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

Abstract

To study finite-dimensional modules of the Lie superalgebras, Kac introduced the Kac-modules and divided them into typical or atypical modules according as they are simple or not. For Lambda being atypical, Hughes et al have an algorithm to determine all the composition factors of a Kac-module; they conjectured that there exists a bijection between the composition factors of a Kac-module and the so-called permissible codes. The aim of this paper is to prove this conjecture. We give a partial proof here, i.e., to any unlinked code, by constructing explicitly the primitive vector, we prove that there corresponds a composition factor of the Kac-module. We will give a full proof of the conjecture in another paper.