= Solution
The first <Young–Jucys–Murphy element> is the empty sum $X_1=0$. Applying it to the nonzero simultaneous <eigenvector> defining a point of the <joint spectrum> gives $0=a_1v$. Hence
$$
\boxed{a_1=0.}
$$
For the remaining spectral arguments use the following local facts, without assuming the <restriction branching rule for a symmetric group>: consecutive <eigenvalues> are distinct; a difference $a_{r+1}-a_r=\pm1$ makes $s_r=(r\ r+1)$ act on the corresponding line as $\pm1$; every other adjacent interchange gives another point of the <joint spectrum> in the same <irreducible representation>. These are the <local spectral rules for Young–Jucys–Murphy elements>. They follow from $s_rX_rs_r+s_r=X_{r+1}$ and the one- or two-dimensional analysis of the two neighboring <eigenspaces>. They are the appropriate local results used below.
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