= Young orthogonal form
{c}
Normalizing the tableau lines in a compatible real phase convention gives the <orthonormal basis> form
$$
s_i\big|_{\operatorname{span}(w_T,w_{s_iT})}
=\begin{pmatrix}d^{-1}&\sqrt{1-d^{-2}}\\\sqrt{1-d^{-2}}&-d^{-1}\end{pmatrix},
\qquad d=c_T(i+1)-c_T(i).
$$
Admissible swaps have $|d|\geq2$. Nonadmissible swaps act by $+1$ within a row and $-1$ within a column. The symmetric matrix is an orthogonal involution and is useful for making the <unitary representation> and its phases explicit.
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