squeezes the original Young diagram $\lambda$ for a given Schur state to some “thinner” $\lambda^{\prime}$’s. To be precise after its action $$\lambda^{\prime}<\lambda \quad \hbox{meaning} \quad \sum_{i=1}^{j}\lambda_{i}^{\prime}<\sum_{i=1}^{j}\lambda_{i},\,\text{for }\, j=1,2,\cdots\,.\label{squeeze}$$ To see this triangular property in a more transparent form, we first reduce the summation area from $r+s<0,k+l>0$ to $k+l>0,k+s<0,k>r$ and $k<r$. The $k<r$ case can be obtained from re-labeling the index ($r\leftrightarrow k$) and exchanging $\psi_{r}$ and $\psi_{k}$ $$\begin{aligned}
\frac{1}{1-b^{2}}H_{t} & = & \sum_{\begin{subarray}{c}
k+l>0, r+s+k+l=0\\
s,k,l\in\mathbb{Z}+\frac{1}{2}
\end{subarray}}\left(\frac{2}{3}(2k-r)\right):\psi_{r}\psi_{s}^{*}\psi_{k}\psi_{l}^{*}:\\
 & = & \sum_{\begin{subarray}{c}
k+l>0,k+s<0\\
k>r,r+s+k+l=0
\end{subarray}}\left(\frac{2}{3}(2k-r-2r+k)\right):\psi_{r}\psi_{s}^{*}\psi_{k}\psi_{l}^{*}:\\
 & = & \sum_{\begin{subarray}{c}
k+l>0,k+s<0\\
k>r,r+s+k+l=0
\end{subarray}}2(k-r):\psi_{r}\psi_{s}^{*}\psi_{k}\psi_{l}^{*}: \, .\end{aligned}$$ $H_{t}$ is simplified and the summation area is decomposed into five cases $$\begin{aligned}
\frac{1}{1-b^{2}}H_{t} & = & 2\sum_{\begin{subarray}{c}
k+l>0,k+s<0\\
k>r,k+l+r+s=0
\end{subarray}}(k-r):\psi_{r}\psi_{s}^{*}\psi_{k}\psi_{l}^{*}:\label{eq:triangularH}\\
 & = & 2\sum_{n=1,k>r>0}^{n=s-1/2}(k-r)\psi_{-k-n}^{*}\psi_{-r+n}^{*}\psi_{r}\psi_{k}\nonumber \\
 &  & +2\sum_{n=1,k>r>0}^{n=[(k-r-1)/2]}(r-k+2n)\psi_{-k+n}\psi_{-r-n}\psi_{r}^{*}\psi_{k}^{*}\nonumber \\
 &  & +2\sum_{n=1,k,r>0}^{n=r-1/2}(k+r-n)\psi_{-r+n}\psi_{-k-n}^{*}\psi_{k}\psi_{r}^{*}\\
 &  & +2\sum_{k>r>0,s>0}(r-k)\psi_{-r-s-k}^{*}\psi_{r}\psi_{k}\psi_{s}^{*}\nonumber \\
 &  & +2\sum_{k>r>0,s>0}(k-r)\psi_{-k}\psi_{-s}^{*}\psi_{-r}\psi_{r+s+k}^{*}\,.\nonumber\end{aligned}$$ Acting on a Schur state corresponding to a Young diagram $H_t$ gives rise to the following five processes.

**Case I:**The first line of the second equality in (\[eq:triangularH\]) annihilates two columns of the corresponding Young diagram of length $r$ and $k$ (k&gt;r) and creates two new columns of length $k+n$ and $r-n$, which makes long column longer and short column shorter simultaneously. So it squeezes the Young 