Discriminants of quantized walled Brauer algebras

In this paper, we compute Gram determinants associated to all cell modules of quantized walled Brauer algebras ${\mathcal{B}}_{r,t}(\rho ,q)$ over an arbitrary field $\kappa$. Suppose $e$ is the quantum characteristic of $q^2$. We classify the blocks of ${\mathcal{B}}_{r,t}(\rho ,q)$ when $e>\max \{r,t\}$ and ${\rho }^2=q^{2n}$, $n\in \mathbb{Z}$. As an application, we give a criterion for a cell module of ${\mathcal{B}}_{r,t}(\rho ,q)$ being equal to its simple head over $\kappa$.