In mathematics, the Jucys–Murphy elements in the group algebra
of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula: They play an important role in the representation theory of the symmetric group.
They generate a commutative subalgebra of
Moreover, Xn commutes with all elements of
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: where ck(U) is the content b − a of the cell (a, b) occupied by k in the standard Young tableau U. Theorem (Jucys): The center
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
holds true: Theorem (Okounkov–Vershik): The subalgebra of