 Jucys–Murphy element

In the group algebra $\mathbb{C}{S}_n$, setThese elements commute: for each triple $a<i<j$, the only overlapping contributions cancel as $[(ai),(ij)+(aj)]=0$. They generate the Gelfand–Tsetlin algebra, and on a standard Young tableau vector $X_r$ acts by the Content of a Young-diagram cell containing $r$.

The Jucys-Murphy elements are a set of operators that arise in the theory of symmetric groups and representations of the symmetric group algebra. They are named after the mathematicians Alexander Jucys and J. D. Murphy, who introduced them in the context of representation theory.