S_{2d}$, let $\widetilde\bft_\sigma\in V^{\otimes
2d}\otimes V^{*\otimes 2d}$ be the element corresponding to $
T^{\sigma^{-1}}_{V,2d}$ via the isomorphism , where $
T^{\sigma^{-1}}_{V,2d}
$ is defined as in . It is easy to verify that $$\widetilde\bft_\sigma=\sum_{i_1,\ldots,i_d\in\mathcal{I}_{m,n}} (-1)_{}^{ \seps%
\left(\sigma;\bfe_{i_1},\ldots,\bfe_{i_{2d}}^{}\right) } \bfe_{i_{\sigma(1)}} \otimes
\cdots \otimes \bfe_{i_{\sigma(2d)}} \otimes \bfe_{i_{2d}}^{*}\otimes
\cdots\otimes \bfe_{i_1}^{*},$$ where $
\seps\left(\sigma;\bfe_{i_1},\ldots,\bfe_{i_{2d}}^{}\right)
$ is defined in . Now set $\sigma_\circ:=(1,2)\cdots (2d-1,2d)\in S_{2d}$ and $$H_{2d}:=\{\sigma\in
S_{2d}\,:\,\sigma\sigma_\circ=
\sigma_\circ\sigma\}.$$ Let $\psf_d^{}:V^{\otimes 2d}\to
V^{\otimes 2d}$ and $\psf_d^{*}:V^{*\otimes 2d}\to
V^{*\otimes 2d}$ be the canonical projections onto $\sS^d(W)$ and $\sS^d(W^*)\cong\sP^d(W)$, so that $$\label{dfpsfdd*}
\psf_d^{}=\frac{1}{2^d d!}\sum_{\sigma\in H_{2d}}T_{V,2d}^{\sigma}
\ \text{ and }\ 
\psf_d^*=\frac{1}{2^d d!}\sum_{\sigma\in H_{2d}}T_{V^*,2d}^{\sigma}.$$ For every $\sigma\in S_{2d}$, let $\bft_\sigma\in \sP^d(W)\otimes \sS^d(W)$ be defined by $$\bft_\sigma:= 
\left(
\braid^{}_{V^{\otimes
2d},V^{*\otimes 2d}}
\circ
(\psf^{}_d\otimes \psf^{*}_d) \right) \big(\widetilde\bft_\sigma\big),$$ where $\braid^{}_{V^{\otimes
2d},V^{*\otimes 2d}}$ is the symmetry isomorphism, defined in . The action of the linear transformation $T_{V^{},2d}^{\sigma_1^{}}\otimes
T_{V^*,2d}^{\sigma_2^{}}$ on $V^{\otimes 2d}\otimes V^{*\otimes 2d}\cong\mathrm{End}_\C(V^{\otimes 2d})$ is given by $$T\mapsto T_{V,2d}^{\sigma_1^{}}T
T_{V,2d}^{\pi\sigma_2^{-1}\pi^{-1}}
\text{ for every }T\in
\mathrm{End}_\C(V^{\otimes 2d}),$$ where $\pi\in S_{2d}$ is defined by $\pi(i)=2d+1-i$ for every $1\leq i\leq 2d$. Thus from it follows that $$\begin{aligned}
\label{tsigmeqqqqn}
\bft_\sigma&= \left(
\frac{1}{d!2^d}\right)^2 \sum_{\sigma^{\prime },\sigma^{\prime \prime
}\in H_{2d}} 
\bft_{\sigma^{\prime }\sigma\sigma^{\prime \prime }}
\\
&= \frac{1}{2^d}\sum_{i_1,\ldots,i_{2d}\in\mathcal{I}_{m,n}} \!\!\!\!\!\!(-1)^{|{i_1}|+\cdots+|{i_{2d}
}|
+
\seps
\left(
\sigma;\bfe_{i_1}^{},\ldots,\bfe_{i_{2d}}^{}\right)} y_{i_{2d-1},i_{2d}}^{}
\cdots y_{i_{1},i_{2}}^{} \otimes x_{i_{\sigma(1)},i_{\sigma(2)}}^{} \cdots
x_{i_{\sigma(2d-1)},i_{\sigma(2d)}}^{}.  