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Jucys–Murphy element

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In mathematics, the Jucys–Murphy elements in the group algebra 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:

X 1 = 0 ,       X k = ( 1 k ) + ( 2 k ) + ⋯ + ( k − 1   k ) ,       k = 2 , … , n .

They play an important role in the representation theory of the symmetric group.

Properties

They generate a commutative subalgebra of C [ S n ] . Moreover, Xn commutes with all elements of 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:

X k v U = c k ( U ) v U ,       k = 1 , … , n ,

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 ] ) of the group algebra 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 ] holds true:

( t + X 1 ) ( t + X 2 ) ⋯ ( t + X n ) = ∑ σ ∈ S n σ t number of cycles of  σ .

Theorem (Okounkov–Vershik): The subalgebra of C [ S n ] generated by the centers

Z ( C [ S 1 ] ) , Z ( C [ S 2 ] ) , … , Z ( C [ S n − 1 ] ) , Z ( C [ S n ] )

is exactly the subalgebra generated by the Jucys–Murphy elements Xk.

References

Jucys–Murphy element Wikipedia
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