Composite supersymmetric S-functions and characters of gl(m|n) representations

It is shown how to associate to a highest weight Λ of the Lie superalgebra gl(m|n) a composite partition ν; μ with composite Young diagram F (ν; μ). The corresponding supersymmetric Schur function sν;μ(x/y) is defined. However, it is known that this S-function does not always coincide with the character of the irreducible representation VΛ with highest weight Λ. Only for covariant, contravariant and typical representations the character and the S-function are known to coincide. Here, the notions of critical composite partitions and critical highest weights are considered. 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 can be computed. Also for this class of representations the character coincides with the composite supersymmetric S-function sν;μ(x/y). This extends considerably the classes of gl(m|n) representations for which the character can be computed by means of Sfunctions.