In mathematics, the Jucys–Murphy elements in the group algebra C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} 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 C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} . Moreover, Xn commutes with all elements of C [ S n − − --> 1 ] {\displaystyle \mathbb {C} [S_{n-1}]} .
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 Z ( C [ S n ] ) {\displaystyle Z(\mathbb {C} [S_{n}])} of the group algebra C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} 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 C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} holds true:
Theorem (Okounkov–Vershik): The subalgebra of C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} generated by the centers
is exactly the subalgebra generated by the Jucys–Murphy elements Xk.
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