= Gelfand–Tsetlin basis of the standard representation
{c}
{title2=$v_j=e_1+\cdots+e_j-je_{j+1}$}
For the sum-zero <standard representation of the symmetric group>, the vectors $v_j=e_1+\cdots+e_j-je_{j+1}$ form a <Gelfand–Tsetlin basis>. The <Young–Jucys–Murphy element> $X_i$ has eigenvalue $i-1$ for $i\leq j$, $-1$ for $i=j+1$, and $i-2$ for $i>j+1$. Normalizing by $\sqrt{j(j+1)}$ gives an <orthonormal basis> corresponding to tableaux of shape $(n-1,1)$ whose second-row entry is $j+1$.
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