The periplectic Brauer algebra

TL;DR

This paper investigates the structure and representation theory of the periplectic Brauer algebra, including its quasi-hereditary properties, block decomposition, and connections to periplectic Lie superalgebras, advancing understanding of its algebraic and categorical features.

Contribution

It develops new theories of Jucys-Murphy elements, Bratteli diagrams, and Murphy bases for the periplectic Brauer algebra, and describes its block decomposition and module categories.

Findings

Determined when the algebra is quasi-hereditary.

Described the block decomposition over fields of characteristic zero.

Established BGG reciprocity and computed some decomposition multiplicities.

Abstract

We study the periplectic Brauer algebra introduced by Moon in the study of invariant theory for periplectic Lie superalgebras. We determine when the algebra is quasi-hereditary and, for fields of characteristic zero, describe the block decomposition. To achieve this, we also develop theories of Jucys-Murphy elements, Bratteli diagrams, Murphy bases, determine when there exist quasi-hereditary 1-covers, obtain a BGG reciprocity relation and determine some decomposition multiplicities of cell modules. As an application, we determine the blocks in the category of finite dimensional weight modules over the periplectic Lie superalgebra.