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We study a commuting family of elements of the walled Brauer algebra$B_{r,s}(\delta)$, called the Jucys-Murphy elements, and show that thesupersymmetric polynomials in these elements belong to the center of the walledBrauer algebra. When $B_{r,s}(\delta)$ is semisimple, we show that thosesupersymmetric polynomials generate the center. Under the same assumption,wedefine a maximal commutative subalgebra of $B_{r,s}(\delta)$, called the\emph{Gelfand-Zetlin subalgebra}, and show that it is generated by theJucys-Murphy elements. As an application, we construct a complete set ofprimitive orthogonal idempotents of $B_{r,s}(\delta)$, when it is semisimple.We also give an alternative proof of a part of the classification theorem ofblocks of $B_{r,s}(\delta)$ in non-semisimple cases, which appeared in the workof Cox-De~Visscher-Doty-Martin.Finally, we present an analogue of Jucys-Murpyelements for the quantized walled Brauer algebra $H_{r,s}(q,\rho)$ over$\mathbb C(q, \rho)$ and by taking the classical limit we show that thesupersymmetric polynomials in these elements generates the center. It followsthat H. Morton conjecture, which appeared in the study of the relation betweenthe framed HOMFLY skein on the annulus and that on the rectangle withdesignated boundary points, holds if we extend the scalar from $\mathbbZ[q^{\pm1},\rho^{\pm1}]_{(q-q^{-1})}$ to $\mathbb C(q, \rho)$.

Supersymmetric polynomials and the center of the walled Brauer algebra