\otimes x_{i_{1},i_{\sigma(2)}}\cdots x_{i_{2d-1},i_{\sigma(2d)}}\\
&
=
\frac{1}{2^d}\sum_{t_1,\ldots,t_{d}\in\mathcal I_{m,n}}
\left(
(-1)_{}^{|t_1|+\sum_{s=1}^d|t_{s}|\cdot |t_{s+1}|}
\prod_{s=1}^d\varphi_{t_{s+1},t_s}
\right)\end{aligned}$$ where $t_s:=i_{2s}$ for $1\leq s\leq d$ and $t_{d+1}:=i_2$. Using and we can write $$\begin{aligned}
\widehat\sfs_d\big(
\check\rho\big(\mathrm{str}\big(\mathbf E^d\big)\big)
\big)&=
\sum_{t_1,\ldots,t_d\in\mathcal I_{m,n}}
\!\!\!\!\!\!(-1)^{|t_1|+\sum_{s=1}^d|t_s|\cdot |t_{s+1}|}
\;\widehat\sfs_d\left(
\check\rho(E_{t_2,t_1})\cdots\check\rho(E_{t_1,t_d})
\right)\\
&=
\sum_{t_1,\ldots,t_d\in\mathcal I_{m,n}}
\!\!\!\!\!\!(-1)^{|t_1|+\sum_{s=1}^d|t_s|\cdot |t_{s+1}|}
\;
\widehat\sfs_1\left(
\check\rho(E_{t_2,t_1})\right)\cdots
\widehat\sfs_1\left(
\check\rho(E_{t_1,t_d})
\right)\\
&=(-2)^d\bft_\sigma.
\qedhere\end{aligned}$$

Let $V:=\C^{m|n}$, $W:=\sS^2(V)$, $\g g:=\gl(V)=\gl(m|n)$, and $\check\rho$ be as above. Our first main result is the following theorem.

\[prpgw\] *(Abstract Capelli Theorem for $W:=\sS^2(V)$.)* For every $d\geq 0$, we have $$\sPD^{ d}(W)^{\g g}=
\check\rho\left(\mathbf{Z}(\g g)\cap \bfU^d(\g g)\right).$$

The inclusion $\supseteq $ is obvious by definition. The proof of the inclusion $\subseteq$ is by induction on $d$. The case $d=0$ is obvious. Next assume that the statement holds for all $d'<d$. Fix $D\in\sPD^{d}(W)^{\g g}$ such that $\mathrm{ord}(D)=d$. It suffices to find $z_D\in\bfZ(\g g)\cap \bfU^{d}(\g g)$ such that $
\widehat\sfs_{d}
\left(\check\rho(z_D)-D\right)=0
$.\
**Step 1.** Choose $\varphi^D\in\sP(W)\otimes \sS(W)$ such that $\sfm(\varphi^D)=D$. We can write $$\varphi^D=\varphi^D_0+
\cdots+\varphi^D_{d}
\text{ where }\varphi^D_r\in\sP(W)\otimes \sS^r(W)
\text{ for }0\leq r\leq d,$$ and Lemma \[prp-g-inv-sPWW\] implies that $\varphi^D_r\in\left(\sP^r(W)\otimes \sS^r(W)\right)^{\g g}$ for $0\leq r\leq d$. Furthermore, $$\widehat\sfs_{d}(D)=
\widehat\sfs_{d}\left(\sfm
\left(\varphi^D\right)\right)
=
\widehat\sfs_{d}\left(\sfm
\left(\varphi_{d}^D\right)\right)=
\varphi_{d}^D.$$ By Lemma \[prp-g-inv-sPWW\], $\varphi^D_{d}$ is a linear combination of the tensors $\bft_{\sigma}$ for $\sigma\in S_{2d}$. Thus to complete 