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

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