Jucys–Murphy element

In mathematics, the Jucys–Murphy elements in the group algebra ${\displaystyle \mathbb {C} [S_{n}]}$ of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula:

${\displaystyle X_{1}=0,~~~X_{k}=(1k)+(2k)+\cdots +(k-1\ k),~~~k=2,\dots ,n.}$

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

Properties

They generate a commutative subalgebra of ${\displaystyle \mathbb {C} [S_{n}]}$. Moreover, X_n commutes with all elements of ${\displaystyle \mathbb {C} [S_{n-1}]}$.

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:

${\displaystyle X_{k}v_{U}=c_{k}(U)v_{U},~~~k=1,\dots ,n,}$

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 ${\displaystyle Z(\mathbb {C} [S_{n}])}$ of the group algebra ${\displaystyle \mathbb {C} [S_{n}]}$ 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 ${\displaystyle \mathbb {C} [S_{n}]}$ holds true:

${\displaystyle (t+X_{1})(t+X_{2})\cdots (t+X_{n})=\sum _{\sigma \in S_{n}}\sigma t^{{\text{number of cycles of }}\sigma }.}$

Theorem (Okounkov–Vershik): The subalgebra of ${\displaystyle \mathbb {C} [S_{n}]}$ generated by the centers

${\displaystyle Z(\mathbb {C} [S_{1}]),Z(\mathbb {C} [S_{2}]),\ldots ,Z(\mathbb {C} [S_{n-1}]),Z(\mathbb {C} [S_{n}])}$

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, arXiv:math.RT/0503040(revised English version).

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

• Jucys, Algimantas Adolfas (1966), "On the Young operators of the symmetric group", Lietuvos Fizikos Rinkinys, 6: 163–180

• Jucys, Algimantas Adolfas (1971), "Factorization of Young projection operators for the symmetric group", Lietuvos Fizikos Rinkinys, 11: 5–10

• 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