Primitive vectors of Kac-modules of the Lie superalgebras sl(m/n)

Abstract

In a study of finite-dimensional modules of simple Lie superalgebras, Kac introduced certain indecomposable modules, now known as Kac-modules V̄(Λ), which are simple if and only if Λ is typical. For Λ atypical, Hughes et al. presented an algorithm to determine all the composition factors of the Kac-module; they conjectured that there exists a bijection between the composition factors of a Kac-module and so-called permissible codes. The aim in this paper is to contribute to the proof of this conjecture. By constructing explicitly the primitive vector, we prove that for any unlinked code there corresponds a composition factor of the Kac-module. It will be proved in another paper that to any linked code there also corresponds a composition factor of the Kac-module. Thus the proof of the Hughes et al. conjecture will be reduced to the problem whether or not each composition factor corresponds to a linked or unlinked code.