Quantum walled Brauer algebra: commuting families, Baxterization, and representations

TL;DR

This paper develops new representations and spectral analysis for the quantum walled Brauer algebra, introduces a Baxterization method, and constructs a universal transfer matrix within a braided monoidal category framework.

Contribution

It constructs Specht and seminormal modules, provides the spectrum of Jucys--Murphy elements, and introduces a Baxterization approach for the algebra.

Findings

Spectrum of Jucys--Murphy elements obtained

Baxterization prescription developed

Universal transfer matrix constructed

Abstract

For the quantum walled Brauer algebra, we construct its Specht modules and (for generic parameters of the algebra) seminormal modules. The latter construction yields the spectrum of a commuting family of Jucys--Murphy elements. We also propose a Baxterization prescription; it involves representing the quantum walled Brauer algebra in terms of morphisms in a braided monoidal category and introducing parameters into these morphisms, which allows constructing a "universal transfer matrix" that generates commuting elements of the algebra.