Computing Super Matrix Invariants

Abstract

In [Trace identities and $\bf {Z}/2\bf {Z}$-graded invariants, {\it Trans. Amer. Math. Soc. \bf309} (1988), 581--589] we generalized the first and second fundamental theorems of invariant theory from the general linear group to the general linear Lie superalgebra. In the current paper we generalize the computations of the the numerical invariants (multiplicities and Poincar\'e series) to the superalgebra case. The results involve either inner products of symmetric functions in two sets of variables, or complex integrals. we generalized the first and second fundamental theorems of invariant theory from the general linear group to the general linear Lie superalgebra. In the current paper we generalize the computations of the the numerical invariants (multiplicities and Poincar\'e series) to the superalgebra case. The results involve either inner products of symmetric functions in two sets of variables, or complex integrals.