Jucys–Murphy element

Elements in representations of the symmetric group

In mathematics, the Jucys–Murphy elements in the group algebra C [ S n ] {\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:

X 1 = 0 ,       X k = ( 1 k ) + ( 2 k ) + ⋯ + ( k − 1 k ) ,       k = 2 , … , n . {\displaystyle 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 C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} . Moreover, Xn commutes with all elements of C [ S n − 1 ] {\displaystyle \mathbb {C} [S_{n-1}]} .

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:

X k v U = c k ( U ) v U ,       k = 1 , … , n , {\displaystyle X_{k}v_{U}=c_{k}(U)v_{U},~~~k=1,\dots ,n,}

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

( t + X 1 ) ( t + X 2 ) ⋯ ( t + X n ) = ∑ σ ∈ S n σ t number of cycles of  σ . {\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 C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} generated by the centers

Z ( C [ S 1 ] ) , Z ( C [ S 2 ] ) , … , Z ( C [ S n − 1 ] ) , Z ( C [ S n ] ) {\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 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, arXiv:math.RT/0503040(revised English version).{{citation}}: CS1 maint: postscript (link)
  • 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" (PDF), Lietuvos Fizikos Rinkinys, 6: 163–180
  • Jucys, Algimantas Adolfas (1971), "Factorization of Young projection operators for the symmetric group" (PDF), 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