Primitive vectors of Kac-modules of the Lie superalgebras \(\text{sl}(m/n)\) (Q2738097)

In his study of finite-dimensional modules of simple Lie superalgebras, \textit{V. G. Kac} [Lect. Notes Math. 676, 597-626 (1978; Zbl 0388.17002)] introduced certain induced highest weight modules \(\overline V(\Lambda)\) now known as Kac-modules. These modules \(\overline V(\Lambda)\) are simple if and only if \(\Lambda\) is typical. For atypical \(\Lambda\), a simple algorithm was conjectured by \textit{J. W. B. Hughes, R. C. King} and \textit{J. Van der Jeugt} [J. Math. Phys. 33, 470-491 (1992; Zbl 0747.17003)] to determine the composition factors of \(\overline V(\Lambda)\). This algorithm describes a bijection between the composition factors (or their highest weights) and so-called permissible codes. It should be mentioned that in the mean time \textit{V. Serganova} [ Sel. Math., New Ser. 2, 607-651 (1996; Zbl 0881.17005)] proved another algorithm with the same purpose of determining the composition factors of \(\overline V(\Lambda)\); her algorithm however is not easy to implement, and so far the relation between the two algorithms is not completely clear. NEWLINENEWLINENEWLINEThe purpose of the present paper is to prove that to certain permissible codes of Hughes et al (the so-called unlinked codes) there corresponds indeed a composition factor of the Kac-module. This is done by constructing a primitive vector corresponding to the code. This construction turns out to be rather heavy. The proof requires a long sequence of technical lemmas and some intricate induction arguments. But in the end it gives in a very explicit way an expression for the highest weight vector of these composition factors.