Discriminants of quantized walled Brauer algebras

Abstract

In this paper, we compute Gram determinants associated to all cell modules of quantized walled Brauer algebras $\mathscr B_{r, t}(\rho, q)$ over an arbitrary field $\kappa$. Suppose $e$ is the quantum characteristic of $q^2$. We classify the blocks of $\mathscr B_{r, t}(\rho, q)$ 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 $\mathscr B_{r, t}(\rho, q)$ being equal to its simple head over $\kappa$.