Jucys–Murphy element
In mathematics, the Jucys–Murphy elements in the group algebra [math]\displaystyle{ \mathbb{C} [S_n] }[/math] of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula:
- [math]\displaystyle{ X_1=0, ~~~ X_k= (1 \; k)+ (2 \; k)+\cdots+(k-1 \; k), ~~~ k=2,\dots,n. }[/math]
They play an important role in the representation theory of the symmetric group.
Properties
They generate a commutative subalgebra of [math]\displaystyle{ \mathbb{C} [ S_n] }[/math]. Moreover, Xn commutes with all elements of [math]\displaystyle{ \mathbb{C} [S_{n-1}] }[/math].
The vectors constituting the basis of Young's "seminormal representation" are eigenvectors for the action of Xn. For any standard Young tableau U we have:
- [math]\displaystyle{ X_k v_U =c_k(U) v_U, ~~~ k=1,\dots,n, }[/math]
where ck(U) is the content b − a of the cell (a, b) occupied by k in the standard Young tableau U.
Theorem (Jucys): The center [math]\displaystyle{ Z(\mathbb{C} [S_n]) }[/math] of the group algebra [math]\displaystyle{ \mathbb{C} [S_n] }[/math] of the symmetric group is generated by the symmetric polynomials in the elements Xk.
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 [math]\displaystyle{ \mathbb{C} [S_n] }[/math] holds true:
- [math]\displaystyle{ (t+X_1) (t+X_2) \cdots (t+X_n)= \sum_{\sigma \in S_n} \sigma t^{\text{number of cycles of }\sigma}. }[/math]
Theorem (Okounkov–Vershik): The subalgebra of [math]\displaystyle{ \mathbb{C} [S_n] }[/math] generated by the centers
- [math]\displaystyle{ Z(\mathbb{C} [ S_1]), Z(\mathbb{C} [ S_2]), \ldots, Z(\mathbb{C} [ S_{n-1}]), Z(\mathbb{C} [S_n]) }[/math]
is exactly the subalgebra generated by the Jucys–Murphy elements Xk.
See also
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
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