Jucys–Murphy element
# Jucys–Murphy element

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

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

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

## Properties

They generate a commutative subalgebra of $\mathbb {C} [S_{n}]$. Moreover, $X_n$ commutes with all elements of $\mathbb {C} [S_{n-1}]$.

The vectors constituting the basis of Young's "seminormal representation" are eigenvectors for the action of $X_n.$ For any standard Young tableau U we have:

    $X_{k}v_{U}=c_{k}(U)v_{U},~~~k=1,\dots ,n,$

where $c_k(U)$ is the content b − a of the cell (a, b) occupied by k in the standard Young tableau U.

Theorem (Jucys): The center $Z(\mathbb {C} [S_{n}])$ of the group algebra $\mathbb {C} [S_{n}]$ of the symmetric group is generated by the symmetric polynomials in the elements $X_k.$

Theorem (Jucys): Let t be a formal variable commuting with everything, then the following identity for polynomials in variable t with values in the group algebra $\mathbb {C} [S_{n}]$ holds true:

    $(t+X_{1})(t+X_{2})\cdots (t+X_{n})=\sum _{\sigma \in S_{n}}\sigma t^{{\text{number of cycles of }}\sigma }.$

Theorem (Okounkov–Vershik): The subalgebra of $\mathbb {C} [S_{n}]$ generated by the centers

    $Z(\mathbb {C} [S_{1}]),Z(\mathbb {C} [S_{2}]),\ldots ,Z(\mathbb {C} [S_{n-1}]),Z(\mathbb {C} [S_{n}])$

is exactly the subalgebra generated by the Jucys–Murphy elements $X_k.$

