Supersymmetric polynomials and the center of the walled Brauer algebra

Abstract

We study a commuting family of elements of the walled Brauer algebra $B_{r,s}(\delta)$, called the Jucys-Murphy elements, and show that the supersymmetric polynomials in these elements belong to the center of the walled Brauer algebra. When $B_{r,s}(\delta)$ is semisimple, we show that those supersymmetric polynomials generate the center. Under the same assumption,we define a maximal commutative subalgebra of $B_{r,s}(\delta)$, called the \emph{Gelfand-Zetlin subalgebra}, and show that it is generated by the Jucys-Murphy elements. As an application, we construct a complete set of primitive orthogonal idempotents of $B_{r,s}(\delta)$, when it is semisimple. We also give an alternative proof of a part of the classification theorem of blocks of $B_{r,s}(\delta)$ in non-semisimple cases, which appeared in the work of Cox-De~Visscher-Doty-Martin.Finally, we present an analogue of Jucys-Murpy elements 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 the supersymmetric polynomials in these elements generates the center. It follows that H. Morton 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 $\mathbb Z[q^{\pm1},\rho^{\pm1}]_{(q-q^{-1})}$ to $\mathbb C(q, \rho)$.