Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 103 / 1 / c / iv / Solution

For $n=4$, take the three vectors from part (iii):Their joint Young–Jucys–Murphy element weights, in the order $(X_1,X_2,X_3,X_4)$, areThey are distinct, and part (ii) says that these operators generate $G{Z}_4$. Therefore a Gelfand–Tsetlin basis isIf an orthonormal basis is desired, use $v_1/\sqrt{2}$, $v_2/\sqrt{6}$, 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.