Jucys–Murphy element explained

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, 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 , ... ,n,

where c_k(U) is the content b - a of the cell (a, b) occupied by k in the standard Young tableau U.

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)} … (t + X _{n)=} \sum _{\sigma \in S _{n}} \sigma t ^{numberofcyclesof\sigma} .

Theorem (Okounkov - Vershik): The subalgebra of

C [S _{n]} generated by the centersZ( C [ S _{1]),} Z( C [ S _{2]),} \ldots, 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