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Jucys–Murphy element

From Wikipedia, the free encyclopedia

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

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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 (ab) 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 Jucys–Murphy elements Xk.

See also

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References

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  • 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).{{citation}}: CS1 maint: postscript (link)
  • 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