Invariant theory of the queer color group

Abstract

In this paper, we introduce the notion of the queer color group, analogous to that of the queer supergroup over the infinite Grassmann algebra. We obtain a Schur-Weyl duality theorem for this group and thereby construct an explicit spanning set of invariants of the associated symmetric algebra of the mixed tensor space of a G-graded vector space where G is a finite abelian group. As a consequence, we obtain a generating set for the polynomial invariants, under the simultaneous action of the queer color group on color analogues of several copies of matrices. We also introduce the notion of concomitants for the queer color group analogous to that of Procesi in [16] and obtain a generating set for the algebra of concomitants. These results generalize those of Berele in [3] to the color setting.