The center of the walled Brauer algebra $B_{r,1}(\delta)$

TL;DR

This paper characterizes the center of the walled Brauer algebra $B_{r,1}(\delta)$ over complex numbers, showing it is generated by supersymmetric polynomials evaluated at Jucys-Murphy elements and that its dimension is independent of the parameter.

Contribution

It establishes the generators of the center of the walled Brauer algebra and proves the dimension's independence from the parameter $\delta$.

Findings

Center generated by supersymmetric polynomials at Jucys-Murphy elements

Dimension of the center is independent of $\delta$

Provides explicit algebraic structure of the center

Abstract

We show that the centre of the walled Brauer algebra $B_{r,1}(\delta)$ over the complex field $\mathbb{C}$, for any parameter $\delta\in \mathbb{C}$, is generated by the supersymmetric polynomials evaluated at the Jucys-Murphy elements. Moreover, we prove that its dimension is independent of the parameter $\delta$.