2}\varsigma_{i-1}\circ\lambda_{(i+r-1\cdots s+i)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots s)(2i\cdots i+r)}$
		$\displaystyle=\sum_{1\leq s<r\leq i}E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots s)(2i\cdots i+r)(i+r-1\cdots i+s)}$

and

$$B=\sum_{\begin{subarray}{c}1\leq r^{\prime}\leq i\\ 1\leq s^{\prime}\leq i-r^{\prime}\end{subarray}}E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r^{\prime})(2i\cdots i+r^{\prime}+s^{\prime})(i+r^{\prime}+s^{\prime}-1\cdots i+r^{\prime})}$$

So $A=B$ and $d((\varsigma^{\prime}_{i})_{12})=0$.

We have shown that $d(\varsigma^{\prime}_{i})=0$,

Fix $r\in\{1,\ldots,i\}$. We put $b_{r}=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}:E_{2}^{i}F_{1}^{i}E_{1}(m)\to E_{2}(m)$.

Consider $s\in\{1,\ldots,i-1\}$.

If $s>r$, we have

	$\displaystyle b_{r}(T_{s}\otimes 1)$	$\displaystyle=E_{2}\varsigma_{i-1}\circ\lambda_{(s,s+1)}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ\lambda_{(i+s-1,i+s)}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(i+s-1,i+s)}\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\lambda_{(i+s,i+s+1)}$
		$\displaystyle=b_{r}(1\otimes T_{s}).$

If $s<r-1$, we have

	$\displaystyle b_{r}(1\otimes T_{s})$	$\displaystyle=E_{2}\varsigma_{i-1}\circ\lambda_{(i+s,i+s+1)}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ\lambda_{(s+1,s+2)}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(s+1,s+2)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(s,s+1)}$
		$\displaystyle=b_{r}(T_{s}\otimes 1).$

We have

$$b_{r}(T_{r-1}\otimes 1)=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(r-1,r)}=0$$

$$b_{r}(1\otimes T_{r})=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(i+r,i+r+1)}=0$$

$$b_{r}(1\otimes T_{r-1})=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r-1)}=b_{r-1}(T_{r-1}\otimes 1).$$

We have shown that $(\varsigma_{i})_{12}(1\otimes T_{s})=(\varsigma_{i})_{12}(T_{s}\otimes 1)$.

We have

	$\displaystyle(\varsigma^{\prime}_{i})_{11}(T_{s}\otimes 1)$	$\displaystyle=E_{2}\varsigma_{i}\circ\lambda_{(s+1,s+2)}\circ\lambda_{(1\cdots 2i+1)}$
		$\displaystyle=E_{2}\varsigma_{i}\circ\lambda_{(i+s+1,i+s+2)}\circ\lambda_{(1\cdots 2i+1)}$
		$\displaystyle=E_{2}\varsigma_{i}\circ\lambda_{(1\cdots 2i+1)}\lambda_{(i+s,i+s+1)}$
		$\displaystyle=(\varsigma^{\prime}_{i})_{11}(1\otimes T_{s}).$

Similarly,

$$(\varsigma^{\prime}_{i})_{22}(T_{s}\otimes 1)=(\varsigma^{\prime}_{i})_{11}(1\otimes T_{s}).$$

So $\varsigma_{i}(1\otimes T_{s})=\varsigma_{i}(T_{s}\otimes 1)$.

Let $l\in\{1,2\}$. We have

$$(\varsigma^{\prime}_{i+j})_{ll}\circ\mu_{ij}=E_{l}\varsigma_{i+j}\circ\lambda_{w}$$

where $w(r)=r$ and $w(i+r)=i+r+j+1$ for $1\leq r\leq i$, $w(2i+r)=i+r$ and $w(2i+j+r)=2i+j+r+1$ for $1\leq r\leq j$ and $w(2i+2j+1)=i+j+1$.

We have

	$\displaystyle(\varsigma^{\prime}_{i})_{ll}\circ(\varsigma^{\prime}_{j})_{ll}$	$\displaystyle=E_{l}\varsigma_{i}\circ E_{l}E_{2}^{i}F_{1}^{i}\varsigma_{j}\circ\lambda_{(1\cdot 2i+1)}\circ\lambda_{(2i+1\cdots 2i+2j+1)}$
		$\displaystyle=E_{l}\varsigma_{i+j}\circ\lambda_{w^{\prime}}\circ\lambda_{(1\cdot 2i+1)}\circ\lambda_{(2i+1\cdots 2i+2j+1)}$

where $w^{\prime}(r)=r$ for $1\leq r\leq i+1$, $w^{\prime}(i+1+r)=i+j+1+r$ for $1\leq r\leq i$, $w^{\prime}(1+2i+r)=1+i+r$ and $w^{\prime}(1+2i+j+r)=1+2i+j+r$ for $1\leq r\leq j$.

It follows that $(\varsigma^{\prime}_{i+j})_{ll}\circ\mu_{ij}=(\varsigma^{\prime}_{i})_{ll}\circ(\varsigma^{\prime}_{j})_{ll}$.

Given $l\leq l^{\prime}\leq 1$, we put $b_{l^{\prime},l}=E_{2}\varsigma_{l^{\prime}-1}\circ E_{2}^{l^{\prime}}F_{l^{\prime}-1}\varepsilon_{1}\circ\lambda_{(1\cdots l)(2l^{\prime}\cdots l^{\prime}+l)}:E_{2}^{l^{\prime}}F_{1}^{l^{\prime}}E_{1}(m)\to E_{2}(m)$. We denote by $w_{l_{1},l_{2}}$ the permutation of ${\mathfrak{S}}_{l_{1}+l_{2}}$ given by $s\mapsto s+l_{2}$ for $1\leq s\leq l_{1}$ and $s\mapsto s-l_{1}$ for $l_{1}+1\leq s\leq l_{1}+l_{2}$.

Consider $r\in\{1,\ldots,i\}$. We have

	$\displaystyle b_{i,r}\circ(\varsigma^{\prime}_{j})_{22}$	$\displaystyle=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varsigma_{j}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{2}^{j}F_{1}^{j}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(2i+1\cdots 2i+2j+1)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ\lambda_{w_{i-1,j}}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{2}^{j}F_{1}^{j}\circ\lambda_{(2i+1\cdots 2i+2j+1)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i}\lambda_{w_{i-1,j}}F_{1}^{j}\circ E_{2}^{i}F_{1}^{i-1}E_{2}^{j}F_{1}^{j-1}\varepsilon_{1}\circ\lambda_{(2i+2j\cdots 2i)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i}\lambda_{w_{i-1,j}}F_{1}^{j}\circ E_{2}^{i}F_{1}^{i-1}E_{2}^{j}F_{1}^{j-1}\varepsilon_{1}\circ\lambda_{(2i+2j\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i+j}F_{1}^{i+j-1}\varepsilon_{1}\circ E_{2}^{i}\lambda_{w_{i-1,j}}F_{1}^{j+1}E_{1}\circ\lambda_{(2i+2j\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i+j}F_{1}^{i+j-1}\varepsilon_{1}\circ\lambda_{(2i+2j\cdots i+j+r)}\circ E_{2}^{i}\lambda_{w_{i,j}}F_{1}^{j}E_{1}$
		$\displaystyle=b_{i+j,r}\circ\mu_{i,j}.$

Consider $r\in\{1,\ldots,j\}$. We have

	$\displaystyle(\varsigma^{\prime}_{i})_{11}\circ b_{j,r}$	$\displaystyle=E_{2}\varsigma_{i}\circ E_{2}^{i+1}F_{1}^{i}\varsigma_{j-1}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}E_{2}^{j}F_{1}^{j-1}\varepsilon_{1}\circ\lambda_{(2i+1\cdots 2i+r)(2i+2j\cdots 2i+j+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i+1}\lambda_{w_{i,j-1}}F_{1}^{j-1}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}E_{2}^{j}F_{1}^{j-1}\varepsilon_{1}\circ\lambda_{(2i+1\cdots 2i+r)(2i+2j\cdots 2i+j+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i+j}F_{1}^{i+j-1}\varepsilon_{1}\circ E_{2}^{i+1}\lambda_{w_{i,j-1}}F_{1}^{j}E_{1}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i+1\cdots 2i+r)(2i+2j\cdots 2i+j+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i+j}F_{1}^{i+j-1}\varepsilon_{1}\circ\lambda_{(1\cdots i+r)(2i+2j\cdots 2i+j+r)}\circ E_{2}^{i}\lambda_{w_{i,j}}F_{1}^{j}E_{1}$
		$\displaystyle=b_{i+j,j+r}\circ\mu_{i,j}.$

It follows that for all $i,j\geq 1$, we have $\varsigma_{i}\circ E_{2}^{i}F_{1}^{i}\varsigma_{j}=\varsigma_{i+j}\circ\mu_{i,j}$. ∎