= Young seminormal form
{c}
Let $T$ be a <standard Young tableau>, $s_i=(i\ i+1)$, and $d=c_T(i+1)-c_T(i)$. A suitable <Gelfand–Tsetlin basis> has, for an admissible pair $R=s_iT$ with tableau length increasing,
$$
s_iv_T=d^{-1}v_T+v_R,\qquad s_iv_R=(1-d^{-2})v_T-d^{-1}v_R.
$$
For a nonadmissible swap the scalar is $+1$ in a row and $-1$ in a column. The diagonal coefficient follows from the <Young–Jucys–Murphy element> relation $s_iX_is_i+s_i=X_{i+1}$, and $s_i^2=1$ forces the product of off-diagonal coefficients. One global normalization is $v_T=P_T\pi_Tv_{T_0}$ from the row-reading tableau: a reduced admissible path makes this vector nonzero and gives coefficient one on every length-increasing edge.
Back to article page
