ON THE COMPOSITION FACTORS OF KAC MODULES FOR THE LIE-SUPERALGEBRAS SL(M/N)

In the classification of finite-dimensional modules of Lie superalgebras, Kac distinguished between typical and atypical modules. Kac introduced an induced module, the so-called Kac module VBAR(LAMBDA) with highest weight-LAMBDA, which was shown to be simple if LAMBDA is a typical highest weight. If LAMBDA is an atypical highest weight, the Kac module is indecomposable and the simple module V(LAMBDA) can be identified with a quotient module of VBAR(LAMBDA). In the present paper the problem of determining the composition factors of the Kac modules for the Lie superalgebra sl(m/n) is considered. An algorithm is given to determine all these composition factors, and conversely, an algorithm is given to determine all the Kac modules containing a given simple module as a composition factor. The two algorithms are presented in the form of conjectures, and illustrated by means of detailed examples. Strong evidence in support of the conjectures is provided. The combinatorial way in which the two algorithms are intertwined is both surprising and interesting, and is a convincing argument in favor of the solution to the composition factor problem presented here.