The periplectic Brauer algebra

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, when it admits a quasi‐hereditary 1‐cover and, for fields of characteristic zero, describes the block decomposition. To achieve this, we also develop theories of Jucys–Murphy elements, Bratteli diagrams, Murphy bases, obtain a Humphreys‐BGG reciprocity relation and determine some decomposition multiplicities of cell modules. As an application, we determine the blocks in the category of finite dimensional integrable modules of the periplectic Lie superalgebra.

The Primitive Spectrum and Category for the Periplectic Lie Superalgebra

“…Following [B+9], we will use the "fake Casimir element" Ω to decompose the functor − ⊗ V . This operator Ω also appeared in [Co2,Section 8.4] and [CP,Section 2]. For the explicit realisation of Ω ∈ g ⊗ gl(n|n), we refer to [B+9,Section 4.1].…”

“…Consequently, the block decomposition of the category of finite dimensional modules and of category O is not controlled by the combination of the centre and the root lattice. For finite dimensional modules, the block decomposition was recently obtained independently in [B+9] and [Co2], with one direction already proved earlier in [Ch]. To determine the blocks in O we establish a BGG reciprocity result and exploit the technique in [B+9] of decomposing the translation functors using a 'fake Casimir operator'.…”

The blocks of the periplectic Brauer algebra in positive characteristic

“…By Proposition 8.3 it is enough to consider the case when λ is obtained from µ by removing two boxes in the same row. For a field K of characteristic zero, we have [Cou18,Proposition 7.2.6] [W K n (λ) : L K n (µ)] = 1. This implies that there exists a submodule M of W K n (λ) such that Hom A K n (W K n (µ), W K n (λ)/M ) = 0.…”

“…In contrast the representation theory of the periplectic Brauer algebra remained unstudied until quite recently. In particular the simple modules have been classified for arbitrary characteristic [KT17] and for characteristic zero (or large characteristic) a classification of the blocks [Cou18] and a complete description of the decomposition multiplicities [CE18] have been obtained.…”

The affine VW supercategory

“…This allows one to apply the Cherednik [13] and Okounkov-Vershik [10], [33] approaches in this context. First steps in this direction were already successfully taken in [3] and [14] from different perspectives to determine the blocks and decomposition numbers in the category of finite dimensional representations of p(n) and of the Brauer superalgebra, and further developed in [15]. A thorough treatment of the corresponding category O is so far missing and will be deferred to subsequent work.…”