Character formulae in category $\mathcal O$ for exceptional Lie superalgebra $G(3)$

Abstract

We classify the blocks, compute the Verma flags of tilting and projective modules in the BGG category O for the exceptional Lie superalgebra G(3). The projective injective modules in O are classified. We also compute the Jordan-Hölder multiplicities of the Verma modules in O. Contents 1. Introduction 1 2. Conditions for nonzero Verma flag multiplicities in tilting modules 4 3. Classification of blocks in the BGG category O for G(3) 6 4. Character formulae for tilting modules in O, I 13 5. Character formulae for tilting modules in O, II 18 6. Character formulae for projective modules in O 22 7. Jordan-Hölder multiplicities of Verma modules in O 25 References 27 2010 Mathematics Subject Classification. Primary 17B10.

Characters for projective modules in the BGG category O for the orthosymplectic Lie superalgebra osp(3|4)

“…By way of translation functors, we explicitly compute standard filtration formulae for projectives. This method is used to solve a similar problem for gl(3|1) and gl(2|2) in [Kan19], for G(3) in [CW18], and D(2|1; ζ) in [CW19].…”

Blocks and characters of $D(2|1;ζ)$-modules of non-integral weights

“…The study of character formulas in the BGG category for exceptional Lie superalgebras was first initiated in [CW18,CW19]. We recall that the Lie superalgebras D(2|1; ζ) = g 0 ⊕ g 1 is a family of simple Lie superalgebras of dimension 17, depending on a parameter ζ ∈ C \ {0, −1} with underlying even subalgebra g 0 ∼ = sl 2 ⊕ sl 2 ⊕ sl 2 .…”

Blocks and characters of $G(3)$-modules of non-integral weights

“…A systematic study of the irreducible character formula in the BGG categories for D(2|1; ζ) and G(3) was initiated by the Weiqiang Wang and the second author in [CW19] and [CW18], where they gave a solution of the irreducible character problem in the case of integral highest weights. Finite-dimensional modules have been studied in [Ger00,Ma14,SZ16].…”

“…1.2. The objective of the present paper is to address irreducible character problem for the remaining cases not treated in [CW18], i.e., for the irreducible G(3)-modules of non-integral highest weights. We solve this problem by providing formulas for Verma flag multiplicities of all tilting modules in O of non-integral highest weight.…”

“…The method of translation functor follows that of [CW19,CW18,CCL20]. Namely, for a given tilting module T whose character we wish to compute, we make a reasonable choice of an initial tilting module.…”