Jucys–Murphy element

In mathematics, the Jucys–Murphy elements in the group algebra $\mathbb{ℂ}[S_n]$ 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 $\mathbb{ℂ}[S_n]$. Moreover, X_n commutes with all elements of $\mathbb{ℂ}[S_{n-1}]$.

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 $Z(\mathbb{ℂ}[S_n])$ of the group algebra $\mathbb{ℂ}[S_n]$ 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 $\mathbb{ℂ}[S_n]$ 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 $\mathbb{ℂ}[S_n]$ generated by the centers

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

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

See also

• Representation theory of the symmetric group
• Young symmetrizer

References

• Okounkov, Andrei; Vershik, Anatoly (2004), "A New Approach to the Representation Theory of the Symmetric Groups. 2", Zapiski Seminarov POMI 307(revised English version). 

• Jucys, Algimantas Adolfas (1974), "Symmetric polynomials and the center of the symmetric group ring", Rep. Mathematical Phys. 5 (1): 107–112, doi:10.1016/0034-4877(74)90019-6, Bibcode: 1974RpMP....5..107J 

• Jucys, Algimantas Adolfas (1966), "On the Young operators of the symmetric group", Lietuvos Fizikos Rinkinys 6: 163–180, https://www.lietuvos-fizikai.lt/chessidr/straipsniai/LietFizRink/LFR-1966-v6-p179-AlgJucys-On_the_Young_operators_of_the_symmetric_groups.pdf 

• Jucys, Algimantas Adolfas (1971), "Factorization of Young projection operators for the symmetric group", Lietuvos Fizikos Rinkinys 11: 5–10, https://www.lietuvos-fizikai.lt/chessidr/straipsniai/LietFizRink/LFR-1971-v11-p5-AlgJucys-Factorization_of_Young_projection_operators_for_the_symmetric_group.pdf 

• 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