Jucys–Murphy element
In mathematics, the Jucys–Murphy elements in the group algebra formula_1 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. Properties. They generate a commutative subalgebra of formula_3. Moreover, "X" commutes with all elements of formula_4. The vectors constituting the basis of Young's "seminormal representation" are eigenvectors for the action of "X". For any standard Young tableau "U" we have: where "c"("U") is the "content" "b" − "a" of  the cell ("a", "b") occupied by "k" in the standard Young tableau "U". Theorem (Jucys): The center formula_6 of the group algebra formula_1 of the symmetric group is generated by the symmetric polynomials in the elements "X". 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 formula_1 holds true: Theorem (Okounkov–Vershik): The subalgebra of formula_1 generated by the centers is exactly the subalgebra generated by the Jucys–Murphy elements "X".