Jucys–Murphy element
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.
Properties
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 is the content ''b − a'' of the cell occupied by k in the standard Young tableau U.
Theorem : The center of the group algebra of the symmetric group is generated by the symmetric polynomials in the elements Xk.
Theorem : 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 : The subalgebra of generated by the centers
is exactly the subalgebra generated by the Jucys-Murphy elements Xk.