A character formula for atypical critical gl(m|n) representations labelled by composite partitions

Let g be the Lie superalgebra gl(m|n). We show how to associate a gl(m|n) weight Λ to a composite partition ν;μ with composite Young diagram F (ν;μ). Based upon the definition of critical representations, the notion of “critical composite partition” is introduced. It is shown that for critical composite partitions (subject to a technical restriction) the corresponding gl(m|n) representation VΛ is tame, so its character formula can be computed. This character is shown to coincide with the composite S-function sν;μ(x/y).