PLethysms for representations of Lie superalgebras with applications to P.I. algebras

We introduce plethysms for representations of general linear Lie superalgebras and of hook symmetric functions and show that the known decompositions of ordinary symmetric functions into a sum of Schur functions can be used for obtaining similar decompositions in the supercase. We apply these results to algebras with polynomial identities. The main result on P.I. algebras is the computation of the double Hilbert (or Poincarè) series of the polynomial identities for the tensor square E E of the Grassmann algebra E of an infinite-dimensional vector space.