On a class of tensor product representations for the orthosymplectic superalgebra

TL;DR

This paper introduces spinor representations for the orthosymplectic superalgebra osp(m|2n), characterizes their uniqueness, and analyzes tensor product decompositions, including criteria for complete reducibility and explicit decompositions.

Contribution

It generalizes spinor representations to osp(m|2n), characterizes their uniqueness, and provides a detailed analysis of tensor product decompositions and reducibility criteria.

Findings

Spinor representations for osp(m|2n) are uniquely characterized.

Criteria for complete reducibility of tensor products are established.

Explicit decomposition of tensor products with spinor spaces is provided.

Abstract

We introduce the spinor representations for osp(m|2n). These generalize the spinors for so(m) and the symplectic spinors for sp(2n) and correspond to representations of the supergroup with supergroup pair (Spin(m) x Mp(2n),osp(m|2n)). We prove that these spinor spaces are uniquely characterized as the completely pointed osp(m|2n)-modules. Then the tensor product of this representation with irreducible finite dimensional osp(m|2n)-modules is studied. Therefore we derive a criterion for complete reducibility of tensor product representations. We calculate the decomposition into irreducible osp(m|2n)-representations of the tensor product of the super spinor space with an extensive class of such representations and also obtain cases where the tensor product is not completely reducible.