Supersymmetric Polynomials and the Center of the Walled Brauer Algebra

We study a commuting family of elements of the walled Brauer algebra B_{r, s}(δ), called the Jucys-Murphy elements, and show that the supersymmetric polynomials in these elements belong to the center of the walled Brauer algebra. When B_r,s(δ) is semisimple, we show that those supersymmetric polynomials generate the center. In addition, we present an analogue of Jucys-Murphy elements for the quantized walled Brauer algebra H_r,s(q,ρ) over ℂ(q, ρ) and by taking the classical limit we show that the supersymmetric polynomials in these elements generates the center. It follows that H. Morton’s conjecture, which appeared in the study of the relation between the framed HOMFLY skein on the annulus and that on the rectangle with designated boundary points, holds if we extend the scalar from ℤ[q±1,ρ±1](q−q−1) to ℂ(q, ρ).