Jucys–Murphy element

In mathematics, the Jucys–Murphy elements in the group algebra ${\displaystyle \mathbb {C} [S_{n}]}$ of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula:

${\displaystyle X_{1}=0,~~~X_{k}=(1\;k)+(2\;k)+\cdots +(k-1\;k),~~~k=2,\dots ,n.}$

They play an important role in the representation theory of the symmetric group.