Tensor Products of Quiver Bundles

TL;DR

We develop a coherent theory of tensor products for twisted quiver representations in the category of $\mathcal{O}_X$-modules, realized through twisted path algebras and representations with relations. The main technical advance is showing that polystability is preserved under tensor products via the quiver vortex equations, yielding a Segre-type embedding for quiver representations and enabling the construction of distinguished closed subschemes in $\mathrm{GL}(n,\mathbb{C})$-character varieties. The framework unifies the path-algebra approach with representations with relations and provides explicit moduli-geometry consequences, including a quiver version of Hitchin-Kobayashi correspondence and collapse/expand operations on quivers. Overall, the paper extends classical quiver representation theory to a rich algebraic-geometric setting with concrete geometric applications, connecting stability, moduli, and tensorial constructions.

Abstract

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.

Figures (9)

• Figure 1:
• Figure 2:
• Figure 3: Tensor product of two $A_3$ quivers
• Figure 4: The Jordan quiver $Q_J$
• Figure 5: Equivalent paths

Theorems & Definitions (54)

• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Definition 2.5
• Definition 2.6
• Remark 2.7
• Example 2.8
• Definition 2.9
• Definition 2.10