Jucys–Murphy Element

In mathematics, the Jucys–Murphy elements in the group algebra \mathbb _n of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula:

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

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

Properties

They generate a commutative subalgebra of $\mathbb$S_n. Moreover, ''X''_''n'' commutes with all elements of \mathbb _.

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(\mathbb _n of the group algebra \mathbb _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 \mathbb _n holds true:

:$(t+X_1) (t+X_2) \cdots (t+X_n)= \sum_ \sigma t^.$

Theorem ( Okounkov–Vershik): The subalgebra of \mathbb _n generated by the centers

: $Z(\mathbb$S_1, Z(\mathbb S_2, \ldots, Z(\mathbb S_, Z(\mathbb _n

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

See also

* Representation theory of the symmetric group
* Young symmetrizer

References

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{{DEFAULTSORT:Jucys-Murphy Element
Permutation groups
Representation theory
Symmetry
Representation theory of finite groups
Symmetric functions