 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 $a_{i+1}-a_i=\pm 1$, the adjacent transposition $s_i$ acts on that line by $\pm 1$. Otherwise interchanging the coordinates gives a spectral vector in the same irreducible representation. These rules follow from $s_i{X}_i{s}_i+s_i=X_{i+1}$ and its two-dimensional eigenspace calculation. Together with the braid relation in a Coxeter group, they exclude the consecutive patterns $(a,a\pm 1,a)$.