Young supertableaux of the basic Lie superalgebras

Abstract

Among the simple Lie superalgebras (LSA), the basic classical ones are well known, together with many of their finite-dimensional representations. This is particularly true for the typical representations which are known to be completely reducible. It is well known that reducible but indecomposable representations of the basic LSA are not completely characterised by their highest weight (HW) or the eigenvalues of their Casimir operators. In fact, there may be several different finite representations characterised by the same HW. Such representations are called non-typical or atypical. The author establishes a complete characterisation of the typical and non-typical representations by means of properly defined Young supertableaux (YST). Therefore, the definition the author uses for the author's YST differs slightly from the standard definitions. The examples given help the illustration of this new interpretation for the YST which makes possible the complete classification of the indecomposable representations characterised by a generalised notion of HW.