= Local spectral rules for Young–Jucys–Murphy elements
For the <joint spectrum> of the <Young–Jucys–Murphy elements>, adjacent coordinates $a_i,a_{i+1}$ are distinct. If $a_{i+1}-a_i=\pm1$, the adjacent <transposition> $s_i$ acts on that line by $\pm1$. Otherwise interchanging the coordinates gives a spectral vector in the same <irreducible representation>. These rules follow from $s_iX_is_i+s_i=X_{i+1}$ and its two-dimensional eigenspace calculation. Together with the <braid relation in a Coxeter group>, they exclude the consecutive patterns $(a,a\pm1,a)$.
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