Extensions of inhomogeneous polynomial representations for $\mathfrak {sl}(m+1|n)$sl(m+1|n)

Abstract

We extend an inhomogeneous polynomial representation of special linear Lie superalgebras $\mathfrak {sl}(m+1|n)$sl(m+1|n) to an inhomogeneous representation on the tensor space of any irreducible tensor representation of the general linear Lie superalgebras $\mathfrak {gl}(m|n)$gl(m|n) with this polynomial space via Shen's mixed product. We find a sufficient and necessary condition for irreducibility of these inhomogeneous representations, and get the Jordan-H$\ddot{\mbox{o}}$ölder series of those which are not irreducible. Furthermore, we point out that all these irreducible modules are lowest weight modules. Their lowest weights and character formulas are obtained.