Jucys–Murphy element

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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= (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[edit]

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

The vectors of the Young basis 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,\dots,n,

where ck(U) is the content b − a of the cell (ab) 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 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 \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 (OkounkovVershik): The subalgebra of \mathbb{C} [S_n] generated by the centers

 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[edit]

References[edit]

  • 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