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(U) is the content b − a of the cell (a, b) occupied by k in the standard Young tableau U.
Theorem (Jucys): The center of the group algebra 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 holds true:
Theorem (Okounkov–Vershik): The subalgebra of generated by the centers
is exactly the subalgebra generated by the Jucys–Murphy elements Xk.
See also Representation theory of the symmetric group Young symmetrizer
References