Jucys–Murphy element

In mathematics, the JucysMurphy 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 (OkounkovVershik): The subalgebra of generated by the centers

is exactly the subalgebra generated by the JucysMurphy elements Xk.

See also

References

  • Okounkov, Andrei; Vershik, Anatoly (2004), "A New Approach to the Representation Theory of the Symmetric Groups. 2", Zapiski Seminarov POMI, 307, arXiv:math.RT/0503040(revised English version).
  • 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
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