Tensor Products of Quiver Bundles

In this work we introduce a notion of tensor product of (twisted) quiver representations with relations in the category of ${\mathcal{O}}_X$-modules. As a first application of our notion, we see that tensor products of polystable quiver bundles are polystable and later we use this to both deduce a quiver version of the Segre embedding and to identify distinguished closed subschemes of $\text{GL}(n,\mathbb{C})$-character varieties of free abelian groups.