Computing Super Matrix Invariants

TL;DR

This paper extends the fundamental theorems of invariant theory to the general linear Lie superalgebra and computes related numerical invariants using symmetric functions and complex integrals.

Contribution

It generalizes the computation of numerical invariants like multiplicities and Poincaré series to the superalgebra case, building on previous work for Lie algebras.

Findings

Derived formulas involving inner products of symmetric functions

Computed Poincaré series for superalgebra invariants

Extended invariant theory results to Lie superalgebras

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.