Jucys–Murphy element

Template:Short descriptionTemplate:No footnotes 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:

$X_1=0,X_k=(1k)+(2k)+\cdots +(k-1k),k=2,\dots ,n.$

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

Properties

They generate a commutative subalgebra of $ℂ[S_n]$. Moreover, X_n 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 X_n. For any standard Young tableau U we have:

$X_k{v}_U=c_k(U)v_U,k=1,\dots ,n,$

where c_k(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 X_k.

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:

$(t+X_1)(t+X_2)\cdots (t+X_n)=\sum _{\sigma \in S_n}\sigma t^{\text{number of cycles of }\sigma }.$

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

$Z(ℂ[S_1]),Z(ℂ[S_2]),\dots ,Z(ℂ[S_{n-1}]),Z(ℂ[S_n])$

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

See also

• Representation theory of the symmetric group
• Young symmetrizer

References

• Template:Citation

• Template:Citation

• Template:Citation

• Template:Citation

• Template:Citation