On the blocks of the walled Brauer algebra

TL;DR

This paper classifies the blocks of the walled Brauer algebra over various characteristics, describing their structure via Weyl group actions and providing criteria for semisimplicity.

Contribution

It offers a complete description of the blocks of the walled Brauer algebra in characteristic zero and positive characteristic, including a linkage principle and semisimplicity classification.

Findings

Blocks are described by Weyl group orbits in characteristic zero.

A linkage principle based on affine Weyl group orbits in positive characteristic.

Complete classification of semisimple walled Brauer algebras across all characteristics.

Abstract

We determine the blocks of the walled Brauer algebra in characteristic zero. These can be described in terms of orbits of the action of a Weyl group of type $A$ on a certain set of weights. In positive characteristic we give a linkage principle in terms of orbits of the corresponding affine Weyl group. We also classify the semisimple walled Brauer algebras in all characteristics.