= Solution
For $n=4$, take the three vectors from part (iii):
$$
v_1=(1,-1,0,0),\qquad
v_2=(1,1,-2,0),\qquad
v_3=(1,1,1,-3).
$$
Their joint <Young–Jucys–Murphy element> weights, in the order $(X_1,X_2,X_3,X_4)$, are
$$
(0,-1,1,2),\qquad(0,1,-1,2),\qquad(0,1,2,-1).
$$
They are distinct, and part (ii) says that these operators generate $GZ_4$. Therefore \b[a <Gelfand–Tsetlin basis> is]
$$
\boxed{\{v_1,v_2,v_3\}}.
$$
If an <orthonormal basis> is desired, use $v_1/\sqrt2$, $v_2/\sqrt6$, and $v_3/\sqrt{12}$. The corresponding <standard Young tableaux> have shape $(3,1)$, with the second-row entry respectively $2,3,4$. Their contents agree with the three displayed weights.
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