Jucys–Murphy element
Introduction 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, ''X''''n'' commutes with all elements of .The vectors constituting   the basis of Young's "seminormal representation"   are eigenvectors for the action of ''X''''n''. For any standard Young tableau ''U'' we have::where ''c''''k''(''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
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