Character formulae in category $\mathcal O$ for exceptional Lie superalgebras $D(2|1;\zeta)$

TL;DR

This paper derives explicit character formulas and detailed module structure descriptions for the category O representations of the exceptional Lie superalgebra family D(2|1;ζ), covering all integral weights and parameters.

Contribution

It provides the first complete character formulae and Verma flag multiplicities for D(2|1;ζ) in category O, including composition factors for all Verma modules.

Findings

Explicit character formulas for D(2|1;ζ) modules

Complete description of Verma flag multiplicities

Determination of composition factors of all Verma modules

Abstract

We establish character formulae for representations of the one-parameter family of simple Lie superalgebras $D(2|1;\zeta)$. We provide a complete description of the Verma flag multiplicities of the tilting modules and the projective modules in the BGG category $\mathcal O$ of $D(2|1;\zeta)$-modules of integral weights, for any complex parameter $\zeta$. The composition factors of all Verma modules in $\mathcal O$ are then obtained.