Mixed tensor invariants of Lie color algebra

TL;DR

This paper studies invariants in mixed tensor spaces of G-graded vector spaces under a color general linear group, identifying generating sets and trace monomials that extend classical invariant theory to color analogues.

Contribution

It introduces a spanning set for invariants of the symmetric algebra under the color general linear group, extending classical results to G-graded and color contexts.

Findings

Identifies a spanning set of invariants for the symmetric algebra.

Establishes that trace monomials generate polynomial invariants.

Shows the coincidence with Berele's generators in the color case.

Abstract

In this paper, we consider the mixed tensor space of a $G$-graded vector space where $G$ is a finite abelian group. We obtain a spanning set of invariants of the associated symmetric algebra under the action of a color analogue of the general linear group which we refer to as the general linear color group. As a consequence, we obtain a generating set for the polynomial invariants, under the simultaneous action of the general linear color group, on color analogues of several copies of matrices. We show that in this special case, this is the set of trace monomials, which coincides with the set of generators obtained by Berele.