Characters for Projective Modules in the BGG Category $\mathcal{O}$ for the Orthosymplectic Lie Superalgebra $\mathfrak{osp}(3|4)$

TL;DR

This paper computes the structure of projective modules in a specific category of representations for the orthosymplectic Lie superalgebra, providing explicit multiplicities and using translation functors and BGG reciprocity.

Contribution

It explicitly determines Verma multiplicities and composition factors for projective modules in the BGG category for rak{osp}(3|4), a novel detailed analysis.

Findings

Verma multiplicities of standard filtrations are explicitly calculated.

Composition factor multiplicities of Verma modules are determined.

Uses translation functors and BGG reciprocity for computations.

Abstract

We determine the Verma multiplicities of standard filtrations of projective modules for integral atypical blocks in the BGG category $\mathcal{O}$ for the orthosymplectic Lie superalgebras $\mathfrak{osp}(3|4)$ by way of translation functors. We then explicitly determine the composition factor multiplicities of Verma modules using BGG reciprocity.