Discriminants of quantized walled Brauer algebras

TL;DR

This paper computes Gram determinants for cell modules of quantized walled Brauer algebras, classifies their blocks under certain conditions, and provides criteria for simple head equivalence.

Contribution

It introduces explicit Gram determinant calculations, block classification criteria, and simple head conditions for quantized walled Brauer algebras.

Findings

Gram determinants for all cell modules computed

Block classification when e > max(r,t) and rho^2=q^{2n}

Criterion for cell module equality to its simple head

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 $\kappa$.