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:

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

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

Properties

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

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

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

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

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

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 ${\displaystyle \mathbb {C} [S_{n}]}$ holds true:

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

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

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

is exactly the subalgebra generated by the Jucys–Murphy elements Xk.