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(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
References
- Okounkov, Andrei; Vershik, Anatoly (2004), "A New Approach to the Representation Theory of the Symmetric Groups. 2", Zapiski Seminarod POMI (In Russian) v., 307, arXiv:math.RT/0503040 (revised English version).
- Jucys, Algimantas Adolfas (1974), "Symmetric polynomials and the center of the symmetric group ring", Rep. Mathematical Phys., 5 (1): 107–112, doi:10.1016/0034-4877(74)90019-6
- Jucys, Algimantas Adolfas (1966), "On the Young operators of the symmetric group", Lietuvos Fizikos Rinkinys, 6: 163–180
- Jucys, Algimantas Adolfas (1971), "Factorization of Young projection operators for the symmetric group", Lietuvos Fizikos Rinkinys, 11: 5–10
- Murphy, G. E. (1981), "A new construction of Young's seminormal representation of the symmetric group", J. Algebra, 69 (2): 287–297, doi:10.1016/0021-8693(81)90205-2