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)%2B (2 k)%2B\cdots%2B(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 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 (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%2BX_1) (t%2BX_2) \cdots (t%2BX_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.

See also

References