Classification of all star and grade star irreps of gl(n‖1)

Abstract

A method for inducing nondegenerate forms on irreducible gl(m‖n) modules that implies some general results on star and grade star representations is investigated. These results are applied to obtain a complete classification, in terms of highest weights, of the irreducible star and grade star representations of gl(n‖1). It is demonstrated that while gl(n‖1) admits a large class of star representations the irreducible grade star representations are comparatively rare. Moreover, for n≠2 all grade star irreducible representations are also star representations and, for n>2, are atypical. The superalgebra gl(2‖1) proves to be a special case and admits a two-parameter family of four-dimensional typical grade star irreducible representations that are not star representations. In particular, typical grade star irreducible representations of gl(n‖1) exist only for n=1,2.