Contravariant Form on Tensor Product of Highest Weight Modules

TL;DR

This paper establishes a criterion for the complete reducibility of tensor products of irreducible highest weight modules over classical or quantum semi-simple groups, based on the non-degeneracy of a specific contravariant form.

Contribution

It introduces a new criterion linking the complete reducibility of tensor products to the non-degeneracy of a contravariant form on highest weight submodules.

Findings

Tensor product $V\otimes Z$ is completely reducible iff the form is non-degenerate on highest weight submodules.

The form is the product of canonical forms on $V$ and $Z$.

Complete reducibility is characterized by the form's non-degeneracy on singular vectors.

Abstract

We give a criterion for complete reducibility of tensor product $V\otimes Z$ of two irreducible highest weight modules $V$ and $Z$ over a classical or quantum semi-simple group in terms of a contravariant symmetric bilinear form on $V\otimes Z$. This form is the product of the canonical contravariant forms on $V$ and $Z$. Then $V\otimes Z$ is completely reducible if and only if the form is non-degenerate when restricted to the sum of all highest weight submodules in $V\otimes Z$ or equivalently to the span of singular vectors.