Representations of $U(d)$. Calculation of Gelfand-Zeitlin patterns for particular vectors.

Following structure is given $\left(\mathbb{C}^d\right)^{\otimes n}$. Consider irreducible representations of $U(d)$. And consider the fully symmetric subspace $T_{\alpha}$ in $\left(\mathbb{C}^d\right)^{\otimes n}$, it has a highest weight vector with a signature $\alpha=(n,0,...,0)$.

Consider a standard orthonormal basis in $\mathbb{C}^d$: $e_i=(0,...,0,1,0,...0)^T$ with $1$ on the $i$-th place.

Now consider a multivector $v=e_1^{\otimes f_1}\otimes e_2^{\otimes f_2}\otimes e_3^{\otimes f_3}$, so that $f_1+f_2+f_3=n$ and construct its projection on $T_{\alpha}$.

What is the Gelfand-Zeitlin pattern of the vector $v_{\alpha}$ (projection of $v$ on $T_{\alpha}$)?

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