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:


 * $$X_1=0,  X_k= (1 \; k)+ (2 \; k)+\cdots+(k-1 \; k),  k=2,\dots,n. $$

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

Properties
They generate a commutative subalgebra of $$\mathbb{C} [ S_n] $$. Moreover, Xn 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 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 &minus; 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 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 (Okounkov–Vershik): 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.