Graded Picture Invariants and polynomial invariants for mixed tensor superspaces

TL;DR

This paper develops a comprehensive set of polynomial invariants for mixed tensor superspaces under supergroup actions, extending classical invariant theory to the superalgebra context with new supertrace-based results.

Contribution

It introduces a spanning set of invariants for mixed tensor superspaces and identifies supertrace monomials as generators, extending classical invariant theory to supergroups.

Findings

Invariants form a spanning set under supergroup actions.

Supertrace monomials generate polynomial invariants.

Results extend classical invariant theory to superalgebra setting.

Abstract

In this paper we consider the mixed tensor space of a $\mathbb Z_2$-graded vector space. We obtain a spanning set of invariants of the associated symmetric algebra under the action of the general linear supergroup as well as the queer supergroup over the Grassmann algebra. As a consequence, we give a generating set of polynomial invariants for the simultaneous adjoint action of the general linear supergroup on several copies of its Lie superalgebra. We show that in this special case, these turn out to be the supertrace monomials which is analogous to the results of Procesi in the classical case. A queer supergroup analogue of these results is also obtained.