Normal form for the walled Brauer algebra: construction and applications

TL;DR

This paper develops a normal form and reduction algorithm for the walled Brauer algebra, enabling efficient calculations of monomials, generators, and annihilator ideals, with applications to representation theory.

Contribution

It introduces a novel normal form and reduction algorithm for the walled Brauer algebra, facilitating explicit computations in its modules.

Findings

Normal form and reduction algorithm for the algebra

Explicit expressions for generators and annihilators

Efficient counting of monomials in generators

Abstract

We construct a normal form for the walled Brauer algebra, together with the reduction algorithm. We apply normal form to calculate the numbers of monomials in generators with minimal length. We further utilize normal form to give explicit expressions for a generating set and annihilator ideal of a particular cyclic vector in a cell module.