 Gelfand–Tsetlin basis of the standard representation

For the sum-zero standard representation of the symmetric group, the vectors $v_j=e_1+\cdots +e_j-j{e}_{j+1}$ form a Gelfand–Tsetlin basis. The Young–Jucys–Murphy element $X_i$ has eigenvalue $i-1$ for $i\le j$, $-1$ 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 $j+1$.